+theorem mod_times_mod : \forall n,m,p:nat. O<p \to O<m \to n \mod p = (n \mod (m*p)) \mod p.
+intros.
+apply (div_mod_spec_to_eq2 n p (n/p) (n \mod p)
+(n/(m*p)*m + (n \mod (m*p)/p))).
+apply div_mod_spec_div_mod.assumption.
+constructor 1.
+apply lt_mod_m_m.assumption.
+rewrite > times_plus_l.
+rewrite > assoc_plus.
+rewrite < div_mod.
+rewrite > assoc_times.
+rewrite < div_mod.
+reflexivity.
+rewrite > (times_n_O O).
+apply lt_times.
+assumption.assumption.assumption.
+qed.
+