Definition Riemann_fine (a b : R) :=
  within (fun ptd => pointed_subdiv ptd /\ SF_h ptd = Rmin a b /\ last (SF_h ptd) (SF_lx ptd) = Rmax a b)
    (locally_dist (fun ptd => seq_step (SF_lx ptd))).Global Instance Riemann_fine_filter :
  forall a b, ProperFilter (Riemann_fine a b).Definition Riemann_sum (f : R -> V) (ptd : SF_seq) : V :=
  foldr plus zero (pairmap (fun x y => (scal (fst y - fst x) (f (snd y)))) (SF_h ptd,zero) (SF_t ptd)).

Lemma Riemann_sum_norm (f : R -> V) (g : R -> R) ptd : pointed_subdiv ptd -> (forall t, SF_h ptd <= t <= last (SF_h ptd) (SF_lx ptd) -> norm (f t) <= g t) -> norm (Riemann_sum f ptd) <= Riemann_sum g ptd.