End Riemann_sum.
Section Riemann_sum_Normed.
Context {V : NormedModule R_AbsRing}.

(* Specifications: *)
Lemma Riemann_sum_Chasles_0
(f : R -> V) (M : R) (x : R) ptd :
forall (eps : posreal),
(forall x, SF_h ptd <= x <= last (SF_h ptd) (SF_lx ptd) -> norm (f x) < M) ->
SF_h ptd <= x <= last (SF_h ptd) (SF_lx ptd) ->
pointed_subdiv ptd ->
seq_step (SF_lx ptd) < eps ->
norm (minus (plus (Riemann_sum f (SF_cut_down ptd x)) (Riemann_sum f (SF_cut_up ptd x)))
(Riemann_sum f ptd)) < 2 * eps * M.
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.