Lemma is_derive_RInt_bound_comp (f : R -> V) (If : R -> R -> V) (a b : R -> R) (da db x : R) : locally (a x, b x) (fun u : R * R => is_RInt f (fst u) (snd u) (If (fst u) (snd u))) -> continuous f (a x) -> continuous f (b x) -> is_derive a x da -> is_derive b x db -> is_derive (fun x => If (a x) (b x)) x (minus (scal db (f (b x))) (scal da (f (a x)))).