repeat rewrite <- N2Nat.inj_mul; apply lt_Nlt; repeat rewrite N2Nat.id; auto.
Qed.

(* Specifications: *)
Theorem Nmult_lt_compat_rev_l2: forall n m p, p * n < p * m -> n < m.
intros n m p H1; apply Nlt_lt; apply mult_lt_compat_rev_l2 with (nat_of_N p).
Theorem Nmult_gt_compat_rev_l1: forall n m p, p * n > p * m -> p > 0.
intros n m p H1; apply Ngt_gt; apply mult_gt_compat_rev_l1 with (nat_of_N n) (nat_of_N m).