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diff --git a/theories/Reals/MVT.v b/theories/Reals/MVT.v
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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+ 
+(*i $Id: MVT.v,v 1.10.2.1 2004/07/16 19:31:10 herbelin Exp $ i*)
+
+Require Import Rbase.
+Require Import Rfunctions.
+Require Import Ranalysis1.
+Require Import Rtopology. Open Local Scope R_scope.
+
+(* The Mean Value Theorem *)
+Theorem MVT :
+ forall (f g:R -> R) (a b:R) (pr1:forall c:R, a < c < b -> derivable_pt f c)
+   (pr2:forall c:R, a < c < b -> derivable_pt g c),
+   a < b ->
+   (forall c:R, a <= c <= b -> continuity_pt f c) ->
+   (forall c:R, a <= c <= b -> continuity_pt g c) ->
+    exists c : R,
+     (exists P : a < c < b,
+        (g b - g a) * derive_pt f c (pr1 c P) =
+        (f b - f a) * derive_pt g c (pr2 c P)).
+intros; assert (H2 := Rlt_le _ _ H).
+set (h := fun y:R => (g b - g a) * f y - (f b - f a) * g y).
+cut (forall c:R, a < c < b -> derivable_pt h c).
+intro; cut (forall c:R, a <= c <= b -> continuity_pt h c).
+intro; assert (H4 := continuity_ab_maj h a b H2 H3).
+assert (H5 := continuity_ab_min h a b H2 H3).
+elim H4; intros Mx H6.
+elim H5; intros mx H7.
+cut (h a = h b).
+intro; set (M := h Mx); set (m := h mx).
+cut
+ (forall (c:R) (P:a < c < b),
+    derive_pt h c (X c P) =
+    (g b - g a) * derive_pt f c (pr1 c P) -
+    (f b - f a) * derive_pt g c (pr2 c P)).
+intro; case (Req_dec (h a) M); intro.
+case (Req_dec (h a) m); intro.
+cut (forall c:R, a <= c <= b -> h c = M).
+intro; cut (a < (a + b) / 2 < b).
+(*** h constant ***)
+intro; exists ((a + b) / 2).
+exists H13.
+apply Rminus_diag_uniq; rewrite <- H9; apply deriv_constant2 with a b.
+elim H13; intros; assumption.
+elim H13; intros; assumption.
+intros; rewrite (H12 ((a + b) / 2)).
+apply H12; split; left; assumption.
+elim H13; intros; split; left; assumption.
+split.
+apply Rmult_lt_reg_l with 2.
+prove_sup0.
+unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ rewrite <- Rinv_r_sym.
+rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H.
+discrR.
+apply Rmult_lt_reg_l with 2.
+prove_sup0.
+unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ rewrite <- Rinv_r_sym.
+rewrite Rmult_1_l; rewrite Rplus_comm; rewrite double;
+ apply Rplus_lt_compat_l; apply H.
+discrR.
+intros; elim H6; intros H13 _.
+elim H7; intros H14 _.
+apply Rle_antisym.
+apply H13; apply H12.
+rewrite H10 in H11; rewrite H11; apply H14; apply H12.
+cut (a < mx < b).
+(*** h admet un minimum global sur [a,b] ***)
+intro; exists mx.
+exists H12.
+apply Rminus_diag_uniq; rewrite <- H9; apply deriv_minimum with a b.
+elim H12; intros; assumption.
+elim H12; intros; assumption.
+intros; elim H7; intros.
+apply H15; split; left; assumption.
+elim H7; intros _ H12; elim H12; intros; split.
+inversion H13.
+apply H15.
+rewrite H15 in H11; elim H11; reflexivity.
+inversion H14.
+apply H15.
+rewrite H8 in H11; rewrite <- H15 in H11; elim H11; reflexivity.
+cut (a < Mx < b).
+(*** h admet un maximum global sur [a,b] ***)
+intro; exists Mx.
+exists H11.
+apply Rminus_diag_uniq; rewrite <- H9; apply deriv_maximum with a b.
+elim H11; intros; assumption.
+elim H11; intros; assumption.
+intros; elim H6; intros; apply H14.
+split; left; assumption.
+elim H6; intros _ H11; elim H11; intros; split.
+inversion H12.
+apply H14.
+rewrite H14 in H10; elim H10; reflexivity.
+inversion H13.
+apply H14.
+rewrite H8 in H10; rewrite <- H14 in H10; elim H10; reflexivity.
+intros; unfold h in |- *;
+ replace
+  (derive_pt (fun y:R => (g b - g a) * f y - (f b - f a) * g y) c (X c P))
+  with
+  (derive_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F) c
+     (derivable_pt_minus _ _ _
+        (derivable_pt_mult _ _ _ (derivable_pt_const (g b - g a) c) (pr1 c P))
+        (derivable_pt_mult _ _ _ (derivable_pt_const (f b - f a) c) (pr2 c P))));
+ [ idtac | apply pr_nu ].
+rewrite derive_pt_minus; do 2 rewrite derive_pt_mult;
+ do 2 rewrite derive_pt_const; do 2 rewrite Rmult_0_l; 
+ do 2 rewrite Rplus_0_l; reflexivity.
+unfold h in |- *; ring.
+intros; unfold h in |- *;
+ change
+   (continuity_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F)
+      c) in |- *.
+apply continuity_pt_minus; apply continuity_pt_mult.
+apply derivable_continuous_pt; apply derivable_const.
+apply H0; apply H3.
+apply derivable_continuous_pt; apply derivable_const.
+apply H1; apply H3.
+intros;
+ change
+   (derivable_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F)
+      c) in |- *.
+apply derivable_pt_minus; apply derivable_pt_mult.
+apply derivable_pt_const.
+apply (pr1 _ H3).
+apply derivable_pt_const.
+apply (pr2 _ H3).
+Qed.
+
+(* Corollaries ... *)
+Lemma MVT_cor1 :
+ forall (f:R -> R) (a b:R) (pr:derivable f),
+   a < b ->
+    exists c : R, f b - f a = derive_pt f c (pr c) * (b - a) /\ a < c < b.
+intros f a b pr H; cut (forall c:R, a < c < b -> derivable_pt f c);
+ [ intro | intros; apply pr ].
+cut (forall c:R, a < c < b -> derivable_pt id c);
+ [ intro | intros; apply derivable_pt_id ].
+cut (forall c:R, a <= c <= b -> continuity_pt f c);
+ [ intro | intros; apply derivable_continuous_pt; apply pr ].
+cut (forall c:R, a <= c <= b -> continuity_pt id c);
+ [ intro | intros; apply derivable_continuous_pt; apply derivable_id ].
+assert (H2 := MVT f id a b X X0 H H0 H1).
+elim H2; intros c H3; elim H3; intros.
+exists c; split.
+cut (derive_pt id c (X0 c x) = derive_pt id c (derivable_pt_id c));
+ [ intro | apply pr_nu ].
+rewrite H5 in H4; rewrite (derive_pt_id c) in H4; rewrite Rmult_1_r in H4;
+ rewrite <- H4; replace (derive_pt f c (X c x)) with (derive_pt f c (pr c));
+ [ idtac | apply pr_nu ]; apply Rmult_comm.
+apply x.
+Qed.
+
+Theorem MVT_cor2 :
+ forall (f f':R -> R) (a b:R),
+   a < b ->
+   (forall c:R, a <= c <= b -> derivable_pt_lim f c (f' c)) ->
+    exists c : R, f b - f a = f' c * (b - a) /\ a < c < b.
+intros f f' a b H H0; cut (forall c:R, a <= c <= b -> derivable_pt f c).
+intro; cut (forall c:R, a < c < b -> derivable_pt f c).
+intro; cut (forall c:R, a <= c <= b -> continuity_pt f c).
+intro; cut (forall c:R, a <= c <= b -> derivable_pt id c).
+intro; cut (forall c:R, a < c < b -> derivable_pt id c).
+intro; cut (forall c:R, a <= c <= b -> continuity_pt id c).
+intro; elim (MVT f id a b X0 X2 H H1 H2); intros; elim H3; clear H3; intros;
+ exists x; split.
+cut (derive_pt id x (X2 x x0) = 1).
+cut (derive_pt f x (X0 x x0) = f' x).
+intros; rewrite H4 in H3; rewrite H5 in H3; unfold id in H3;
+ rewrite Rmult_1_r in H3; rewrite Rmult_comm; symmetry  in |- *; 
+ assumption.
+apply derive_pt_eq_0; apply H0; elim x0; intros; split; left; assumption.
+apply derive_pt_eq_0; apply derivable_pt_lim_id.
+assumption.
+intros; apply derivable_continuous_pt; apply X1; assumption.
+intros; apply derivable_pt_id.
+intros; apply derivable_pt_id.
+intros; apply derivable_continuous_pt; apply X; assumption.
+intros; elim H1; intros; apply X; split; left; assumption.
+intros; unfold derivable_pt in |- *; apply existT with (f' c); apply H0;
+ apply H1.
+Qed.
+
+Lemma MVT_cor3 :
+ forall (f f':R -> R) (a b:R),
+   a < b ->
+   (forall x:R, a <= x -> x <= b -> derivable_pt_lim f x (f' x)) ->
+    exists c : R, a <= c /\ c <= b /\ f b = f a + f' c * (b - a).
+intros f f' a b H H0;
+ assert (H1 :  exists c : R, f b - f a = f' c * (b - a) /\ a < c < b);
+ [ apply MVT_cor2; [ apply H | intros; elim H1; intros; apply (H0 _ H2 H3) ]
+ | elim H1; intros; exists x; elim H2; intros; elim H4; intros; split;
+    [ left; assumption | split; [ left; assumption | rewrite <- H3; ring ] ] ].
+Qed.
+
+Lemma Rolle :
+ forall (f:R -> R) (a b:R) (pr:forall x:R, a < x < b -> derivable_pt f x),
+   (forall x:R, a <= x <= b -> continuity_pt f x) ->
+   a < b ->
+   f a = f b ->
+    exists c : R, (exists P : a < c < b, derive_pt f c (pr c P) = 0).
+intros; assert (H2 : forall x:R, a < x < b -> derivable_pt id x).
+intros; apply derivable_pt_id.
+assert (H3 := MVT f id a b pr H2 H0 H);
+ assert (H4 : forall x:R, a <= x <= b -> continuity_pt id x).
+intros; apply derivable_continuous; apply derivable_id.
+elim (H3 H4); intros; elim H5; intros; exists x; exists x0; rewrite H1 in H6;
+ unfold id in H6; unfold Rminus in H6; rewrite Rplus_opp_r in H6;
+ rewrite Rmult_0_l in H6; apply Rmult_eq_reg_l with (b - a);
+ [ rewrite Rmult_0_r; apply H6
+ | apply Rminus_eq_contra; red in |- *; intro; rewrite H7 in H0;
+    elim (Rlt_irrefl _ H0) ].
+Qed.
+
+(**********)
+Lemma nonneg_derivative_1 :
+ forall (f:R -> R) (pr:derivable f),
+   (forall x:R, 0 <= derive_pt f x (pr x)) -> increasing f.
+intros.
+unfold increasing in |- *.
+intros.
+case (total_order_T x y); intro.
+elim s; intro.
+apply Rplus_le_reg_l with (- f x).
+rewrite Rplus_opp_l; rewrite Rplus_comm.
+assert (H1 := MVT_cor1 f _ _ pr a).
+elim H1; intros.
+elim H2; intros.
+unfold Rminus in H3.
+rewrite H3.
+apply Rmult_le_pos.
+apply H.
+apply Rplus_le_reg_l with x.
+rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ].
+rewrite b; right; reflexivity.
+elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 r)).
+Qed.
+ 
+(**********)
+Lemma nonpos_derivative_0 :
+ forall (f:R -> R) (pr:derivable f),
+   decreasing f -> forall x:R, derive_pt f x (pr x) <= 0.
+intros f pr H x; assert (H0 := H); unfold decreasing in H0;
+ generalize (derivable_derive f x (pr x)); intro; elim H1; 
+ intros l H2.
+rewrite H2; case (Rtotal_order l 0); intro.
+left; assumption.
+elim H3; intro.
+right; assumption.
+generalize (derive_pt_eq_1 f x l (pr x) H2); intros; cut (0 < l / 2).
+intro; elim (H5 (l / 2) H6); intros delta H7;
+ cut (delta / 2 <> 0 /\ 0 < delta / 2 /\ Rabs (delta / 2) < delta).
+intro; decompose [and] H8; intros; generalize (H7 (delta / 2) H9 H12);
+ cut ((f (x + delta / 2) - f x) / (delta / 2) <= 0).
+intro; cut (0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)).
+intro; unfold Rabs in |- *;
+ case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)).
+intros;
+ generalize
+  (Rplus_lt_compat_r (- l) (- ((f (x + delta / 2) - f x) / (delta / 2) - l))
+     (l / 2) H14); unfold Rminus in |- *.
+replace (l / 2 + - l) with (- (l / 2)).
+replace (- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) + - l) with
+ (- ((f (x + delta / 2) + - f x) / (delta / 2))).
+intro.
+generalize
+ (Ropp_lt_gt_contravar (- ((f (x + delta / 2) + - f x) / (delta / 2)))
+    (- (l / 2)) H15). 
+repeat rewrite Ropp_involutive.
+intro.
+generalize
+ (Rlt_trans 0 (l / 2) ((f (x + delta / 2) - f x) / (delta / 2)) H6 H16);
+ intro.
+elim
+ (Rlt_irrefl 0
+    (Rlt_le_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) 0 H17 H10)).
+ring.
+pattern l at 3 in |- *; rewrite double_var.
+ring.
+intros.
+generalize
+ (Ropp_ge_le_contravar ((f (x + delta / 2) - f x) / (delta / 2) - l) 0 r).
+rewrite Ropp_0.
+intro.
+elim
+ (Rlt_irrefl 0
+    (Rlt_le_trans 0 (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) 0 H13
+       H15)).
+replace (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) with
+ ((f x - f (x + delta / 2)) / (delta / 2) + l).
+unfold Rminus in |- *.
+apply Rplus_le_lt_0_compat.
+unfold Rdiv in |- *; apply Rmult_le_pos.
+cut (x <= x + delta * / 2).
+intro; generalize (H0 x (x + delta * / 2) H13); intro;
+ generalize
+  (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H14);
+ rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
+pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
+ left; assumption.
+left; apply Rinv_0_lt_compat; assumption.
+assumption.
+rewrite Ropp_minus_distr.
+unfold Rminus in |- *.
+rewrite (Rplus_comm l).
+unfold Rdiv in |- *.
+rewrite <- Ropp_mult_distr_l_reverse.
+rewrite Ropp_plus_distr.
+rewrite Ropp_involutive.
+rewrite (Rplus_comm (f x)).
+reflexivity.
+replace ((f (x + delta / 2) - f x) / (delta / 2)) with
+ (- ((f x - f (x + delta / 2)) / (delta / 2))).
+rewrite <- Ropp_0.
+apply Ropp_ge_le_contravar.
+apply Rle_ge.
+unfold Rdiv in |- *; apply Rmult_le_pos.
+cut (x <= x + delta * / 2).
+intro; generalize (H0 x (x + delta * / 2) H10); intro.
+generalize
+ (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H13);
+ rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
+pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
+ left; assumption.
+left; apply Rinv_0_lt_compat; assumption.
+unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse.
+rewrite Ropp_minus_distr.
+reflexivity.
+split.
+unfold Rdiv in |- *; apply prod_neq_R0.
+generalize (cond_pos delta); intro; red in |- *; intro H9; rewrite H9 in H8;
+ elim (Rlt_irrefl 0 H8).
+apply Rinv_neq_0_compat; discrR.
+split.
+unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+ [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
+rewrite Rabs_right.
+unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2.
+prove_sup0.
+rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
+rewrite Rmult_1_l; rewrite double; pattern (pos delta) at 1 in |- *;
+ rewrite <- Rplus_0_r.
+apply Rplus_lt_compat_l; apply (cond_pos delta).
+discrR.
+apply Rle_ge; unfold Rdiv in |- *; left; apply Rmult_lt_0_compat.
+apply (cond_pos delta).
+apply Rinv_0_lt_compat; prove_sup0.
+unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+ [ apply H4 | apply Rinv_0_lt_compat; prove_sup0 ].
+Qed.
+
+(**********)
+Lemma increasing_decreasing_opp :
+ forall f:R -> R, increasing f -> decreasing (- f)%F.
+unfold increasing, decreasing, opp_fct in |- *; intros; generalize (H x y H0);
+ intro; apply Ropp_ge_le_contravar; apply Rle_ge; assumption.
+Qed.
+
+(**********)
+Lemma nonpos_derivative_1 :
+ forall (f:R -> R) (pr:derivable f),
+   (forall x:R, derive_pt f x (pr x) <= 0) -> decreasing f.
+intros.
+cut (forall h:R, - - f h = f h).
+intro.
+generalize (increasing_decreasing_opp (- f)%F).
+unfold decreasing in |- *.
+unfold opp_fct in |- *.
+intros.
+rewrite <- (H0 x); rewrite <- (H0 y).
+apply H1.
+cut (forall x:R, 0 <= derive_pt (- f) x (derivable_opp f pr x)).
+intros.
+replace (fun x:R => - f x) with (- f)%F; [ idtac | reflexivity ].
+apply (nonneg_derivative_1 (- f)%F (derivable_opp f pr) H3).
+intro.
+assert (H3 := derive_pt_opp f x0 (pr x0)).
+cut
+ (derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
+  derive_pt (- f) x0 (derivable_opp f pr x0)).
+intro.
+rewrite <- H4.
+rewrite H3.
+rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; apply (H x0).
+apply pr_nu.
+assumption.
+intro; ring.
+Qed.
+ 
+(**********)
+Lemma positive_derivative :
+ forall (f:R -> R) (pr:derivable f),
+   (forall x:R, 0 < derive_pt f x (pr x)) -> strict_increasing f.
+intros.
+unfold strict_increasing in |- *.
+intros.
+apply Rplus_lt_reg_r with (- f x).
+rewrite Rplus_opp_l; rewrite Rplus_comm.
+assert (H1 := MVT_cor1 f _ _ pr H0).
+elim H1; intros.
+elim H2; intros.
+unfold Rminus in H3.
+rewrite H3.
+apply Rmult_lt_0_compat.
+apply H.
+apply Rplus_lt_reg_r with x.
+rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ].
+Qed.
+
+(**********)
+Lemma strictincreasing_strictdecreasing_opp :
+ forall f:R -> R, strict_increasing f -> strict_decreasing (- f)%F.
+unfold strict_increasing, strict_decreasing, opp_fct in |- *; intros;
+ generalize (H x y H0); intro; apply Ropp_lt_gt_contravar; 
+ assumption.
+Qed.
+ 
+(**********)
+Lemma negative_derivative :
+ forall (f:R -> R) (pr:derivable f),
+   (forall x:R, derive_pt f x (pr x) < 0) -> strict_decreasing f.
+intros.
+cut (forall h:R, - - f h = f h).
+intros.
+generalize (strictincreasing_strictdecreasing_opp (- f)%F).
+unfold strict_decreasing, opp_fct in |- *.
+intros.
+rewrite <- (H0 x).
+rewrite <- (H0 y).
+apply H1; [ idtac | assumption ].
+cut (forall x:R, 0 < derive_pt (- f) x (derivable_opp f pr x)).
+intros; eapply positive_derivative; apply H3.
+intro.
+assert (H3 := derive_pt_opp f x0 (pr x0)).
+cut
+ (derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) =
+  derive_pt (- f) x0 (derivable_opp f pr x0)).
+intro.
+rewrite <- H4; rewrite H3.
+rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; apply (H x0).
+apply pr_nu.
+intro; ring.
+Qed.
+ 
+(**********)
+Lemma null_derivative_0 :
+ forall (f:R -> R) (pr:derivable f),
+   constant f -> forall x:R, derive_pt f x (pr x) = 0. 
+intros.
+unfold constant in H.
+apply derive_pt_eq_0.
+intros; exists (mkposreal 1 Rlt_0_1); simpl in |- *; intros.
+rewrite (H x (x + h)); unfold Rminus in |- *; unfold Rdiv in |- *;
+ rewrite Rplus_opp_r; rewrite Rmult_0_l; rewrite Rplus_opp_r; 
+ rewrite Rabs_R0; assumption.
+Qed.
+
+(**********)
+Lemma increasing_decreasing :
+ forall f:R -> R, increasing f -> decreasing f -> constant f.
+unfold increasing, decreasing, constant in |- *; intros;
+ case (Rtotal_order x y); intro.
+generalize (Rlt_le x y H1); intro;
+ apply (Rle_antisym (f x) (f y) (H x y H2) (H0 x y H2)).
+elim H1; intro.
+rewrite H2; reflexivity.
+generalize (Rlt_le y x H2); intro; symmetry  in |- *;
+ apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)).
+Qed.
+
+(**********)
+Lemma null_derivative_1 :
+ forall (f:R -> R) (pr:derivable f),
+   (forall x:R, derive_pt f x (pr x) = 0) -> constant f.
+intros.
+cut (forall x:R, derive_pt f x (pr x) <= 0).
+cut (forall x:R, 0 <= derive_pt f x (pr x)).
+intros.
+assert (H2 := nonneg_derivative_1 f pr H0).
+assert (H3 := nonpos_derivative_1 f pr H1).
+apply increasing_decreasing; assumption.
+intro; right; symmetry  in |- *; apply (H x).
+intro; right; apply (H x).
+Qed.
+
+(**********)
+Lemma derive_increasing_interv_ax :
+ forall (a b:R) (f:R -> R) (pr:derivable f),
+   a < b ->
+   ((forall t:R, a < t < b -> 0 < derive_pt f t (pr t)) ->
+    forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x < f y) /\
+   ((forall t:R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
+    forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x <= f y).
+intros.
+split; intros.
+apply Rplus_lt_reg_r with (- f x).
+rewrite Rplus_opp_l; rewrite Rplus_comm.
+assert (H4 := MVT_cor1 f _ _ pr H3).
+elim H4; intros.
+elim H5; intros.
+unfold Rminus in H6.
+rewrite H6.
+apply Rmult_lt_0_compat.
+apply H0.
+elim H7; intros.
+split.
+elim H1; intros.
+apply Rle_lt_trans with x; assumption.
+elim H2; intros.
+apply Rlt_le_trans with y; assumption.
+apply Rplus_lt_reg_r with x.
+rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ].
+apply Rplus_le_reg_l with (- f x).
+rewrite Rplus_opp_l; rewrite Rplus_comm.
+assert (H4 := MVT_cor1 f _ _ pr H3).
+elim H4; intros.
+elim H5; intros.
+unfold Rminus in H6.
+rewrite H6.
+apply Rmult_le_pos.
+apply H0.
+elim H7; intros.
+split.
+elim H1; intros.
+apply Rle_lt_trans with x; assumption.
+elim H2; intros.
+apply Rlt_le_trans with y; assumption.
+apply Rplus_le_reg_l with x.
+rewrite Rplus_0_r; replace (x + (y + - x)) with y;
+ [ left; assumption | ring ].
+Qed.
+
+(**********)
+Lemma derive_increasing_interv :
+ forall (a b:R) (f:R -> R) (pr:derivable f),
+   a < b ->
+   (forall t:R, a < t < b -> 0 < derive_pt f t (pr t)) ->
+   forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x < f y.
+intros.
+generalize (derive_increasing_interv_ax a b f pr H); intro.
+elim H4; intros H5 _; apply (H5 H0 x y H1 H2 H3).
+Qed.
+
+(**********)
+Lemma derive_increasing_interv_var :
+ forall (a b:R) (f:R -> R) (pr:derivable f),
+   a < b ->
+   (forall t:R, a < t < b -> 0 <= derive_pt f t (pr t)) ->
+   forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x <= f y.
+intros a b f pr H H0 x y H1 H2 H3;
+ generalize (derive_increasing_interv_ax a b f pr H); 
+ intro; elim H4; intros _ H5; apply (H5 H0 x y H1 H2 H3).
+Qed.
+
+(**********)
+(**********)
+Theorem IAF :
+ forall (f:R -> R) (a b k:R) (pr:derivable f),
+   a <= b ->
+   (forall c:R, a <= c <= b -> derive_pt f c (pr c) <= k) ->
+   f b - f a <= k * (b - a).
+intros.
+case (total_order_T a b); intro.
+elim s; intro.
+assert (H1 := MVT_cor1 f _ _ pr a0).
+elim H1; intros.
+elim H2; intros.
+rewrite H3.
+do 2 rewrite <- (Rmult_comm (b - a)).
+apply Rmult_le_compat_l.
+apply Rplus_le_reg_l with a; rewrite Rplus_0_r.
+replace (a + (b - a)) with b; [ assumption | ring ].
+apply H0.
+elim H4; intros.
+split; left; assumption.
+rewrite b0.
+unfold Rminus in |- *; do 2 rewrite Rplus_opp_r.
+rewrite Rmult_0_r; right; reflexivity.
+elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)).
+Qed.
+
+Lemma IAF_var :
+ forall (f g:R -> R) (a b:R) (pr1:derivable f) (pr2:derivable g),
+   a <= b ->
+   (forall c:R, a <= c <= b -> derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)) ->
+   g b - g a <= f b - f a.
+intros.
+cut (derivable (g - f)).
+intro.
+cut (forall c:R, a <= c <= b -> derive_pt (g - f) c (X c) <= 0).
+intro. 
+assert (H2 := IAF (g - f)%F a b 0 X H H1).
+rewrite Rmult_0_l in H2; unfold minus_fct in H2.
+apply Rplus_le_reg_l with (- f b + f a).
+replace (- f b + f a + (f b - f a)) with 0; [ idtac | ring ].
+replace (- f b + f a + (g b - g a)) with (g b - f b - (g a - f a));
+ [ apply H2 | ring ].
+intros.
+cut
+ (derive_pt (g - f) c (X c) =
+  derive_pt (g - f) c (derivable_pt_minus _ _ _ (pr2 c) (pr1 c))).
+intro.
+rewrite H2.
+rewrite derive_pt_minus.
+apply Rplus_le_reg_l with (derive_pt f c (pr1 c)).
+rewrite Rplus_0_r.
+replace
+ (derive_pt f c (pr1 c) + (derive_pt g c (pr2 c) - derive_pt f c (pr1 c)))
+ with (derive_pt g c (pr2 c)); [ idtac | ring ].
+apply H0; assumption.
+apply pr_nu.
+apply derivable_minus; assumption.
+Qed.
+
+(* If f has a null derivative in ]a,b[ and is continue in [a,b], *)
+(* then f is constant on [a,b] *)
+Lemma null_derivative_loc :
+ forall (f:R -> R) (a b:R) (pr:forall x:R, a < x < b -> derivable_pt f x),
+   (forall x:R, a <= x <= b -> continuity_pt f x) ->
+   (forall (x:R) (P:a < x < b), derive_pt f x (pr x P) = 0) ->
+   constant_D_eq f (fun x:R => a <= x <= b) (f a).
+intros; unfold constant_D_eq in |- *; intros; case (total_order_T a b); intro.
+elim s; intro.
+assert (H2 : forall y:R, a < y < x -> derivable_pt id y).
+intros; apply derivable_pt_id.
+assert (H3 : forall y:R, a <= y <= x -> continuity_pt id y).
+intros; apply derivable_continuous; apply derivable_id.
+assert (H4 : forall y:R, a < y < x -> derivable_pt f y).
+intros; apply pr; elim H4; intros; split.
+assumption.
+elim H1; intros; apply Rlt_le_trans with x; assumption.
+assert (H5 : forall y:R, a <= y <= x -> continuity_pt f y).
+intros; apply H; elim H5; intros; split.
+assumption.
+elim H1; intros; apply Rle_trans with x; assumption.
+elim H1; clear H1; intros; elim H1; clear H1; intro.
+assert (H7 := MVT f id a x H4 H2 H1 H5 H3).
+elim H7; intros; elim H8; intros; assert (H10 : a < x0 < b).
+elim x1; intros; split.
+assumption.
+apply Rlt_le_trans with x; assumption.
+assert (H11 : derive_pt f x0 (H4 x0 x1) = 0).
+replace (derive_pt f x0 (H4 x0 x1)) with (derive_pt f x0 (pr x0 H10));
+ [ apply H0 | apply pr_nu ].
+assert (H12 : derive_pt id x0 (H2 x0 x1) = 1).
+apply derive_pt_eq_0; apply derivable_pt_lim_id.
+rewrite H11 in H9; rewrite H12 in H9; rewrite Rmult_0_r in H9;
+ rewrite Rmult_1_r in H9; apply Rminus_diag_uniq; symmetry  in |- *;
+ assumption.
+rewrite H1; reflexivity.
+assert (H2 : x = a).
+rewrite <- b0 in H1; elim H1; intros; apply Rle_antisym; assumption.
+rewrite H2; reflexivity.
+elim H1; intros;
+ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H2 H3) r)).
+Qed.
+
+(* Unicity of the antiderivative *)
+Lemma antiderivative_Ucte :
+ forall (f g1 g2:R -> R) (a b:R),
+   antiderivative f g1 a b ->
+   antiderivative f g2 a b ->
+    exists c : R, (forall x:R, a <= x <= b -> g1 x = g2 x + c).
+unfold antiderivative in |- *; intros; elim H; clear H; intros; elim H0;
+ clear H0; intros H0 _; exists (g1 a - g2 a); intros;
+ assert (H3 : forall x:R, a <= x <= b -> derivable_pt g1 x).
+intros; unfold derivable_pt in |- *; apply existT with (f x0); elim (H x0 H3);
+ intros; eapply derive_pt_eq_1; symmetry  in |- *; 
+ apply H4.
+assert (H4 : forall x:R, a <= x <= b -> derivable_pt g2 x).
+intros; unfold derivable_pt in |- *; apply existT with (f x0);
+ elim (H0 x0 H4); intros; eapply derive_pt_eq_1; symmetry  in |- *; 
+ apply H5.
+assert (H5 : forall x:R, a < x < b -> derivable_pt (g1 - g2) x).
+intros; elim H5; intros; apply derivable_pt_minus;
+ [ apply H3; split; left; assumption | apply H4; split; left; assumption ].
+assert (H6 : forall x:R, a <= x <= b -> continuity_pt (g1 - g2) x).
+intros; apply derivable_continuous_pt; apply derivable_pt_minus;
+ [ apply H3 | apply H4 ]; assumption.
+assert (H7 : forall (x:R) (P:a < x < b), derive_pt (g1 - g2) x (H5 x P) = 0).
+intros; elim P; intros; apply derive_pt_eq_0; replace 0 with (f x0 - f x0);
+ [ idtac | ring ].
+assert (H9 : a <= x0 <= b).
+split; left; assumption.
+apply derivable_pt_lim_minus; [ elim (H _ H9) | elim (H0 _ H9) ]; intros;
+ eapply derive_pt_eq_1; symmetry  in |- *; apply H10.
+assert (H8 := null_derivative_loc (g1 - g2)%F a b H5 H6 H7);
+ unfold constant_D_eq in H8; assert (H9 := H8 _ H2); 
+ unfold minus_fct in H9; rewrite <- H9; ring.
+Qed.