End Comp.

(* Specifications: *)
Lemma RInt_Chasles_bound_comp_l_loc (f : R -> R -> R) (a : R -> R) (b x : R) :
locally x (fun y => ex_RInt (f y) (a x) b) ->
(exists eps : posreal, locally x (fun y => ex_RInt (f y) (a x - eps) (a x + eps))) ->
continuous a x ->
locally x (fun x' => RInt (f x') (a x') (a x) + RInt (f x') (a x) b =
RInt (f x') (a x') b).
Lemma RInt_Chasles_bound_comp_loc (f : R -> R -> R) (a b : R -> R) (x : R) :
locally x (fun y => ex_RInt (f y) (a x) (b x)) ->
(exists eps : posreal, locally x (fun y => ex_RInt (f y) (a x - eps) (a x + eps))) ->
(exists eps : posreal, locally x (fun y => ex_RInt (f y) (b x - eps) (b x + eps))) ->
continuous a x ->
continuous b x ->
locally x (fun x' => RInt (f x') (a x') (a x) + RInt (f x') (a x) (b x') =
RInt (f x') (a x') (b x')).
Lemma Derive_comp (f g : R -> R) (x : R) :
ex_derive f (g x) -> ex_derive g x
-> Derive (fun x => f (g x)) x = Derive g x * Derive f (g x).