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