+apply divides_to_div.
+assumption.
+qed.
+
+theorem div_div: \forall n,d:nat. O < n \to divides d n \to
+n/(n/d) = d.
+intros.
+apply (inj_times_l1 (n/d))
+ [apply (lt_times_n_to_lt d)
+ [apply (divides_to_lt_O ? ? H H1).
+ |rewrite > divides_to_div;assumption
+ ]
+ |rewrite > divides_to_div
+ [rewrite > sym_times.
+ rewrite > divides_to_div
+ [reflexivity
+ |assumption
+ ]
+ |apply (witness ? ? d).
+ apply sym_eq.
+ apply divides_to_div.
+ assumption
+ ]
+ ]