Definition mkposreal_lb_ub (x lb ub:R) (lb_lt_x:lb<x) (x_lt_ub:x<ub) : posreal.

Lemma derivable_pt_lim_CVU (fn fn':nat -> R -> R) (f g:R->R) (x:R) : forall c r, Boule c r x -> (forall y n, Boule c r y -> derivable_pt_lim (fn n) y (fn' n y)) -> (forall y, Boule c r y -> Un_cv (fun n => fn n y) (f y)) -> (CVU fn' g c r) -> (forall y, Boule c r y -> continuity_pt g y) -> derivable_pt_lim f x (g x).