+
+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.
+
+theorem le_S_times_SSO: \forall n,m.O < m \to
+n \le m \to S n \le (S(S O))*m.
+intros.
+simplify.
+rewrite > plus_n_O.
+simplify.rewrite > plus_n_Sm.
+apply le_plus
+ [assumption
+ |rewrite < plus_n_O.
+ assumption
+ ]
+qed.
+(*0 and times *)
+theorem O_lt_const_to_le_times_const: \forall a,c:nat.
+O \lt c \to a \le a*c.
+intros.
+rewrite > (times_n_SO a) in \vdash (? % ?).
+apply le_times
+[ apply le_n
+| assumption
+]
+qed.