Lemma derive_increasing_interv : forall (a b : R) (f : R -> R), a < b -> forall (pr:forall x, a < x < b -> derivable_pt f x), (forall t:R, forall (t_encad : a < t < b), 0 < derive_pt f t (pr t t_encad)) -> forall x y:R, a < x < b -> a < y < b -> x < y -> f x < f y.