Proof with Geometry.
unfold on_inter_line_parallel.
intros.
DecompAndAll.
assert (P<>Q).
unfold not;intro;subst P...
assert (~ Col A R Y).
unfold not;intro.
assert (U<>V).
unfold not;intro;subst U...
assert (Col U A Y).
eapply col_trans_1 with (B:=V)...
assert (Col Y R U).
eapply col_trans_1 with (B:=A)...
assert (parallel P Q Y U).
eapply col_par_par.
apply H3.
Geometry.
Geometry.
assert (U <> Y).
unfold not;intro.
subst Y.
clear H13 H12 H11 H7.
assert (parallel P Q U A).
eapply col_par_par.
apply H3.
Geometry.
Geometry.
assert (parallel P Q U V).
eapply col_par_par.
apply H2.
Geometry.
Geometry.
intuition.
assert (parallel P Q U V).
eapply col_par_par.
apply H14.
Geometry.
Geometry.
intuition.
eapply elim_length_ratio_on_parallel_d_2...
apply on_parallel_to_on_parallel_d.
unfold on_parallel;repeat split...
Qed.

(* Specifications: *)
Lemma elim_length_ratio_on_inter_parallel_parallel_1 :
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 <> R ->
C <> D -> parallel A Y C D -> A ** Y / C ** D = S4 A P R Q / S4 C P D Q.