Proof with auto.
intros.
assert (on_parallel_d Y R P Q (R ** Y / P ** Q)).
apply on_parallel_to_on_parallel_d...
apply elim_area_on_parallel_d...
Qed.
(** We prove parallel transitivity using the elimination lemmas *)

Lemma elim_area_on_inter_line_parallel :
forall P Q R U V Y A B : Point,
on_inter_line_parallel Y R U V P Q ->
R <>Y ->
S A B Y = (S4 P U Q R * S A B V - S4 P V Q R * S A B U) / S4 P U Q V.