End Comp.

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')).