intros f Hmon x y Hlt; case (le_gt_dec y x); auto.
intros Hle; elim (le_lt_or_eq _ _ Hle).
intros Hlt'; elim (lt_asym _ _ Hlt); apply Hmon; auto.
intros Heq; elim (lt_neq _ _ Hlt); rewrite Heq; auto.
Qed.

Theorem mult_lt : forall a b c : nat, c <> 0 -> a < b -> a * c < b * c.
intros a b c; elim c.