+lemma divides_to_prime_divides : \forall n,m.1 < m \to m < n \to m \divides n \to
+ \exists p.p \leq m \land prime p \land p \divides n.
+intros;apply (ex_intro ? ? (nth_prime (max_prime_factor m)));split
+ [split
+ [apply divides_to_le
+ [apply lt_to_le;assumption
+ |apply divides_max_prime_factor_n;assumption]
+ |apply prime_nth_prime;]
+ |apply (transitive_divides ? ? ? ? H2);apply divides_max_prime_factor_n;
+ assumption]
+qed.
+