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).