apply le_trans with (a * d).
apply le_mult_cst; assumption.
elim mult_comm; elim mult_comm with d b.
apply le_mult_cst; assumption.
Qed.
Hint Resolve le_mult_csts.

Lemma lt_mult_n_Sn : forall n m : nat, 0 < m -> n * m < S n * m.
simple induction n; simple induction m; auto with arith.