proposition: 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).
proof: Proof. 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.
imports: Rbase Rfunctions Ranalysis1 Rtopology Lra