Ltac cases_per A B C := elim (per_dec A B C);intros.

Lemma l3_9_aux : forall B D P Q R S Y ,
Col Y B D ->
B<>D ->
B ** Y = Q ** S ->
weak_3_parallelogram B Y S Q ->
Py4 P Q R S = Q ** S / B ** D * Py4 P B R D.