Section RInt_comp.
Context {V : NormedModule R_AbsRing}.

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)))).