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_ext :
forall (m n : nat) (f g : nat -> nat),
(forall x : nat, x <= n -> f (m + x) = g (m + x)) ->
sum_nm n m f = sum_nm n m g.