Lemma f_incr_implies_g_incr_interv : forall f g:R->R, forall lb ub, lb < ub -> (forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) -> (forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x) -> (forall x , f lb <= x -> x <= f ub -> lb <= g x <= ub) -> (forall x y, f lb <= x -> x < y -> y <= f ub -> g x < g y).