Proof with auto with real.
unfold RsumOver.
intros.
induction l...
simpl.
apply Rplus_le_compat.
apply H.
left...
apply IHl.
intros.
apply H.
right...
Qed.

(* Specifications: *)
Lemma RsumOver_mult_constant (X: Set) (f: X -> R) c (l: list X):
c * RsumOver l f = RsumOver l (Rmult c ∘ f).