apply (mkposreal ((ub-lb)/2) (ub_lt_2_pos x ub lb lb_lt_x x_lt_ub)).
Defined.
(* end hide *)

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).