simplify.rewrite < sym_plus.
apply le_plus_n.
elim le_to_or_lt_eq O n2.
-assumption.apply le_O_n.
+assumption.
absurd O<m.assumption.
rewrite > H2.rewrite < H3.rewrite < times_n_O.
apply not_le_Sn_n O.
+apply le_O_n.
qed.
theorem divides_to_lt_O : \forall n,m. O < m \to divides n m \to O < n.