Proof with auto with real.
intros.
rewrite map_app.
rewrite Rsum_app.
rewrite Rsum_map_plus.
do 2 rewrite map_map.
reflexivity.
Qed.
Instance elemsR_le_Rsum: Proper (elemsR Rle ==> Rle) Rsum.
Proof with auto with real.
repeat intro.
induction H...
simpl...
Qed.

Lemma Rsum_map_le X f g (l: list X): (forall x, In x l -> f x <= g x) -> Rsum (map f l) <= Rsum (map g l).