Proof with Geometry.
unfold parallel, S4 in |- *.
intros.
uniformize_signed_areas.
RewriteVar (S A C B) H.
ring.
Qed.
Hint Resolve par_1: Geom.

(* Specifications: *)
Theorem par_2 : forall A B C D : Point,
parallel A B C D -> parallel C D A B.