End Riemann_sum_Normed.

(* Specifications: *)
Lemma Riemann_sum_le (f : R -> R) (g : R -> R) ptd :
pointed_subdiv ptd ->
(forall t, SF_h ptd <= t <= last (SF_h ptd) (SF_lx ptd) -> f t <= g t) ->
Riemann_sum f ptd <= Riemann_sum g ptd.