+theorem lt_times_r1:
+\forall n,m,p. O < n \to m < p \to n*m < n*p.
+intros.
+elim H;apply lt_times_r;assumption.
+qed.
+
+theorem lt_times_l1:
+\forall n,m,p. O < n \to m < p \to m*n < p*n.
+intros.
+elim H;apply lt_times_l;assumption.
+qed.
+
+theorem lt_to_le_to_lt_times :
+\forall n,n1,m,m1. n < n1 \to m \le m1 \to O < m1 \to n*m < n1*m1.
+intros.
+apply (le_to_lt_to_lt ? (n*m1))
+ [apply le_times_r.assumption
+ |apply lt_times_l1
+ [assumption|assumption]
+ ]
+qed.
+