+
+theorem le_times_to_le:
+\forall a,n,m. S O \le a \to a * n \le a * m \to n \le m.
+intro.
+apply nat_elim2;intros
+ [apply le_O_n
+ |apply False_ind.
+ rewrite < times_n_O in H1.
+ generalize in match H1.
+ apply (lt_O_n_elim ? H).
+ intros.
+ simplify in H2.
+ apply (le_to_not_lt ? ? H2).
+ apply lt_O_S
+ |apply le_S_S.
+ apply H
+ [assumption
+ |rewrite < times_n_Sm in H2.
+ rewrite < times_n_Sm in H2.
+ apply (le_plus_to_le a).
+ assumption
+ ]
+ ]
+qed.
\ No newline at end of file