+
+theorem lt_exp_to_lt1:
+\forall a,n,m. O < a \to exp n a < exp m a \to n < m.
+intros.
+elim (le_to_or_lt_eq n m)
+ [assumption
+ |apply False_ind.
+ apply (lt_to_not_eq ? ? H1).
+ rewrite < H2.
+ reflexivity
+ |apply (le_exp_to_le1 ? ? a)
+ [assumption
+ |apply lt_to_le.
+ assumption
+ ]
+ ]
+qed.