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 parallel_transitivity : forall A B C D E F,
C<> D ->
parallel A B C D ->
parallel C D E F ->
parallel A B E F.