Definition SP (fn:nat -> R -> R) (N:nat) (x:R) : R :=
sum_f_R0 (fun k:nat => fn k x) N.

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Lemma sum_incr :
forall (An:nat -> R) (N:nat) (l:R),
Un_cv (fun n:nat => sum_f_R0 An n) l ->
(forall n:nat, 0 <= An n) -> sum_f_R0 An N <= l.