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

Lemma l3_9 : forall P Q R S A B C D,
parallel P R A C ->
parallel Q S B D ->
B<>D -> A<>C -> ~ perp A C B D ->
Py4 P Q R S / Py4 A B C D = (P**R / A**C) * (Q**S / B**D).