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 is_RInt_fct_extend_pair (f : R -> U * V) (a b : R) lu lv : is_RInt (fun t => fst (f t)) a b lu -> is_RInt (fun t => snd (f t)) a b lv -> is_RInt f a b (lu,lv).