Definition is_RInt (f : R -> V) (a b : R) (If : V) :=
  filterlim (fun ptd => scal (sign (b-a)) (Riemann_sum f ptd)) (Riemann_fine a b) (locally If).Definition ex_RInt (f : R -> V) (a b : R) :=
  exists If : V, is_RInt f a b If.

Lemma ex_RInt_inside {V : CompleteNormedModule R_AbsRing} : forall (f : R -> V) (a b x e : R), ex_RInt f (x-e) (x+e) -> Rabs (a-x) <= e -> Rabs (b-x) <= e -> ex_RInt f a b.