(* RsumOver *)

Definition RsumOver (X: Set) (l: list X) (f: X -> R): R := Rsum (map f l).

Lemma RsumOver_nats_le x y z: (y <= z)%nat -> forall f, (forall q, (x <= q)%nat -> 0 <= f q) -> RsumOver (nats x y) f <= RsumOver (nats x z) f.