Proof with Geometry.
intros.
replace (S4 P D A C) with (S P D C - S A D C)...
apply l2_12a_strong_3...
Qed.

Theorem pascalian_ax :
forall A B C P Q R : Point,
Col A B C -> Col P Q R ->
parallel A Q R B -> parallel B P Q C ->
parallel A P R C.