intros m n f; generalize m; clear m; elim n; simpl in |- *; intros.
rewrite (plus_comm m); auto.
rewrite H; repeat (rewrite (plus_comm m); simpl in |- *); auto with arith.
Qed.

Lemma sum_nm_times :
forall (m n x : nat) (f : nat -> nat),
x * sum_nm n m f = sum_nm n m (fun i : nat => x * f i).