Proof with Geometry.
unfold on_inter_parallel_parallel.
intros.
DecompAndAll.
assert (P<>Q).
unfold not;intro;subst P...
eapply elim_length_ratio_on_parallel_d_2...
apply on_parallel_to_on_parallel_d...
unfold on_parallel;repeat split...
unfold not;intro.
assert (U<>V).
unfold not;intro;subst U...
assert (A<>Y).
unfold not;intro;subst A...
assert (parallel P Q Y A).
eapply col_par_par.
apply H1.
Geometry.
Geometry.
intuition.
Qed.

(* Specifications: *)
Lemma elim_length_ratio_on_inter_parallel_parallel_2 :
forall A C D P Q R U V W Y : Point,
on_inter_parallel_parallel Y R P Q W U V ->
parallel A Y P Q ->
Y <> A ->
C <> D ->
Y <> W ->
parallel A Y C D ->
A ** Y / C ** D = S4 A U W V / S4 C U D V.