Ltac double A B A' :=
   assert (mp:= symmetric_point_construction A B);
   elim mp; intros A' ; intro; clear mp.

Lemma col_per2_cases : forall A B C D B', B <> C -> B' <> C -> C <> D -> Col B C D -> Per A B C -> Per A B' C -> B = B' \/ ~Col B' C D.