(************************************************************************)
(*         *   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 SeqSeries.
Require Import Rtrigo1.
Require Import Ranalysis.
Local Open Scope R_scope.

(*******************************************)
(*            Newton's Integral            *)
(*******************************************)

Definition Newton_integrable (f:R -> R) (a b:R) : Type :=
  { g:R -> R | antiderivative f g a b \/ antiderivative f g b a }.

Definition NewtonInt (f:R -> R) (a b:R) (pr:Newton_integrable f a b) : R :=
  let (g,_) := pr in g b - g a.

(* If f is differentiable, then f' is Newton integrable (Tautology ?) *)
Lemma FTCN_step1 :
  forall (f:Differential) (a b:R),
    Newton_integrable (fun x:R => derive_pt f x (cond_diff f x)) a b.
forall (f : Differential) (a b : R),
Newton_integrable
  (fun x : R => derive_pt f x (cond_diff f x)) a b
Proof.
forall (f : Differential) (a b : R),
Newton_integrable
  (fun x : R => derive_pt f x (cond_diff f x)) a b
  intros f a b; unfold Newton_integrable; exists (d1 f);
    unfold antiderivative; intros; case (Rle_dec a b);
      intro;
        [ left; split; [ intros; exists (cond_diff f x); reflexivity | assumption ]
          | right; split;
            [ intros; exists (cond_diff f x); reflexivity | auto with real ] ].
Defined.

(* By definition, we have the Fondamental Theorem of Calculus *)
Lemma FTC_Newton :
  forall (f:Differential) (a b:R),
    NewtonInt (fun x:R => derive_pt f x (cond_diff f x)) a b
    (FTCN_step1 f a b) = f b - f a.
forall (f : Differential) (a b : R),
NewtonInt (fun x : R => derive_pt f x (cond_diff f x))
  a b (FTCN_step1 f a b) = f b - f a
Proof.
forall (f : Differential) (a b : R),
NewtonInt (fun x : R => derive_pt f x (cond_diff f x))
  a b (FTCN_step1 f a b) = f b - f a
  intros; unfold NewtonInt; reflexivity.
Qed.

(* $\int_a^a f$ exists forall a:R and f:R->R *)
Lemma NewtonInt_P1 : forall (f:R -> R) (a:R), Newton_integrable f a a.
forall (f : R -> R) (a : R), Newton_integrable f a a
Proof.
forall (f : R -> R) (a : R), Newton_integrable f a a
  intros f a; unfold Newton_integrable;
    exists (fct_cte (f a) * id)%F; left;
      unfold antiderivative; split.
f:R -> R
a:R
forall x : R,
a <= x <= a ->
exists pr : derivable_pt (fct_cte (f a) * id) x,
  f x = derive_pt (fct_cte (f a) * id) x pr
f:R -> R
a:R
a <= a
  intros; assert (H1 : derivable_pt (fct_cte (f a) * id) x).
f:R -> R
a, x:R
H:a <= x <= a
derivable_pt (fct_cte (f a) * id) x
f:R -> R
a, x:R
H:a <= x <= a
H1:derivable_pt (fct_cte (f a) * id) x
exists pr : derivable_pt (fct_cte (f a) * id) x,
  f x = derive_pt (fct_cte (f a) * id) x pr
f:R -> R
a:R
a <= a
  apply derivable_pt_mult.
f:R -> R
a, x:R
H:a <= x <= a
derivable_pt (fct_cte (f a)) x
f:R -> R
a, x:R
H:a <= x <= a
derivable_pt id x
f:R -> R
a, x:R
H:a <= x <= a
H1:derivable_pt (fct_cte (f a) * id) x
exists pr : derivable_pt (fct_cte (f a) * id) x,
  f x = derive_pt (fct_cte (f a) * id) x pr
f:R -> R
a:R
a <= a
  apply derivable_pt_const.
f:R -> R
a, x:R
H:a <= x <= a
derivable_pt id x
f:R -> R
a, x:R
H:a <= x <= a
H1:derivable_pt (fct_cte (f a) * id) x
exists pr : derivable_pt (fct_cte (f a) * id) x,
  f x = derive_pt (fct_cte (f a) * id) x pr
f:R -> R
a:R
a <= a
  apply derivable_pt_id.
f:R -> R
a, x:R
H:a <= x <= a
H1:derivable_pt (fct_cte (f a) * id) x
exists pr : derivable_pt (fct_cte (f a) * id) x,
  f x = derive_pt (fct_cte (f a) * id) x pr
f:R -> R
a:R
a <= a
  exists H1; assert (H2 : x = a).
f:R -> R
a, x:R
H:a <= x <= a
H1:derivable_pt (fct_cte (f a) * id) x
x = a
f:R -> R
a, x:R
H:a <= x <= a
H1:derivable_pt (fct_cte (f a) * id) x
H2:x = a
f x = derive_pt (fct_cte (f a) * id) x H1
f:R -> R
a:R
a <= a
  elim H; intros; apply Rle_antisym; assumption.
f:R -> R
a, x:R
H:a <= x <= a
H1:derivable_pt (fct_cte (f a) * id) x
H2:x = a
f x = derive_pt (fct_cte (f a) * id) x H1
f:R -> R
a:R
a <= a
  symmetry ; apply derive_pt_eq_0;
    replace (f x) with (0 * id x + fct_cte (f a) x * 1);
    [ apply (derivable_pt_lim_mult (fct_cte (f a)) id x);
      [ apply derivable_pt_lim_const | apply derivable_pt_lim_id ]
      | unfold id, fct_cte; rewrite H2; ring ].
f:R -> R
a:R
a <= a
  right; reflexivity.
Qed.

(* $\int_a^a f = 0$ *)
Lemma NewtonInt_P2 :
  forall (f:R -> R) (a:R), NewtonInt f a a (NewtonInt_P1 f a) = 0.
forall (f : R -> R) (a : R),
NewtonInt f a a (NewtonInt_P1 f a) = 0
Proof.
forall (f : R -> R) (a : R),
NewtonInt f a a (NewtonInt_P1 f a) = 0
  intros; unfold NewtonInt; simpl;
    unfold mult_fct, fct_cte, id.
f:R -> R
a:R
(let (g, _) := NewtonInt_P1 f a in g a - g a) = 0
  destruct NewtonInt_P1 as [g _].
f:R -> R
a:R
g:R -> R
g a - g a = 0
  now apply Rminus_diag_eq.
Qed.

(* If $\int_a^b f$ exists, then $\int_b^a f$ exists too *)
Lemma NewtonInt_P3 :
  forall (f:R -> R) (a b:R) (X:Newton_integrable f a b),
    Newton_integrable f b a.
forall (f : R -> R) (a b : R),
Newton_integrable f a b -> Newton_integrable f b a
Proof.
forall (f : R -> R) (a b : R),
Newton_integrable f a b -> Newton_integrable f b a
  unfold Newton_integrable; intros; elim X; intros g H;
    exists g; tauto.
Defined.

(* $\int_a^b f = -\int_b^a f$ *)
Lemma NewtonInt_P4 :
  forall (f:R -> R) (a b:R) (pr:Newton_integrable f a b),
    NewtonInt f a b pr = - NewtonInt f b a (NewtonInt_P3 f a b pr).
forall (f : R -> R) (a b : R)
  (pr : Newton_integrable f a b),
NewtonInt f a b pr =
- NewtonInt f b a (NewtonInt_P3 f a b pr)
Proof.
forall (f : R -> R) (a b : R)
  (pr : Newton_integrable f a b),
NewtonInt f a b pr =
- NewtonInt f b a (NewtonInt_P3 f a b pr)
  intros f a b (x,H).
f:R -> R
a, b:R
x:R -> R
H:antiderivative f x a b \/ antiderivative f x b a
NewtonInt f a b
  (exist
     (fun g : R -> R =>
      antiderivative f g a b \/ antiderivative f g b a)
     x H) =
-
NewtonInt f b a
  (NewtonInt_P3 f a b
     (exist
        (fun g : R -> R =>
         antiderivative f g a b \/
         antiderivative f g b a) x H)) unfold NewtonInt, NewtonInt_P3; simpl; ring.
Qed.

(* The set of Newton integrable functions is a vectorial space *)
Lemma NewtonInt_P5 :
  forall (f g:R -> R) (l a b:R),
    Newton_integrable f a b ->
    Newton_integrable g a b ->
    Newton_integrable (fun x:R => l * f x + g x) a b.
forall (f g : R -> R) (l a b : R),
Newton_integrable f a b ->
Newton_integrable g a b ->
Newton_integrable (fun x : R => l * f x + g x) a b
Proof.
forall (f g : R -> R) (l a b : R),
Newton_integrable f a b ->
Newton_integrable g a b ->
Newton_integrable (fun x : R => l * f x + g x) a b
  unfold Newton_integrable; intros f g l a b X X0;
    elim X; intros x p; elim X0; intros x0 p0;
      exists (fun y:R => l * x y + x0 y).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  elim p; intro.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  elim p0; intro.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
H0:antiderivative g x0 a b
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  left; unfold antiderivative; unfold antiderivative in H, H0; elim H;
    clear H; intros; elim H0; clear H0; intros H0 _.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
(forall x1 : R,
 a <= x1 <= b ->
 exists
   pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
   l * f x1 + g x1 =
   derive_pt (fun y : R => l * x y + x0 y) x1 pr) /\
a <= b
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  split.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
forall x1 : R,
a <= x1 <= b ->
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
a <= b
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  intros; elim (H _ H2); elim (H0 _ H2); intros.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt x x4,
  f x4 = derive_pt x x4 pr
H1:a <= b
H0:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt x0 x4,
  g x4 = derive_pt x0 x4 pr
x1:R
H2:a <= x1 <= b
x2:derivable_pt x0 x1
H3:g x1 = derive_pt x0 x1 x2
x3:derivable_pt x x1
H4:f x1 = derive_pt x x1 x3
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
a <= b
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt x x4,
  f x4 = derive_pt x x4 pr
H1:a <= b
H0:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt x0 x4,
  g x4 = derive_pt x0 x4 pr
x1:R
H2:a <= x1 <= b
x2:derivable_pt x0 x1
H3:g x1 = derive_pt x0 x1 x2
x3:derivable_pt x x1
H4:f x1 = derive_pt x x1 x3
derivable_pt (fun y : R => l * x y + x0 y) x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt x x4,
  f x4 = derive_pt x x4 pr
H1:a <= b
H0:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt x0 x4,
  g x4 = derive_pt x0 x4 pr
x1:R
H2:a <= x1 <= b
x2:derivable_pt x0 x1
H3:g x1 = derive_pt x0 x1 x2
x3:derivable_pt x x1
H4:f x1 = derive_pt x x1 x3
H5:derivable_pt (fun y : R => l * x y + x0 y) x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
a <= b
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  reg.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt x x4,
  f x4 = derive_pt x x4 pr
H1:a <= b
H0:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt x0 x4,
  g x4 = derive_pt x0 x4 pr
x1:R
H2:a <= x1 <= b
x2:derivable_pt x0 x1
H3:g x1 = derive_pt x0 x1 x2
x3:derivable_pt x x1
H4:f x1 = derive_pt x x1 x3
H5:derivable_pt (fun y : R => l * x y + x0 y) x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
a <= b
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  exists H5; symmetry ; reg; rewrite <- H3; rewrite <- H4; reflexivity.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:a <= b
H0:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
a <= b
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assumption.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x a b
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a < b
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  left; rewrite <- H5; unfold antiderivative; split.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
forall x1 : R,
a <= x1 <= a ->
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  intros; elim H6; intros; assert (H9 : x1 = a).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
x1 = a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  apply Rle_antisym; assumption.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H10 : a <= x1 <= b).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
a <= x1 <= b
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  split; right; [ symmetry ; assumption | rewrite <- H5; assumption ].
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H11 : b <= x1 <= a).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
b <= x1 <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  split; right; [ rewrite <- H5; symmetry ; assumption | assumption ].
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H12 : derivable_pt x x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
derivable_pt x x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  unfold derivable_pt; exists (f x1); elim (H3 _ H10); intros;
    eapply derive_pt_eq_1; symmetry ; apply H12.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H13 : derivable_pt x0 x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
derivable_pt x0 x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  unfold derivable_pt; exists (g x1); elim (H1 _ H11); intros;
    eapply derive_pt_eq_1; symmetry ; apply H13.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
derivable_pt (fun y : R => l * x y + x0 y) x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  reg.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  exists H14; symmetry ; reg.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H15 : derive_pt x0 x1 H13 = g x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
derive_pt x0 x1 H13 = g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  elim (H1 _ H11); intros; rewrite H15; apply pr_nu.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H16 : derive_pt x x1 H12 = f x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
derive_pt x x1 H12 = f x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
H16:derive_pt x x1 H12 = f x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  elim (H3 _ H10); intros; rewrite H16; apply pr_nu.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ a <= b
H0:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ b <= a
H1:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:b <= a
H3:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:a <= b
H5:a = b
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
H16:derive_pt x x1 H12 = f x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  rewrite H15; rewrite H16; ring.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ a <= b
H0:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ b <= a
H1:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:b <= a
H3:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:a <= b
H5:a = b
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  right; reflexivity.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  elim p0; intro.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 a b
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b < a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  left; rewrite H5; unfold antiderivative; split.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
forall x1 : R,
a <= x1 <= a ->
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  intros; elim H6; intros; assert (H9 : x1 = a).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
x1 = a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  apply Rle_antisym; assumption.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H10 : a <= x1 <= b).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
a <= x1 <= b
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  split; right; [ symmetry ; assumption | rewrite H5; assumption ].
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H11 : b <= x1 <= a).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
b <= x1 <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  split; right; [ rewrite H5; symmetry ; assumption | assumption ].
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H12 : derivable_pt x x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
derivable_pt x x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  unfold derivable_pt; exists (f x1); elim (H3 _ H11); intros;
    eapply derive_pt_eq_1; symmetry ; apply H12.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H13 : derivable_pt x0 x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
derivable_pt x0 x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  unfold derivable_pt; exists (g x1); elim (H1 _ H10); intros;
    eapply derive_pt_eq_1; symmetry ; apply H13.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
derivable_pt (fun y : R => l * x y + x0 y) x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  reg.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  exists H14; symmetry ; reg.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H15 : derive_pt x0 x1 H13 = g x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
derive_pt x0 x1 H13 = g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  elim (H1 _ H10); intros; rewrite H15; apply pr_nu.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  assert (H16 : derive_pt x x1 H12 = f x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
derive_pt x x1 H12 = f x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
H16:derive_pt x x1 H12 = f x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  elim (H3 _ H11); intros; rewrite H16; apply pr_nu.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x2 : R,
 b <= x2 <= a ->
 exists pr : derivable_pt x x2,
   f x2 = derive_pt x x2 pr) /\ b <= a
H0:(forall x2 : R,
 a <= x2 <= b ->
 exists pr : derivable_pt x0 x2,
   g x2 = derive_pt x0 x2 pr) /\ a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt x0 x2,
  g x2 = derive_pt x0 x2 pr
H2:a <= b
H3:forall x2 : R,
b <= x2 <= a ->
exists pr : derivable_pt x x2,
  f x2 = derive_pt x x2 pr
H4:b <= a
H5:b = a
x1:R
H6:a <= x1 <= a
H7:a <= x1
H8:x1 <= a
H9:x1 = a
H10:a <= x1 <= b
H11:b <= x1 <= a
H12:derivable_pt x x1
H13:derivable_pt x0 x1
H14:derivable_pt (fun y : R => l * x y + x0 y) x1
H15:derive_pt x0 x1 H13 = g x1
H16:derive_pt x x1 H12 = f x1
l * derive_pt x x1 H12 + derive_pt x0 x1 H13 =
l * f x1 + g x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  rewrite H15; rewrite H16; ring.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:(forall x1 : R,
 b <= x1 <= a ->
 exists pr : derivable_pt x x1,
   f x1 = derive_pt x x1 pr) /\ b <= a
H0:(forall x1 : R,
 a <= x1 <= b ->
 exists pr : derivable_pt x0 x1,
   g x1 = derive_pt x0 x1 pr) /\ a <= b
H1:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
H2:a <= b
H3:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H4:b <= a
H5:b = a
a <= a
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  right; reflexivity.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:antiderivative f x b a
H0:antiderivative g x0 b a
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) a b \/
antiderivative (fun x1 : R => l * f x1 + g x1)
  (fun y : R => l * x y + x0 y) b a
  right; unfold antiderivative; unfold antiderivative in H, H0; elim H;
    clear H; intros; elim H0; clear H0; intros H0 _; split.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:b <= a
H0:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
forall x1 : R,
b <= x1 <= a ->
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:b <= a
H0:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
b <= a
  intros; elim (H _ H2); elim (H0 _ H2); intros.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x4 : R,
b <= x4 <= a ->
exists pr : derivable_pt x x4,
  f x4 = derive_pt x x4 pr
H1:b <= a
H0:forall x4 : R,
b <= x4 <= a ->
exists pr : derivable_pt x0 x4,
  g x4 = derive_pt x0 x4 pr
x1:R
H2:b <= x1 <= a
x2:derivable_pt x0 x1
H3:g x1 = derive_pt x0 x1 x2
x3:derivable_pt x x1
H4:f x1 = derive_pt x x1 x3
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:b <= a
H0:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
b <= a
  assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1).
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x4 : R,
b <= x4 <= a ->
exists pr : derivable_pt x x4,
  f x4 = derive_pt x x4 pr
H1:b <= a
H0:forall x4 : R,
b <= x4 <= a ->
exists pr : derivable_pt x0 x4,
  g x4 = derive_pt x0 x4 pr
x1:R
H2:b <= x1 <= a
x2:derivable_pt x0 x1
H3:g x1 = derive_pt x0 x1 x2
x3:derivable_pt x x1
H4:f x1 = derive_pt x x1 x3
derivable_pt (fun y : R => l * x y + x0 y) x1
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x4 : R,
b <= x4 <= a ->
exists pr : derivable_pt x x4,
  f x4 = derive_pt x x4 pr
H1:b <= a
H0:forall x4 : R,
b <= x4 <= a ->
exists pr : derivable_pt x0 x4,
  g x4 = derive_pt x0 x4 pr
x1:R
H2:b <= x1 <= a
x2:derivable_pt x0 x1
H3:g x1 = derive_pt x0 x1 x2
x3:derivable_pt x x1
H4:f x1 = derive_pt x x1 x3
H5:derivable_pt (fun y : R => l * x y + x0 y) x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:b <= a
H0:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
b <= a
  reg.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x4 : R,
b <= x4 <= a ->
exists pr : derivable_pt x x4,
  f x4 = derive_pt x x4 pr
H1:b <= a
H0:forall x4 : R,
b <= x4 <= a ->
exists pr : derivable_pt x0 x4,
  g x4 = derive_pt x0 x4 pr
x1:R
H2:b <= x1 <= a
x2:derivable_pt x0 x1
H3:g x1 = derive_pt x0 x1 x2
x3:derivable_pt x x1
H4:f x1 = derive_pt x x1 x3
H5:derivable_pt (fun y : R => l * x y + x0 y) x1
exists
  pr : derivable_pt (fun y : R => l * x y + x0 y) x1,
  l * f x1 + g x1 =
  derive_pt (fun y : R => l * x y + x0 y) x1 pr
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:b <= a
H0:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
b <= a
  exists H5; symmetry ; reg; rewrite <- H3; rewrite <- H4; reflexivity.
f, g:R -> R
l, a, b:R
X:{g0 : R -> R |
antiderivative f g0 a b \/ antiderivative f g0 b a}
X0:{g0 : R -> R |
antiderivative g g0 a b \/
antiderivative g g0 b a}
x:R -> R
p:antiderivative f x a b \/ antiderivative f x b a
x0:R -> R
p0:antiderivative g x0 a b \/
antiderivative g x0 b a
H:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x x1,
  f x1 = derive_pt x x1 pr
H1:b <= a
H0:forall x1 : R,
b <= x1 <= a ->
exists pr : derivable_pt x0 x1,
  g x1 = derive_pt x0 x1 pr
b <= a
  assumption.
Defined.

(**********)
Lemma antiderivative_P1 :
  forall (f g F G:R -> R) (l a b:R),
    antiderivative f F a b ->
    antiderivative g G a b ->
    antiderivative (fun x:R => l * f x + g x) (fun x:R => l * F x + G x) a b.
forall (f g F G : R -> R) (l a b : R),
antiderivative f F a b ->
antiderivative g G a b ->
antiderivative (fun x : R => l * f x + g x)
  (fun x : R => l * F x + G x) a b
Proof.
forall (f g F G : R -> R) (l a b : R),
antiderivative f F a b ->
antiderivative g G a b ->
antiderivative (fun x : R => l * f x + g x)
  (fun x : R => l * F x + G x) a b
  unfold antiderivative; intros; elim H; elim H0; clear H H0; intros;
    split.
f, g, F, G:R -> R
l, a, b:R
H:forall x : R,
a <= x <= b ->
exists pr : derivable_pt G x,
  g x = derive_pt G x pr
H0:a <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F x,
  f x = derive_pt F x pr
H2:a <= b
forall x : R,
a <= x <= b ->
exists
  pr : derivable_pt (fun x0 : R => l * F x0 + G x0) x,
  l * f x + g x =
  derive_pt (fun x0 : R => l * F x0 + G x0) x pr
f, g, F, G:R -> R
l, a, b:R
H:forall x : R,
a <= x <= b ->
exists pr : derivable_pt G x,
  g x = derive_pt G x pr
H0:a <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F x,
  f x = derive_pt F x pr
H2:a <= b
a <= b
  intros; elim (H _ H3); elim (H1 _ H3); intros.
f, g, F, G:R -> R
l, a, b:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt G x2,
  g x2 = derive_pt G x2 pr
H0:a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F x2,
  f x2 = derive_pt F x2 pr
H2:a <= b
x:R
H3:a <= x <= b
x0:derivable_pt F x
H4:f x = derive_pt F x x0
x1:derivable_pt G x
H5:g x = derive_pt G x x1
exists
  pr : derivable_pt (fun x2 : R => l * F x2 + G x2) x,
  l * f x + g x =
  derive_pt (fun x2 : R => l * F x2 + G x2) x pr
f, g, F, G:R -> R
l, a, b:R
H:forall x : R,
a <= x <= b ->
exists pr : derivable_pt G x,
  g x = derive_pt G x pr
H0:a <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F x,
  f x = derive_pt F x pr
H2:a <= b
a <= b
  assert (H6 : derivable_pt (fun x:R => l * F x + G x) x).
f, g, F, G:R -> R
l, a, b:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt G x2,
  g x2 = derive_pt G x2 pr
H0:a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F x2,
  f x2 = derive_pt F x2 pr
H2:a <= b
x:R
H3:a <= x <= b
x0:derivable_pt F x
H4:f x = derive_pt F x x0
x1:derivable_pt G x
H5:g x = derive_pt G x x1
derivable_pt (fun x2 : R => l * F x2 + G x2) x
f, g, F, G:R -> R
l, a, b:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt G x2,
  g x2 = derive_pt G x2 pr
H0:a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F x2,
  f x2 = derive_pt F x2 pr
H2:a <= b
x:R
H3:a <= x <= b
x0:derivable_pt F x
H4:f x = derive_pt F x x0
x1:derivable_pt G x
H5:g x = derive_pt G x x1
H6:derivable_pt (fun x2 : R => l * F x2 + G x2) x
exists
  pr : derivable_pt (fun x2 : R => l * F x2 + G x2) x,
  l * f x + g x =
  derive_pt (fun x2 : R => l * F x2 + G x2) x pr
f, g, F, G:R -> R
l, a, b:R
H:forall x : R,
a <= x <= b ->
exists pr : derivable_pt G x,
  g x = derive_pt G x pr
H0:a <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F x,
  f x = derive_pt F x pr
H2:a <= b
a <= b
  reg.
f, g, F, G:R -> R
l, a, b:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt G x2,
  g x2 = derive_pt G x2 pr
H0:a <= b
H1:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F x2,
  f x2 = derive_pt F x2 pr
H2:a <= b
x:R
H3:a <= x <= b
x0:derivable_pt F x
H4:f x = derive_pt F x x0
x1:derivable_pt G x
H5:g x = derive_pt G x x1
H6:derivable_pt (fun x2 : R => l * F x2 + G x2) x
exists
  pr : derivable_pt (fun x2 : R => l * F x2 + G x2) x,
  l * f x + g x =
  derive_pt (fun x2 : R => l * F x2 + G x2) x pr
f, g, F, G:R -> R
l, a, b:R
H:forall x : R,
a <= x <= b ->
exists pr : derivable_pt G x,
  g x = derive_pt G x pr
H0:a <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F x,
  f x = derive_pt F x pr
H2:a <= b
a <= b
  exists H6; symmetry ; reg; rewrite <- H4; rewrite <- H5; ring.
f, g, F, G:R -> R
l, a, b:R
H:forall x : R,
a <= x <= b ->
exists pr : derivable_pt G x,
  g x = derive_pt G x pr
H0:a <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F x,
  f x = derive_pt F x pr
H2:a <= b
a <= b
  assumption.
Qed.

(* $\int_a^b \lambda f + g = \lambda \int_a^b f + \int_a^b f *)
Lemma NewtonInt_P6 :
  forall (f g:R -> R) (l a b:R) (pr1:Newton_integrable f a b)
    (pr2:Newton_integrable g a b),
    NewtonInt (fun x:R => l * f x + g x) a b (NewtonInt_P5 f g l a b pr1 pr2) =
    l * NewtonInt f a b pr1 + NewtonInt g a b pr2.
forall (f g : R -> R) (l a b : R)
  (pr1 : Newton_integrable f a b)
  (pr2 : Newton_integrable g a b),
NewtonInt (fun x : R => l * f x + g x) a b
  (NewtonInt_P5 f g l a b pr1 pr2) =
l * NewtonInt f a b pr1 + NewtonInt g a b pr2
Proof.
forall (f g : R -> R) (l a b : R)
  (pr1 : Newton_integrable f a b)
  (pr2 : Newton_integrable g a b),
NewtonInt (fun x : R => l * f x + g x) a b
  (NewtonInt_P5 f g l a b pr1 pr2) =
l * NewtonInt f a b pr1 + NewtonInt g a b pr2
  intros f g l a b pr1 pr2; unfold NewtonInt;
    destruct (NewtonInt_P5 f g l a b pr1 pr2) as (x,o); destruct pr1 as (x0,o0);
      destruct pr2 as (x1,o1); destruct (total_order_T a b) as [[Hlt|Heq]|Hgt].
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  elim o; intro.
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  elim o0; intro.
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 a b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  elim o1; intro.
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 a b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  assert (H2 := antiderivative_P1 f g x0 x1 l a b H0 H1);
    assert (H3 := antiderivative_Ucte _ _ _ _ _ H H2);
      elim H3; intros; assert (H5 : a <= a <= b).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hlt:a < b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 a b
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) a b
H3:exists c : R,
  forall x3 : R,
  a <= x3 <= b ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
a <= x3 <= b -> x x3 = l * x0 x3 + x1 x3 + x2
a <= a <= b
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hlt:a < b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 a b
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) a b
H3:exists c : R,
  forall x3 : R,
  a <= x3 <= b ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
a <= x3 <= b -> x x3 = l * x0 x3 + x1 x3 + x2
H5:a <= a <= b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  split; [ right; reflexivity | left; assumption ].
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hlt:a < b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 a b
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) a b
H3:exists c : R,
  forall x3 : R,
  a <= x3 <= b ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
a <= x3 <= b -> x x3 = l * x0 x3 + x1 x3 + x2
H5:a <= a <= b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  assert (H6 : a <= b <= b).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hlt:a < b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 a b
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) a b
H3:exists c : R,
  forall x3 : R,
  a <= x3 <= b ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
a <= x3 <= b -> x x3 = l * x0 x3 + x1 x3 + x2
H5:a <= a <= b
a <= b <= b
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hlt:a < b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 a b
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) a b
H3:exists c : R,
  forall x3 : R,
  a <= x3 <= b ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
a <= x3 <= b -> x x3 = l * x0 x3 + x1 x3 + x2
H5:a <= a <= b
H6:a <= b <= b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  split; [ left; assumption | right; reflexivity ].
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hlt:a < b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 a b
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) a b
H3:exists c : R,
  forall x3 : R,
  a <= x3 <= b ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
a <= x3 <= b -> x x3 = l * x0 x3 + x1 x3 + x2
H5:a <= a <= b
H6:a <= b <= b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  assert (H7 := H4 _ H5); assert (H8 := H4 _ H6); rewrite H7; rewrite H8; ring.
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 a b
H1:antiderivative g x1 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  unfold antiderivative in H1; elim H1; intros;
    elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H3 Hlt)).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  unfold antiderivative in H0; elim H0; intros;
    elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hlt)).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hlt:a < b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  unfold antiderivative in H; elim H; intros;
    elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 Hlt)).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Heq:a = b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  rewrite Heq; ring.
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  elim o; intro.
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  unfold antiderivative in H; elim H; intros;
    elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 Hgt)).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  elim o0; intro.
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
H0:antiderivative f x0 a b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  unfold antiderivative in H0; elim H0; intros;
    elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hgt)).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
H0:antiderivative f x0 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  elim o1; intro.
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
H0:antiderivative f x0 b a
H1:antiderivative g x1 a b
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
H0:antiderivative f x0 b a
H1:antiderivative g x1 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  unfold antiderivative in H1; elim H1; intros;
    elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H3 Hgt)).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x2 : R => l * f x2 + g x2) x a
  b \/
antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
Hgt:a > b
H:antiderivative (fun x2 : R => l * f x2 + g x2) x b
  a
H0:antiderivative f x0 b a
H1:antiderivative g x1 b a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  assert (H2 := antiderivative_P1 f g x0 x1 l b a H0 H1);
    assert (H3 := antiderivative_Ucte _ _ _ _ _ H H2);
      elim H3; intros; assert (H5 : b <= a <= a).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hgt:a > b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
H0:antiderivative f x0 b a
H1:antiderivative g x1 b a
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) b a
H3:exists c : R,
  forall x3 : R,
  b <= x3 <= a ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
b <= x3 <= a -> x x3 = l * x0 x3 + x1 x3 + x2
b <= a <= a
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hgt:a > b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
H0:antiderivative f x0 b a
H1:antiderivative g x1 b a
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) b a
H3:exists c : R,
  forall x3 : R,
  b <= x3 <= a ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
b <= x3 <= a -> x x3 = l * x0 x3 + x1 x3 + x2
H5:b <= a <= a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  split; [ left; assumption | right; reflexivity ].
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hgt:a > b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
H0:antiderivative f x0 b a
H1:antiderivative g x1 b a
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) b a
H3:exists c : R,
  forall x3 : R,
  b <= x3 <= a ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
b <= x3 <= a -> x x3 = l * x0 x3 + x1 x3 + x2
H5:b <= a <= a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  assert (H6 : b <= b <= a).
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hgt:a > b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
H0:antiderivative f x0 b a
H1:antiderivative g x1 b a
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) b a
H3:exists c : R,
  forall x3 : R,
  b <= x3 <= a ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
b <= x3 <= a -> x x3 = l * x0 x3 + x1 x3 + x2
H5:b <= a <= a
b <= b <= a
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hgt:a > b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
H0:antiderivative f x0 b a
H1:antiderivative g x1 b a
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) b a
H3:exists c : R,
  forall x3 : R,
  b <= x3 <= a ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
b <= x3 <= a -> x x3 = l * x0 x3 + x1 x3 + x2
H5:b <= a <= a
H6:b <= b <= a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  split; [ right; reflexivity | left; assumption ].
f, g:R -> R
l, a, b:R
x0:R -> R
o0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
o1:antiderivative g x1 a b \/
antiderivative g x1 b a
x:R -> R
o:antiderivative (fun x3 : R => l * f x3 + g x3) x a
  b \/
antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
Hgt:a > b
H:antiderivative (fun x3 : R => l * f x3 + g x3) x b
  a
H0:antiderivative f x0 b a
H1:antiderivative g x1 b a
H2:antiderivative (fun x3 : R => l * f x3 + g x3)
  (fun x3 : R => l * x0 x3 + x1 x3) b a
H3:exists c : R,
  forall x3 : R,
  b <= x3 <= a ->
  x x3 = (fun x4 : R => l * x0 x4 + x1 x4) x3 + c
x2:R
H4:forall x3 : R,
b <= x3 <= a -> x x3 = l * x0 x3 + x1 x3 + x2
H5:b <= a <= a
H6:b <= b <= a
x b - x a = l * (x0 b - x0 a) + (x1 b - x1 a)
  assert (H7 := H4 _ H5); assert (H8 := H4 _ H6); rewrite H7; rewrite H8; ring.
Qed.

Lemma antiderivative_P2 :
  forall (f F0 F1:R -> R) (a b c:R),
    antiderivative f F0 a b ->
    antiderivative f F1 b c ->
    antiderivative f
    (fun x:R =>
      match Rle_dec x b with
        | left _ => F0 x
        | right _ => F1 x + (F0 b - F1 b)
      end) a c.
forall (f F0 F1 : R -> R) (a b c : R),
antiderivative f F0 a b ->
antiderivative f F1 b c ->
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c
Proof.
forall (f F0 F1 : R -> R) (a b c : R),
antiderivative f F0 a b ->
antiderivative f F1 b c ->
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c
  intros; destruct H as (H,H1), H0 as (H0,H2); split.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H1:a <= b
H0:forall x : R,
b <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H2:b <= c
forall x : R,
a <= x <= c ->
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H1:a <= b
H0:forall x : R,
b <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H2:b <= c
a <= c
  2: apply Rle_trans with b; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H1:a <= b
H0:forall x : R,
b <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H2:b <= c
forall x : R,
a <= x <= c ->
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  intros x (H3,H4); destruct (total_order_T x b) as [[Hlt|Heq]|Hgt].
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert (H5 : a <= x <= b).
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
a <= x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  split; [ assumption | left; assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  destruct (H _ H5) as (x0,H6).
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
    assert
      (H7 :
        derivable_pt_lim
        (fun x:R =>
          match Rle_dec x b with
            | left _ => F0 x
            | right _ => F1 x + (F0 b - F1 b)
          end) x (f x)).
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  unfold derivable_pt_lim.
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr intros eps H9.
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert (H7 : derive_pt F0 x x0 = f x) by (symmetry; assumption).
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  destruct (derive_pt_eq_1 F0 x (f x) x0 H7 _ H9) as (x1,H10); set (D := Rmin x1 (b - x)).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x1 (b - x):R
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert (H11 : 0 < D).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x1 (b - x):R
0 < D
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  unfold D, Rmin; case (Rle_dec x1 (b - x)); intro.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x1 (b - x):R
r:x1 <= b - x
0 < x1
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x1 (b - x):R
n:~ x1 <= b - x
0 < b - x
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply (cond_pos x1).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x1 (b - x):R
n:~ x1 <= b - x
0 < b - x
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply Rlt_Rminus; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  exists (mkposreal _ H11); intros h H12 H13.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Rabs
  (((if Rle_dec (x + h) b
     then F0 (x + h)
     else F1 (x + h) + (F0 b - F1 b)) -
    (if Rle_dec x b
     then F0 x
     else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr case (Rle_dec x b) as [|[]].
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
Rabs
  (((if Rle_dec (x + h) b
     then F0 (x + h)
     else F1 (x + h) + (F0 b - F1 b)) - F0 x) / h -
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  case (Rle_dec (x + h) b) as [|[]].
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
r0:x + h <= b
Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
x + h <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply H10.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
r0:x + h <= b
h <> 0
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
r0:x + h <= b
Rabs h < x1
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
x + h <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
r0:x + h <= b
Rabs h < x1
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
x + h <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply Rlt_le_trans with D; [ assumption | unfold D; apply Rmin_l ].
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
x + h <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  left; apply Rlt_le_trans with (x + D).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
x + h < x + D
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
x + D <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply Rplus_lt_compat_l; apply Rle_lt_trans with (Rabs h).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
h <= Rabs h
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
Rabs h < D
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
x + D <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply RRle_abs.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
Rabs h < D
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
x + D <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply H13.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
r:x <= b
x + D <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply Rplus_le_reg_l with (- x); rewrite <- Rplus_assoc; rewrite Rplus_opp_l;
    rewrite Rplus_0_l; rewrite Rplus_comm; unfold D;
      apply Rmin_r.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
eps:R
H9:0 < eps
H7:derive_pt F0 x x0 = f x
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x1 (b - x):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  left; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert
    (H8 :
      derivable_pt
      (fun x:R =>
        match Rle_dec x b with
          | left _ => F0 x
          | right _ => F1 x + (F0 b - F1 b)
        end) x).
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
derivable_pt
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
H8:derivable_pt
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  unfold derivable_pt; exists (f x); apply H7.
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hlt:x < b
H5:a <= x <= b
x0:derivable_pt F0 x
H6:f x = derive_pt F0 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
H8:derivable_pt
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  exists H8; symmetry ; apply derive_pt_eq_0; apply H7.
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert (H5 : a <= x <= b).
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
a <= x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  split; [ assumption | right; assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert (H6 : b <= x <= c).
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
b <= x <= c
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  split; [ right; symmetry ; assumption | assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  elim (H _ H5); elim (H0 _ H6); intros; assert (H9 : derive_pt F0 x x1 = f x).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
derive_pt F0 x x1 = f x
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  symmetry ; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert (H10 : derive_pt F1 x x0 = f x).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
derive_pt F1 x x0 = f x
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  symmetry ; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert (H11 := derive_pt_eq_1 F0 x (f x) x1 H9);
    assert (H12 := derive_pt_eq_1 F1 x (f x) x0 H10);
      assert
        (H13 :
          derivable_pt_lim
          (fun x:R =>
            match Rle_dec x b with
              | left _ => F0 x
              | right _ => F1 x + (F0 b - F1 b)
            end) x (f x)).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
derivable_pt_lim
  (fun x2 : R =>
   if Rle_dec x2 b
   then F0 x2
   else F1 x2 + (F0 b - F1 b)) x (f x)
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  unfold derivable_pt_lim; unfold derivable_pt_lim in H11, H12; intros;
    elim (H11 _ H13); elim (H12 _ H13); intros; set (D := Rmin x2 x3);
      assert (H16 : 0 < D).
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h : R,
h <> 0 -> Rabs h < x2 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
x3:posreal
H15:forall h : R,
h <> 0 -> Rabs h < x3 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x2 x3:R
0 < D
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h : R,
h <> 0 -> Rabs h < x2 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
x3:posreal
H15:forall h : R,
h <> 0 -> Rabs h < x3 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  unfold D; unfold Rmin; case (Rle_dec x2 x3); intro.
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h : R,
h <> 0 -> Rabs h < x2 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
x3:posreal
H15:forall h : R,
h <> 0 -> Rabs h < x3 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x2 x3:R
r:x2 <= x3
0 < x2
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h : R,
h <> 0 -> Rabs h < x2 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
x3:posreal
H15:forall h : R,
h <> 0 -> Rabs h < x3 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x2 x3:R
n:~ x2 <= x3
0 < x3
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h : R,
h <> 0 -> Rabs h < x2 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
x3:posreal
H15:forall h : R,
h <> 0 -> Rabs h < x3 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply (cond_pos x2).
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h : R,
h <> 0 -> Rabs h < x2 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
x3:posreal
H15:forall h : R,
h <> 0 -> Rabs h < x3 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x2 x3:R
n:~ x2 <= x3
0 < x3
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h : R,
h <> 0 -> Rabs h < x2 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
x3:posreal
H15:forall h : R,
h <> 0 -> Rabs h < x3 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply (cond_pos x3).
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h : R,
h <> 0 -> Rabs h < delta -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h : R,
h <> 0 -> Rabs h < x2 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
x3:posreal
H15:forall h : R,
h <> 0 -> Rabs h < x3 -> Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  exists (mkposreal _ H16); intros; case (Rle_dec x b) as [|[]].
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
Rabs
  (((if Rle_dec (x + h) b
     then F0 (x + h)
     else F1 (x + h) + (F0 b - F1 b)) - F0 x) / h -
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  case (Rle_dec (x + h) b); intro.
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
r0:x + h <= b
Rabs ((F0 (x + h) - F0 x) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
Rabs ((F1 (x + h) + (F0 b - F1 b) - F0 x) / h - f x) <
eps
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply H15.
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
r0:x + h <= b
h <> 0
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
r0:x + h <= b
Rabs h < x3
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
Rabs ((F1 (x + h) + (F0 b - F1 b) - F0 x) / h - f x) <
eps
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
r0:x + h <= b
Rabs h < x3
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
Rabs ((F1 (x + h) + (F0 b - F1 b) - F0 x) / h - f x) <
eps
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply Rlt_le_trans with D; [ assumption | unfold D; apply Rmin_r ].
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
Rabs ((F1 (x + h) + (F0 b - F1 b) - F0 x) / h - f x) <
eps
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  replace (F1 (x + h) + (F0 b - F1 b) - F0 x) with (F1 (x + h) - F1 x).
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
F1 (x + h) - F1 x = F1 (x + h) + (F0 b - F1 b) - F0 x
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply H14.
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
h <> 0
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
Rabs h < x2
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
F1 (x + h) - F1 x = F1 (x + h) + (F0 b - F1 b) - F0 x
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
Rabs h < x2
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
F1 (x + h) - F1 x = F1 (x + h) + (F0 b - F1 b) - F0 x
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  apply Rlt_le_trans with D; [ assumption | unfold D; apply Rmin_l ].
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
r:x <= b
n:~ x + h <= b
F1 (x + h) - F1 x = F1 (x + h) + (F0 b - F1 b) - F0 x
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  rewrite Heq; ring.
f, F0, F1:R -> R
a, b, c:R
H:forall x4 : R,
a <= x4 <= b ->
exists pr : derivable_pt F0 x4,
  f x4 = derive_pt F0 x4 pr
H1:a <= b
H0:forall x4 : R,
b <= x4 <= c ->
exists pr : derivable_pt F1 x4,
  f x4 = derive_pt F1 x4 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps0
H12:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
forall h0 : R,
h0 <> 0 -> Rabs h0 < delta -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H13:0 < eps
x2:posreal
H14:forall h0 : R,
h0 <> 0 -> Rabs h0 < x2 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
x3:posreal
H15:forall h0 : R,
h0 <> 0 -> Rabs h0 < x3 -> Rabs ((F0 (x + h0) - F0 x) / h0 - f x) < eps
D:=Rmin x2 x3:R
H16:0 < D
h:R
H17:h <> 0
H18:Rabs h < {| pos := D; cond_pos := H16 |}
x <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  right; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert
    (H14 :
      derivable_pt
      (fun x:R =>
        match Rle_dec x b with
          | left _ => F0 x
          | right _ => F1 x + (F0 b - F1 b)
        end) x).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
derivable_pt
  (fun x2 : R =>
   if Rle_dec x2 b
   then F0 x2
   else F1 x2 + (F0 b - F1 b)) x
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
H14:derivable_pt
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  unfold derivable_pt; exists (f x); apply H13.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Heq:x = b
H5:a <= x <= b
H6:b <= x <= c
x0:derivable_pt F1 x
H7:f x = derive_pt F1 x x0
x1:derivable_pt F0 x
H8:f x = derive_pt F0 x x1
H9:derive_pt F0 x x1 = f x
H10:derive_pt F1 x x0 = f x
H11:derivable_pt_lim F0 x (f x)
H12:derivable_pt_lim F1 x (f x)
H13:derivable_pt_lim
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
(f x)
H14:derivable_pt
(fun x2 : R => if Rle_dec x2 b then F0 x2 else F1 x2 + (F0 b - F1 b)) x
exists
  pr : derivable_pt
         (fun x2 : R =>
          if Rle_dec x2 b
          then F0 x2
          else F1 x2 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x2 : R =>
     if Rle_dec x2 b
     then F0 x2
     else F1 x2 + (F0 b - F1 b)) x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  exists H14; symmetry ; apply derive_pt_eq_0; apply H13.
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert (H5 : b <= x <= c).
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
b <= x <= c
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  split; [ left; assumption | assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x0 : R,
a <= x0 <= b ->
exists pr : derivable_pt F0 x0,
  f x0 = derive_pt F0 x0 pr
H1:a <= b
H0:forall x0 : R,
b <= x0 <= c ->
exists pr : derivable_pt F1 x0,
  f x0 = derive_pt F1 x0 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
exists
  pr : derivable_pt
         (fun x0 : R =>
          if Rle_dec x0 b
          then F0 x0
          else F1 x0 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x0 : R =>
     if Rle_dec x0 b
     then F0 x0
     else F1 x0 + (F0 b - F1 b)) x pr
  assert (H6 := H0 _ H5); elim H6; clear H6; intros;
    assert
      (H7 :
        derivable_pt_lim
        (fun x:R =>
          match Rle_dec x b with
            | left _ => F0 x
            | right _ => F1 x + (F0 b - F1 b)
          end) x (f x)).
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  unfold derivable_pt_lim; assert (H7 : derive_pt F1 x x0 = f x).
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
derive_pt F1 x x0 = f x
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  symmetry ; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
forall eps : R,
0 < eps ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  assert (H8 := derive_pt_eq_1 F1 x (f x) x0 H7); unfold derivable_pt_lim in H8;
    intros; elim (H8 _ H9); intros; set (D := Rmin x1 (x - b));
      assert (H11 : 0 < D).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
D:=Rmin x1 (x - b):R
0 < D
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  unfold D; unfold Rmin; case (Rle_dec x1 (x - b)); intro.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
D:=Rmin x1 (x - b):R
r:x1 <= x - b
0 < x1
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
D:=Rmin x1 (x - b):R
n:~ x1 <= x - b
0 < x - b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  apply (cond_pos x1).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
D:=Rmin x1 (x - b):R
n:~ x1 <= x - b
0 < x - b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  apply Rlt_Rminus; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((F1 (x + h) - F1 x) / h - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h : R,
h <> 0 -> Rabs h < x1 -> Rabs ((F1 (x + h) - F1 x) / h - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs
    (((if Rle_dec (x + h) b
       then F0 (x + h)
       else F1 (x + h) + (F0 b - F1 b)) -
      (if Rle_dec x b
       then F0 x
       else F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  exists (mkposreal _ H11); intros; destruct (Rle_dec x b) as [Hle|Hnle].
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hle:x <= b
Rabs
  (((if Rle_dec (x + h) b
     then F0 (x + h)
     else F1 (x + h) + (F0 b - F1 b)) - F0 x) / h -
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 -> Rabs h0 < x1 -> Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Rabs
  (((if Rle_dec (x + h) b
     then F0 (x + h)
     else F1 (x + h) + (F0 b - F1 b)) -
    (F1 x + (F0 b - F1 b))) / h - f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ Hle Hgt)).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Rabs
  (((if Rle_dec (x + h) b
     then F0 (x + h)
     else F1 (x + h) + (F0 b - F1 b)) -
    (F1 x + (F0 b - F1 b))) / h - 
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  destruct (Rle_dec (x + h) b) as [Hle'|Hnle'].
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
Rabs ((F0 (x + h) - (F1 x + (F0 b - F1 b))) / h - f x) <
eps
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs
  ((F1 (x + h) + (F0 b - F1 b) -
    (F1 x + (F0 b - F1 b))) / h - 
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  cut (b < x + h).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
b < x + h ->
Rabs ((F0 (x + h) - (F1 x + (F0 b - F1 b))) / h - f x) <
eps
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
b < x + h
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs
  ((F1 (x + h) + (F0 b - F1 b) -
    (F1 x + (F0 b - F1 b))) / h - 
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ Hle' H14)).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
b < x + h
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs
  ((F1 (x + h) + (F0 b - F1 b) -
    (F1 x + (F0 b - F1 b))) / h - 
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  apply Rplus_lt_reg_l with (- h - b); replace (- h - b + b) with (- h);
    [ idtac | ring ]; replace (- h - b + (x + h)) with (x - b);
    [ idtac | ring ]; apply Rle_lt_trans with (Rabs h).
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
- h <= Rabs h
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
Rabs h < x - b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs
  ((F1 (x + h) + (F0 b - F1 b) -
    (F1 x + (F0 b - F1 b))) / h - 
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  rewrite <- Rabs_Ropp; apply RRle_abs.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
Rabs h < x - b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs
  ((F1 (x + h) + (F0 b - F1 b) -
    (F1 x + (F0 b - F1 b))) / h - 
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  apply Rlt_le_trans with D.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
Rabs h < D
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
D <= x - b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs
  ((F1 (x + h) + (F0 b - F1 b) -
    (F1 x + (F0 b - F1 b))) / h - 
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  apply H13.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hle':x + h <= b
D <= x - b
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs
  ((F1 (x + h) + (F0 b - F1 b) -
    (F1 x + (F0 b - F1 b))) / h - 
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  unfold D; apply Rmin_r.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs
  ((F1 (x + h) + (F0 b - F1 b) -
    (F1 x + (F0 b - F1 b))) / h - 
   f x) < eps
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  replace (F1 (x + h) + (F0 b - F1 b) - (F1 x + (F0 b - F1 b))) with
  (F1 (x + h) - F1 x); [ idtac | ring ]; apply H10.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
h <> 0
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs h < x1
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs h < x1
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  apply Rlt_le_trans with D.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
Rabs h < D
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
D <= x1
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x2 : R,
a <= x2 <= b ->
exists pr : derivable_pt F0 x2,
  f x2 = derive_pt F0 x2 pr
H1:a <= b
H0:forall x2 : R,
b <= x2 <= c ->
exists pr : derivable_pt F1 x2,
  f x2 = derive_pt F1 x2 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derive_pt F1 x x0 = f x
H8:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps0
eps:R
H9:0 < eps
x1:posreal
H10:forall h0 : R,
h0 <> 0 ->
Rabs h0 < x1 ->
Rabs ((F1 (x + h0) - F1 x) / h0 - f x) < eps
D:=Rmin x1 (x - b):R
H11:0 < D
h:R
H12:h <> 0
H13:Rabs h < {| pos := D; cond_pos := H11 |}
Hnle:~ x <= b
Hnle':~ x + h <= b
D <= x1
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  unfold D; apply Rmin_l.
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  assert
    (H8 :
      derivable_pt
      (fun x:R =>
        match Rle_dec x b with
          | left _ => F0 x
          | right _ => F1 x + (F0 b - F1 b)
        end) x).
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
derivable_pt
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
H8:derivable_pt
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  unfold derivable_pt; exists (f x); apply H7.
f, F0, F1:R -> R
a, b, c:R
H:forall x1 : R,
a <= x1 <= b ->
exists pr : derivable_pt F0 x1,
  f x1 = derive_pt F0 x1 pr
H1:a <= b
H0:forall x1 : R,
b <= x1 <= c ->
exists pr : derivable_pt F1 x1,
  f x1 = derive_pt F1 x1 pr
H2:b <= c
x:R
H3:a <= x
H4:x <= c
Hgt:x > b
H5:b <= x <= c
x0:derivable_pt F1 x
H6:f x = derive_pt F1 x x0
H7:derivable_pt_lim
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x 
  (f x)
H8:derivable_pt
  (fun x1 : R =>
   if Rle_dec x1 b
   then F0 x1
   else F1 x1 + (F0 b - F1 b)) x
exists
  pr : derivable_pt
         (fun x1 : R =>
          if Rle_dec x1 b
          then F0 x1
          else F1 x1 + (F0 b - F1 b)) x,
  f x =
  derive_pt
    (fun x1 : R =>
     if Rle_dec x1 b
     then F0 x1
     else F1 x1 + (F0 b - F1 b)) x pr
  exists H8; symmetry ; apply derive_pt_eq_0; apply H7.
Qed.

Lemma antiderivative_P3 :
  forall (f F0 F1:R -> R) (a b c:R),
    antiderivative f F0 a b ->
    antiderivative f F1 c b ->
    antiderivative f F1 c a \/ antiderivative f F0 a c.
forall (f F0 F1 : R -> R) (a b c : R),
antiderivative f F0 a b ->
antiderivative f F1 c b ->
antiderivative f F1 c a \/ antiderivative f F0 a c
Proof.
forall (f F0 F1 : R -> R) (a b c : R),
antiderivative f F0 a b ->
antiderivative f F1 c b ->
antiderivative f F1 c a \/ antiderivative f F0 a c
  intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0;
    intros; destruct (total_order_T a c) as [[Hle|Heq]|Hgt].
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hle:a < c
antiderivative f F1 c a \/ antiderivative f F0 a c
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:a = c
antiderivative f F1 c a \/ antiderivative f F0 a c
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
antiderivative f F1 c a \/ antiderivative f F0 a c
  right; unfold antiderivative; split.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hle:a < c
forall x : R,
a <= x <= c ->
exists pr : derivable_pt F0 x, f x = derive_pt F0 x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hle:a < c
a <= c
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:a = c
antiderivative f F1 c a \/ antiderivative f F0 a c
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
antiderivative f F1 c a \/ antiderivative f F0 a c
  intros; apply H1; elim H3; intros; split;
    [ assumption | apply Rle_trans with c; assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hle:a < c
a <= c
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:a = c
antiderivative f F1 c a \/ antiderivative f F0 a c
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
antiderivative f F1 c a \/ antiderivative f F0 a c
  left; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:a = c
antiderivative f F1 c a \/ antiderivative f F0 a c
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
antiderivative f F1 c a \/ antiderivative f F0 a c
  right; unfold antiderivative; split.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:a = c
forall x : R,
a <= x <= c ->
exists pr : derivable_pt F0 x, f x = derive_pt F0 x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:a = c
a <= c
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
antiderivative f F1 c a \/ antiderivative f F0 a c
  intros; apply H1; elim H3; intros; split;
    [ assumption | apply Rle_trans with c; assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:a = c
a <= c
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
antiderivative f F1 c a \/ antiderivative f F0 a c
  right; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
antiderivative f F1 c a \/ antiderivative f F0 a c
  left; unfold antiderivative; split.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
forall x : R,
c <= x <= a ->
exists pr : derivable_pt F1 x, f x = derive_pt F1 x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
c <= a
  intros; apply H; elim H3; intros; split;
    [ assumption | apply Rle_trans with a; assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
c <= x <= b ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:c <= b
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:a > c
c <= a
  left; assumption.
Qed.

Lemma antiderivative_P4 :
  forall (f F0 F1:R -> R) (a b c:R),
    antiderivative f F0 a b ->
    antiderivative f F1 a c ->
    antiderivative f F1 b c \/ antiderivative f F0 c b.
forall (f F0 F1 : R -> R) (a b c : R),
antiderivative f F0 a b ->
antiderivative f F1 a c ->
antiderivative f F1 b c \/ antiderivative f F0 c b
Proof.
forall (f F0 F1 : R -> R) (a b c : R),
antiderivative f F0 a b ->
antiderivative f F1 a c ->
antiderivative f F1 b c \/ antiderivative f F0 c b
  intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0;
    intros; destruct (total_order_T c b) as [[Hlt|Heq]|Hgt].
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hlt:c < b
antiderivative f F1 b c \/ antiderivative f F0 c b
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:c = b
antiderivative f F1 b c \/ antiderivative f F0 c b
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
antiderivative f F1 b c \/ antiderivative f F0 c b
  right; unfold antiderivative; split.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hlt:c < b
forall x : R,
c <= x <= b ->
exists pr : derivable_pt F0 x, f x = derive_pt F0 x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hlt:c < b
c <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:c = b
antiderivative f F1 b c \/ antiderivative f F0 c b
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
antiderivative f F1 b c \/ antiderivative f F0 c b
  intros; apply H1; elim H3; intros; split;
    [ apply Rle_trans with c; assumption | assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hlt:c < b
c <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:c = b
antiderivative f F1 b c \/ antiderivative f F0 c b
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
antiderivative f F1 b c \/ antiderivative f F0 c b
  left; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:c = b
antiderivative f F1 b c \/ antiderivative f F0 c b
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
antiderivative f F1 b c \/ antiderivative f F0 c b
  right; unfold antiderivative; split.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:c = b
forall x : R,
c <= x <= b ->
exists pr : derivable_pt F0 x, f x = derive_pt F0 x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:c = b
c <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
antiderivative f F1 b c \/ antiderivative f F0 c b
  intros; apply H1; elim H3; intros; split;
    [ apply Rle_trans with c; assumption | assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Heq:c = b
c <= b
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
antiderivative f F1 b c \/ antiderivative f F0 c b
  right; assumption.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
antiderivative f F1 b c \/ antiderivative f F0 c b
  left; unfold antiderivative; split.
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
forall x : R,
b <= x <= c ->
exists pr : derivable_pt F1 x, f x = derive_pt F1 x pr
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
b <= c
  intros; apply H; elim H3; intros; split;
    [ apply Rle_trans with b; assumption | assumption ].
f, F0, F1:R -> R
a, b, c:R
H:forall x : R,
a <= x <= c ->
exists pr : derivable_pt F1 x,
  f x = derive_pt F1 x pr
H0:a <= c
H1:forall x : R,
a <= x <= b ->
exists pr : derivable_pt F0 x,
  f x = derive_pt F0 x pr
H2:a <= b
Hgt:c > b
b <= c
  left; assumption.
Qed.

Lemma NewtonInt_P7 :
  forall (f:R -> R) (a b c:R),
    a < b ->
    b < c ->
    Newton_integrable f a b ->
    Newton_integrable f b c -> Newton_integrable f a c.
forall (f : R -> R) (a b c : R),
a < b ->
b < c ->
Newton_integrable f a b ->
Newton_integrable f b c -> Newton_integrable f a c
Proof.
forall (f : R -> R) (a b c : R),
a < b ->
b < c ->
Newton_integrable f a b ->
Newton_integrable f b c -> Newton_integrable f a c
  unfold Newton_integrable; intros f a b c Hab Hbc X X0; elim X;
    clear X; intros F0 H0; elim X0; clear X0; intros F1 H1;
      set
        (g :=
          fun x:R =>
            match Rle_dec x b with
              | left _ => F0 x
              | right _ => F1 x + (F0 b - F1 b)
            end); exists g; left; unfold g;
        apply antiderivative_P2.
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
antiderivative f F0 a b
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
antiderivative f F1 b c
  elim H0; intro.
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
H:antiderivative f F0 a b
antiderivative f F0 a b
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
H:antiderivative f F0 b a
antiderivative f F0 a b
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
antiderivative f F1 b c
  assumption.
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
H:antiderivative f F0 b a
antiderivative f F0 a b
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
antiderivative f F1 b c
  unfold antiderivative in H; elim H; clear H; intros;
    elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hab)).
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
antiderivative f F1 b c
  elim H1; intro.
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
H:antiderivative f F1 b c
antiderivative f F1 b c
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
H:antiderivative f F1 c b
antiderivative f F1 b c
  assumption.
f:R -> R
a, b, c:R
Hab:a < b
Hbc:b < c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
g:=fun x : R =>
if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b):R -> R
H:antiderivative f F1 c b
antiderivative f F1 b c
  unfold antiderivative in H; elim H; clear H; intros;
    elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hbc)).
Qed.

Lemma NewtonInt_P8 :
  forall (f:R -> R) (a b c:R),
    Newton_integrable f a b ->
    Newton_integrable f b c -> Newton_integrable f a c.
forall (f : R -> R) (a b c : R),
Newton_integrable f a b ->
Newton_integrable f b c -> Newton_integrable f a c
Proof.
forall (f : R -> R) (a b c : R),
Newton_integrable f a b ->
Newton_integrable f b c -> Newton_integrable f a c
  intros.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
Newton_integrable f a c
  elim X; intros F0 H0.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
Newton_integrable f a c
  elim X0; intros F1 H1.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Newton_integrable f a c
  destruct (total_order_T a b) as [[Hlt|Heq]|Hgt].
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  destruct (total_order_T b c) as [[Hlt'|Heq']|Hgt'].
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
(* a<b & b<c *)
  unfold Newton_integrable;
    exists
      (fun x:R =>
        match Rle_dec x b with
          | left _ => F0 x
          | right _ => F1 x + (F0 b - F1 b)
        end).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim H0; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 a b
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 b a
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim H1; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 a b
H2:antiderivative f F1 b c
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 a b
H2:antiderivative f F1 c b
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 b a
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  left; apply antiderivative_P2; assumption.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 a b
H2:antiderivative f F1 c b
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 b a
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold antiderivative in H2; elim H2; clear H2; intros _ H2.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 a b
H2:c <= b
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 b a
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hlt')).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:antiderivative f F0 b a
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hlt':b < c
H:b <= a
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  a c \/
antiderivative f
  (fun x : R =>
   if Rle_dec x b then F0 x else F1 x + (F0 b - F1 b))
  c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hlt)).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
(* a<b & b=c *)
  rewrite Heq' in X; apply X.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
(* a<b & b>c *)
  destruct (total_order_T a c) as [[Hlt''|Heq'']|Hgt''].
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold Newton_integrable; exists F0.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
antiderivative f F0 a c \/ antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  left.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim H1; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 b c
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:b <= c
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hgt')).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim H0; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  assert (H3 := antiderivative_P3 f F0 F1 a b c H2 H).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim H3; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
H4:antiderivative f F1 c a
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
H4:antiderivative f F0 a c
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold antiderivative in H4; elim H4; clear H4; intros _ H4.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
H4:c <= a
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
H4:antiderivative f F0 a c
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 Hlt'')).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
H4:antiderivative f F0 a c
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  assumption.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold antiderivative in H2; elim H2; clear H2; intros _ H2.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hlt'':a < c
H:antiderivative f F1 c b
H2:b <= a
antiderivative f F0 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hlt)).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  rewrite Heq''; apply NewtonInt_P1.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold Newton_integrable; exists F1.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
antiderivative f F1 a c \/ antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  right.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim H1; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 b c
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:b <= c
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hgt')).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim H0; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  assert (H3 := antiderivative_P3 f F0 F1 a b c H2 H).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim H3; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
H4:antiderivative f F1 c a
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
H4:antiderivative f F0 a c
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  assumption.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
H4:antiderivative f F0 a c
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold antiderivative in H4; elim H4; clear H4; intros _ H4.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 a b
H3:antiderivative f F1 c a \/
antiderivative f F0 a c
H4:a <= c
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 Hgt'')).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:antiderivative f F0 b a
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  unfold antiderivative in H2; elim H2; clear H2; intros _ H2.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hlt:a < b
Hgt':b > c
Hgt'':a > c
H:antiderivative f F1 c b
H2:b <= a
antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hlt)).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Heq:a = b
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
(* a=b *)
  rewrite Heq; apply X0.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Newton_integrable f a c
  destruct (total_order_T b c) as [[Hlt'|Heq']|Hgt'].
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
(* a>b & b<c *)
  destruct (total_order_T a c) as [[Hlt''|Heq'']|Hgt''].
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  unfold Newton_integrable; exists F1.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
antiderivative f F1 a c \/ antiderivative f F1 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  left.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim H1; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
(*****************)
  elim H0; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:antiderivative f F0 a b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:antiderivative f F0 b a
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  unfold antiderivative in H2; elim H2; clear H2; intros _ H2.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:a <= b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:antiderivative f F0 b a
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hgt)).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:antiderivative f F0 b a
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  assert (H3 := antiderivative_P4 f F0 F1 b a c H2 H).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:antiderivative f F0 b a
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim H3; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:antiderivative f F0 b a
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
H4:antiderivative f F1 a c
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:antiderivative f F0 b a
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
H4:antiderivative f F0 c a
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  assumption.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:antiderivative f F0 b a
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
H4:antiderivative f F0 c a
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  unfold antiderivative in H4; elim H4; clear H4; intros _ H4.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 b c
H2:antiderivative f F0 b a
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
H4:c <= a
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 Hlt'')).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:antiderivative f F1 c b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hlt'':a < c
H:c <= b
antiderivative f F1 a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hlt')).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Heq'':a = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  rewrite Heq''; apply NewtonInt_P1.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  unfold Newton_integrable; exists F0.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
antiderivative f F0 a c \/ antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  right.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim H0; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 a b
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:a <= b
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hgt)).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim H1; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 b c
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 c b
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  assert (H3 := antiderivative_P4 f F0 F1 b a c H H2).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 b c
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 c b
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim H3; intro.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 b c
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
H4:antiderivative f F1 a c
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 b c
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
H4:antiderivative f F0 c a
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 c b
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  unfold antiderivative in H4; elim H4; clear H4; intros _ H4.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 b c
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
H4:a <= c
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 b c
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
H4:antiderivative f F0 c a
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 c b
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 Hgt'')).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 b c
H3:antiderivative f F1 a c \/
antiderivative f F0 c a
H4:antiderivative f F0 c a
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 c b
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  assumption.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:antiderivative f F1 c b
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  unfold antiderivative in H2; elim H2; clear H2; intros _ H2.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hlt':b < c
Hgt'':a > c
H:antiderivative f F0 b a
H2:c <= b
antiderivative f F0 c a
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hlt')).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Heq':b = c
Newton_integrable f a c
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
(* a>b & b=c *)
  rewrite Heq' in X; apply X.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
Newton_integrable f a c
(* a>b & b>c *)
  assert (X1 := NewtonInt_P3 f a b X).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
X1:Newton_integrable f b a
Newton_integrable f a c
  assert (X2 := NewtonInt_P3 f b c X0).
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
X1:Newton_integrable f b a
X2:Newton_integrable f c b
Newton_integrable f a c
  apply NewtonInt_P3.
f:R -> R
a, b, c:R
X:Newton_integrable f a b
X0:Newton_integrable f b c
F0:R -> R
H0:antiderivative f F0 a b \/
antiderivative f F0 b a
F1:R -> R
H1:antiderivative f F1 b c \/
antiderivative f F1 c b
Hgt:a > b
Hgt':b > c
X1:Newton_integrable f b a
X2:Newton_integrable f c b
Newton_integrable f c a
  apply NewtonInt_P7 with b; assumption.
Qed.

(* Chasles' relation *)
Lemma NewtonInt_P9 :
  forall (f:R -> R) (a b c:R) (pr1:Newton_integrable f a b)
    (pr2:Newton_integrable f b c),
    NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2) =
    NewtonInt f a b pr1 + NewtonInt f b c pr2.
forall (f : R -> R) (a b c : R)
  (pr1 : Newton_integrable f a b)
  (pr2 : Newton_integrable f b c),
NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2) =
NewtonInt f a b pr1 + NewtonInt f b c pr2
Proof.
forall (f : R -> R) (a b c : R)
  (pr1 : Newton_integrable f a b)
  (pr2 : Newton_integrable f b c),
NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2) =
NewtonInt f a b pr1 + NewtonInt f b c pr2
  intros; unfold NewtonInt.
f:R -> R
a, b, c:R
pr1:Newton_integrable f a b
pr2:Newton_integrable f b c
(let (g, _) := NewtonInt_P8 f a b c pr1 pr2 in
 g c - g a) =
(let (g, _) := pr1 in g b - g a) +
(let (g, _) := pr2 in g c - g b)
  case (NewtonInt_P8 f a b c pr1 pr2) as (x,Hor).
f:R -> R
a, b, c:R
pr1:Newton_integrable f a b
pr2:Newton_integrable f b c
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
x c - x a =
(let (g, _) := pr1 in g b - g a) +
(let (g, _) := pr2 in g c - g b)
  case pr1 as (x0,Hor0).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
pr2:Newton_integrable f b c
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
x c - x a =
x0 b - x0 a + (let (g, _) := pr2 in g c - g b)
  case pr2 as (x1,Hor1).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  destruct (total_order_T a b) as [[Hlt|Heq]|Hgt].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  destruct (total_order_T b c) as [[Hlt'|Heq']|Hgt'].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
(* a<b & b<c *)
  case Hor0; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  case Hor1; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  case Hor; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H2 := antiderivative_P2 f x0 x1 a b c H H0).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 b
   then x0 x2
   else x1 x2 + (x0 b - x1 b)) a c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert
    (H3 :=
      antiderivative_Ucte f x
      (fun x:R =>
        match Rle_dec x b with
          | left _ => x0 x
          | right _ => x1 x + (x0 b - x1 b)
        end) a c H1 H2).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 b
   then x0 x2
   else x1 x2 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x2 : R,
  a <= x2 <= c ->
  x x2 =
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) x2 + c0
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H3; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H5 : a <= a <= c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
a <= a <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ right; reflexivity | left; apply Rlt_trans with b; assumption ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H6 : a <= c <= c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
a <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ left; apply Rlt_trans with b; assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 _ H5); rewrite (H4 _ H6).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
(if Rle_dec c b then x0 c else x1 c + (x0 b - x1 b)) +
x2 -
((if Rle_dec a b then x0 a else x1 a + (x0 b - x1 b)) +
 x2) = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  destruct (Rle_dec a b) as [Hlea|Hnlea].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
Hlea:a <= b
(if Rle_dec c b then x0 c else x1 c + (x0 b - x1 b)) +
x2 - (x0 a + x2) = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
Hnlea:~ a <= b
(if Rle_dec c b then x0 c else x1 c + (x0 b - x1 b)) +
x2 - (x1 a + (x0 b - x1 b) + x2) =
x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  destruct (Rle_dec c b) as [Hlec|Hnlec].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
Hlea:a <= b
Hlec:c <= b
x0 c + x2 - (x0 a + x2) = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
Hlea:a <= b
Hnlec:~ c <= b
x1 c + (x0 b - x1 b) + x2 - (x0 a + x2) =
x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
Hnlea:~ a <= b
(if Rle_dec c b then x0 c else x1 c + (x0 b - x1 b)) +
x2 - (x1 a + (x0 b - x1 b) + x2) =
x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ Hlec Hlt')).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
Hlea:a <= b
Hnlec:~ c <= b
x1 c + (x0 b - x1 b) + x2 - (x0 a + x2) =
x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
Hnlea:~ a <= b
(if Rle_dec c b then x0 c else x1 c + (x0 b - x1 b)) +
x2 - (x1 a + (x0 b - x1 b) + x2) =
x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x0 x3
   else x1 x3 + (x0 b - x1 b)) a c
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= c ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x0 x4
   else x1 x4 + (x0 b - x1 b)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= c ->
x x3 =
(if Rle_dec x3 b
 then x0 x3
 else x1 x3 + (x0 b - x1 b)) + x2
H5:a <= a <= c
H6:a <= c <= c
Hnlea:~ a <= b
(if Rle_dec c b then x0 c else x1 c + (x0 b - x1 b)) +
x2 - (x1 a + (x0 b - x1 b) + x2) =
x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hnlea; left; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H1; elim H1; clear H1; intros _ H1.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 b c
H1:c <= a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 (Rlt_trans _ _ _ Hlt Hlt'))).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H0; elim H0; clear H0; intros _ H0.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 a b
H0:c <= b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 Hlt')).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hlt':b < c
H:b <= a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hlt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
(* a<b & b=c *)
  rewrite <- Heq'.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x b - x a = x0 b - x0 a + (x1 b - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Heq':b = c
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite <- Heq' in Hor.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor0; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H1 := antiderivative_Ucte f x x0 a b H0 H).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x2 : R,
  a <= x2 <= b -> x x2 = x0 x2 + c0
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H1; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, a <= x3 <= b -> x x3 = x0 x3 + x2
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H2 b).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, a <= x3 <= b -> x x3 = x0 x3 + x2
x0 b + x2 + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, a <= x3 <= b -> x x3 = x0 x3 + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H2 a).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, a <= x3 <= b -> x x3 = x0 x3 + x2
x0 b + x2 + - (x0 a + x2) = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, a <= x3 <= b -> x x3 = x0 x3 + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, a <= x3 <= b -> x x3 = x0 x3 + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, a <= x3 <= b -> x x3 = x0 x3 + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, a <= x3 <= b -> x x3 = x0 x3 + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ right; reflexivity | left; assumption ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x a b
H1:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, a <= x3 <= b -> x x3 = x0 x3 + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ left; assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H0; elim H0; clear H0; intros _ H0.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 a b
H0:b <= a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 Hlt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hlt:a < b
Heq':b = c
H:b <= a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hlt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
(* a<b & b>c *)
  elim Hor1; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 b c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:b <= c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hgt')).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor0; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H2 := antiderivative_P2 f x x1 a c b H1 H).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 c
   then x x2
   else x1 x2 + (x c - x1 c)) a b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H3 := antiderivative_Ucte _ _ _ a b H0 H2).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 c
   then x x2
   else x1 x2 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x2 : R,
  a <= x2 <= b ->
  x0 x2 =
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) x2 + c0
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H3; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 a).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
x c - x a =
x0 b -
((if Rle_dec a c then x a else x1 a + (x c - x1 c)) +
 x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 b).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
x c - x a =
(if Rle_dec b c then x b else x1 b + (x c - x1 c)) +
x2 -
((if Rle_dec a c then x a else x1 a + (x c - x1 c)) +
 x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  destruct (Rle_dec b c) as [Hle|Hnle].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
Hle:b <= c
x c - x a =
x b + x2 -
((if Rle_dec a c then x a else x1 a + (x c - x1 c)) +
 x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
Hnle:~ b <= c
x c - x a =
x1 b + (x c - x1 c) + x2 -
((if Rle_dec a c then x a else x1 a + (x c - x1 c)) +
 x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ Hle Hgt')).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
Hnle:~ b <= c
x c - x a =
x1 b + (x c - x1 c) + x2 -
((if Rle_dec a c then x a else x1 a + (x c - x1 c)) +
 x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  destruct (Rle_dec a c) as [Hle'|Hnle'].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
Hnle:~ b <= c
Hle':a <= c
x c - x a =
x1 b + (x c - x1 c) + x2 - (x a + x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
Hnle:~ b <= c
Hnle':~ a <= c
x c - x a =
x1 b + (x c - x1 c) + x2 - (x1 a + (x c - x1 c) + x2) +
(x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
Hnle:~ b <= c
Hnle':~ a <= c
x c - x a =
x1 b + (x c - x1 c) + x2 - (x1 a + (x c - x1 c) + x2) +
(x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hnle'; unfold antiderivative in H1; elim H1; intros; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ left; assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x x3
   else x1 x3 + (x c - x1 c)) a b
H3:exists c0 : R,
  forall x3 : R,
  a <= x3 <= b ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x x4
   else x1 x4 + (x c - x1 c)) x3 + c0
x2:R
H4:forall x3 : R,
a <= x3 <= b ->
x0 x3 =
(if Rle_dec x3 c
 then x x3
 else x1 x3 + (x c - x1 c)) + x2
a <= a <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ right; reflexivity | left; assumption ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H2 := antiderivative_P2 _ _ _ _ _ _ H1 H0).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 a
   then x x2
   else x0 x2 + (x a - x0 a)) c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H3 := antiderivative_Ucte _ _ _ c b H H2).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 a
   then x x2
   else x0 x2 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x2 : R,
  c <= x2 <= b ->
  x1 x2 =
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) x2 + c0
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H3; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
x c - x a =
x0 b - x0 a +
((if Rle_dec c a then x c else x0 c + (x a - x0 a)) +
 x2 - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 b).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
x c - x a =
x0 b - x0 a +
((if Rle_dec c a then x c else x0 c + (x a - x0 a)) +
 x2 -
 ((if Rle_dec b a then x b else x0 b + (x a - x0 a)) +
  x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  destruct (Rle_dec b a) as [Hle|Hnle].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
Hle:b <= a
x c - x a =
x0 b - x0 a +
((if Rle_dec c a then x c else x0 c + (x a - x0 a)) +
 x2 - (x b + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
Hnle:~ b <= a
x c - x a =
x0 b - x0 a +
((if Rle_dec c a then x c else x0 c + (x a - x0 a)) +
 x2 - (x0 b + (x a - x0 a) + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ Hle Hlt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
Hnle:~ b <= a
x c - x a =
x0 b - x0 a +
((if Rle_dec c a then x c else x0 c + (x a - x0 a)) +
 x2 - (x0 b + (x a - x0 a) + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  destruct (Rle_dec c a) as [Hle'|[]].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
Hnle:~ b <= a
Hle':c <= a
x c - x a =
x0 b - x0 a + (x c + x2 - (x0 b + (x a - x0 a) + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
Hnle:~ b <= a
c <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
Hnle:~ b <= a
c <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H1; elim H1; intros; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ left; assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 a b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x x3
   else x0 x3 + (x a - x0 a)) c b
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x x4
   else x0 x4 + (x a - x0 a)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= b ->
x1 x3 =
(if Rle_dec x3 a
 then x x3
 else x0 x3 + (x a - x0 a)) + x2
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ right; reflexivity | left; assumption ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H0; elim H0; clear H0; intros _ H0.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hlt:a < b
Hgt':b > c
H:antiderivative f x1 c b
H0:b <= a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 Hlt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Heq:a = b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
(* a=b *)
  rewrite Heq in Hor |- *.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor1; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H1 := antiderivative_Ucte _ _ _ b c H H0).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x2 : R,
  b <= x2 <= c -> x x2 = x1 x2 + c0
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H1; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H3 : b <= c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H; elim H; intros; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H2 b).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
x c - (x1 b + x2) = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H2 c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
x1 c + x2 - (x1 b + x2) = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 b c
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= c -> x x3 = x1 x3 + x2
H3:b <= c
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ right; reflexivity | assumption ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H1 : b = c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
b = c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
H1:b = c
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H, H0; elim H; elim H0; intros; apply Rle_antisym;
    assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x b c
H0:antiderivative f x1 c b
H1:b = c
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite H1; ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor1; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 b c
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H1 : b = c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 b c
b = c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 b c
H1:b = c
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H, H0; elim H; elim H0; intros; apply Rle_antisym;
    assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 b c
H1:b = c
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite H1; ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H1 := antiderivative_Ucte _ _ _ c b H H0).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x2 : R,
  c <= x2 <= b -> x x2 = x1 x2 + c0
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H1; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H3 : c <= b).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H; elim H; intros; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
x c - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H2 c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
x1 c + x2 - x b = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H2 b).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
x1 c + x2 - (x1 b + x2) = x0 b - x0 b + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
c <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
c <= b <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x b c \/ antiderivative f x c b
Heq:a = b
H:antiderivative f x c b
H0:antiderivative f x1 c b
H1:exists c0 : R,
  forall x3 : R,
  c <= x3 <= b -> x x3 = x1 x3 + c0
x2:R
H2:forall x3 : R, c <= x3 <= b -> x x3 = x1 x3 + x2
H3:c <= b
c <= c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ right; reflexivity | assumption ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
(* a>b & b<c *)
  destruct (total_order_T b c) as [[Hlt'|Heq']|Hgt'].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor0; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 a b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:a <= b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hgt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor1; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H2 := antiderivative_P2 _ _ _ _ _ _ H H1).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 a
   then x0 x2
   else x x2 + (x0 a - x a)) b c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H3 := antiderivative_Ucte _ _ _ b c H0 H2).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 a
   then x0 x2
   else x x2 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x2 : R,
  b <= x2 <= c ->
  x1 x2 =
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) x2 + c0
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H3; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 b).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
x c - x a =
x0 b - x0 a +
(x1 c -
 ((if Rle_dec b a then x0 b else x b + (x0 a - x a)) +
  x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
x c - x a =
x0 b - x0 a +
((if Rle_dec c a then x0 c else x c + (x0 a - x a)) +
 x2 -
 ((if Rle_dec b a then x0 b else x b + (x0 a - x a)) +
  x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  case (Rle_dec b a) as [|[]].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
r:b <= a
x c - x a =
x0 b - x0 a +
((if Rle_dec c a then x0 c else x c + (x0 a - x a)) +
 x2 - (x0 b + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  case (Rle_dec c a) as [|].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
r:b <= a
r0:c <= a
x c - x a = x0 b - x0 a + (x0 c + x2 - (x0 b + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
r:b <= a
n:~ c <= a
x c - x a =
x0 b - x0 a + (x c + (x0 a - x a) + x2 - (x0 b + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H5 : a = c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
r:b <= a
r0:c <= a
a = c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
r:b <= a
r0:c <= a
H5:a = c
x c - x a = x0 b - x0 a + (x0 c + x2 - (x0 b + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
r:b <= a
n:~ c <= a
x c - x a =
x0 b - x0 a + (x c + (x0 a - x a) + x2 - (x0 b + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H1; elim H1; intros; apply Rle_antisym; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
r:b <= a
r0:c <= a
H5:a = c
x c - x a = x0 b - x0 a + (x0 c + x2 - (x0 b + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
r:b <= a
n:~ c <= a
x c - x a =
x0 b - x0 a + (x c + (x0 a - x a) + x2 - (x0 b + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite H5; ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
r:b <= a
n:~ c <= a
x c - x a =
x0 b - x0 a + (x c + (x0 a - x a) + x2 - (x0 b + x2))
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  left; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= c <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ left; assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x a c
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 a
   then x0 x3
   else x x3 + (x0 a - x a)) b c
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= c ->
  x1 x3 =
  (fun x4 : R =>
   if Rle_dec x4 a
   then x0 x4
   else x x4 + (x0 a - x a)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= c ->
x1 x3 =
(if Rle_dec x3 a
 then x0 x3
 else x x3 + (x0 a - x a)) + x2
b <= b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ right; reflexivity | left; assumption ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H2 := antiderivative_P2 _ _ _ _ _ _ H0 H1).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 c
   then x1 x2
   else x x2 + (x1 c - x c)) b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H3 := antiderivative_Ucte _ _ _ b a H H2).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 c
   then x1 x2
   else x x2 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x2 : R,
  b <= x2 <= a ->
  x0 x2 =
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) x2 + c0
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H3; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 a).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
x c - x a =
x0 b -
((if Rle_dec a c then x1 a else x a + (x1 c - x c)) +
 x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 b).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
x c - x a =
(if Rle_dec b c then x1 b else x b + (x1 c - x c)) +
x2 -
((if Rle_dec a c then x1 a else x a + (x1 c - x c)) +
 x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  case (Rle_dec b c) as [|[]].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
r:b <= c
x c - x a =
x1 b + x2 -
((if Rle_dec a c then x1 a else x a + (x1 c - x c)) +
 x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  case (Rle_dec a c) as [|].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
r:b <= c
r0:a <= c
x c - x a = x1 b + x2 - (x1 a + x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
r:b <= c
n:~ a <= c
x c - x a =
x1 b + x2 - (x a + (x1 c - x c) + x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H5 : a = c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
r:b <= c
r0:a <= c
a = c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
r:b <= c
r0:a <= c
H5:a = c
x c - x a = x1 b + x2 - (x1 a + x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
r:b <= c
n:~ a <= c
x c - x a =
x1 b + x2 - (x a + (x1 c - x c) + x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H1; elim H1; intros; apply Rle_antisym; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
r:b <= c
r0:a <= c
H5:a = c
x c - x a = x1 b + x2 - (x1 a + x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
r:b <= c
n:~ a <= c
x c - x a =
x1 b + x2 - (x a + (x1 c - x c) + x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite H5; ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
r:b <= c
n:~ a <= c
x c - x a =
x1 b + x2 - (x a + (x1 c - x c) + x2) + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= c
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  left; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ right; reflexivity | left; assumption ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 c
   then x1 x3
   else x x3 + (x1 c - x c)) b a
H3:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a ->
  x0 x3 =
  (fun x4 : R =>
   if Rle_dec x4 c
   then x1 x4
   else x x4 + (x1 c - x c)) x3 + c0
x2:R
H4:forall x3 : R,
b <= x3 <= a ->
x0 x3 =
(if Rle_dec x3 c
 then x1 x3
 else x x3 + (x1 c - x c)) + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ left; assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H0; elim H0; clear H0; intros _ H0.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hlt':b < c
H:antiderivative f x0 b a
H0:c <= b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 Hlt')).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
(* a>b & b=c *)
  rewrite <- Heq'.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x b - x a = x0 b - x0 a + (x1 b - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Heq':b = c
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite <- Heq' in Hor.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor0; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 a b
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:a <= b
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hgt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x a b
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H0; elim H0; clear H0; intros _ H0.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:a <= b
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 Hgt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H1 := antiderivative_Ucte f x x0 b a H0 H).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x2 : R,
  b <= x2 <= a -> x x2 = x0 x2 + c0
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H1; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= a -> x x3 = x0 x3 + x2
x b + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H2 b).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= a -> x x3 = x0 x3 + x2
x0 b + x2 + - x a = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= a -> x x3 = x0 x3 + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H2 a).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= a -> x x3 = x0 x3 + x2
x0 b + x2 + - (x0 a + x2) = x0 b + - x0 a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= a -> x x3 = x0 x3 + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= a -> x x3 = x0 x3 + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= a -> x x3 = x0 x3 + x2
b <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= a -> x x3 = x0 x3 + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ left; assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a b \/ antiderivative f x b a
Hgt:a > b
Heq':b = c
H:antiderivative f x0 b a
H0:antiderivative f x b a
H1:exists c0 : R,
  forall x3 : R,
  b <= x3 <= a -> x x3 = x0 x3 + c0
x2:R
H2:forall x3 : R, b <= x3 <= a -> x x3 = x0 x3 + x2
b <= b <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  split; [ right; reflexivity | left; assumption ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
(* a>b & b>c *)
  elim Hor0; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 a b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H; elim H; clear H; intros _ H.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:a <= b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hgt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor1; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 b c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H0; elim H0; clear H0; intros _ H0.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:b <= c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 Hgt')).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim Hor; intro.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x a c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H1; elim H1; clear H1; intros _ H1.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:a <= c
x c - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 (Rlt_trans _ _ _ Hgt' Hgt))).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H2 := antiderivative_P2 _ _ _ _ _ _ H0 H).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 b
   then x1 x2
   else x0 x2 + (x1 b - x0 b)) c a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H3 := antiderivative_Ucte _ _ _ c a H1 H2).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x2 : R =>
   if Rle_dec x2 b
   then x1 x2
   else x0 x2 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x2 : R,
  c <= x2 <= a ->
  x x2 =
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) x2 + c0
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  elim H3; intros.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  assert (H5 : c <= a).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
c <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  unfold antiderivative in H1; elim H1; intros; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
x c - x a = x0 b - x0 a + (x1 c - x1 b)
  rewrite (H4 c).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
(if Rle_dec c b then x1 c else x0 c + (x1 b - x0 b)) +
x2 - x a = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= c <= a
  rewrite (H4 a).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
(if Rle_dec c b then x1 c else x0 c + (x1 b - x0 b)) +
x2 -
((if Rle_dec a b then x1 a else x0 a + (x1 b - x0 b)) +
 x2) = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= c <= a
  destruct (Rle_dec a b) as [Hle|Hnle].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
Hle:a <= b
(if Rle_dec c b then x1 c else x0 c + (x1 b - x0 b)) +
x2 - (x1 a + x2) = x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
Hnle:~ a <= b
(if Rle_dec c b then x1 c else x0 c + (x1 b - x0 b)) +
x2 - (x0 a + (x1 b - x0 b) + x2) =
x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= c <= a
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ Hle Hgt)).
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
Hnle:~ a <= b
(if Rle_dec c b then x1 c else x0 c + (x1 b - x0 b)) +
x2 - (x0 a + (x1 b - x0 b) + x2) =
x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= c <= a
  destruct (Rle_dec c b) as [|[]].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
Hnle:~ a <= b
r:c <= b
x1 c + x2 - (x0 a + (x1 b - x0 b) + x2) =
x0 b - x0 a + (x1 c - x1 b)
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
Hnle:~ a <= b
c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= c <= a
  ring.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
Hnle:~ a <= b
c <= b
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= c <= a
  left; assumption.
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= a <= a
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= c <= a
  split; [ assumption | right; reflexivity ].
f:R -> R
a, b, c:R
x0:R -> R
Hor0:antiderivative f x0 a b \/
antiderivative f x0 b a
x1:R -> R
Hor1:antiderivative f x1 b c \/
antiderivative f x1 c b
x:R -> R
Hor:antiderivative f x a c \/ antiderivative f x c a
Hgt:a > b
Hgt':b > c
H:antiderivative f x0 b a
H0:antiderivative f x1 c b
H1:antiderivative f x c a
H2:antiderivative f
  (fun x3 : R =>
   if Rle_dec x3 b
   then x1 x3
   else x0 x3 + (x1 b - x0 b)) c a
H3:exists c0 : R,
  forall x3 : R,
  c <= x3 <= a ->
  x x3 =
  (fun x4 : R =>
   if Rle_dec x4 b
   then x1 x4
   else x0 x4 + (x1 b - x0 b)) x3 + c0
x2:R
H4:forall x3 : R,
c <= x3 <= a ->
x x3 =
(if Rle_dec x3 b
 then x1 x3
 else x0 x3 + (x1 b - x0 b)) + x2
H5:c <= a
c <= c <= a
  split; [ right; reflexivity | assumption ].
Qed.