apply lt_le_trans with (1 * n).
auto with arith.
simpl in |- *.
elim plus_n_O.
apply le_mult_r; auto with arith.
Qed.

Lemma plus_m_mult_n_m : forall n m : nat, m + n * m = S n * m.
simple induction n; simple induction m; simpl in |- *; auto with arith.