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_add :
forall (m n : nat) (f g : nat -> nat),
sum_nm n m f + sum_nm n m g = sum_nm n m (fun i : nat => f i + g i).