intros m H'0; rewrite <- minus_n_n; elim H'0.
rewrite <- minus_Sn_m; auto with arith.
intros m0 H'1 H'2; rewrite <- minus_Sn_m; auto with arith.
intros m H'0 H'1 m0 H'2; repeat rewrite <- minus_Sn_m; auto with arith.
apply le_trans with n; auto with arith.
Qed.

Theorem lt_mult_anti_compatibility :
forall n n1 n2 : nat, 0 < n -> n * n1 < n * n2 -> n1 < n2.