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_plus : forall (f g : R -> V) (a b : R), ex_RInt f a b -> ex_RInt g a b -> ex_RInt (fun y => plus (f y) (g y)) a b.