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 filterlim_RInt {U} {V : CompleteNormedModule R_AbsRing} : forall (f : U -> R -> V) (a b : R) F (FF : ProperFilter F) g h, (forall x, is_RInt (f x) a b (h x)) -> (filterlim f F (locally g)) -> exists If, filterlim h F (locally If) /\ is_RInt g a b If.