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 RInt_fct_extend_pair (U_RInt : (R -> U) -> R -> R -> U) (V_RInt : (R -> V) -> R -> R -> V) : (forall f a b l, is_RInt f a b l -> U_RInt f a b = l) -> (forall f a b l, is_RInt f a b l -> V_RInt f a b = l) -> forall f a b l, is_RInt f a b l -> (U_RInt (fun t => fst (f t)) a b, V_RInt (fun t => snd (f t)) a b) = l.