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).