Proof with Geometry.
intros.
unfold on_inter_parallel_parallel in *.
DecompAndAll.
assert ((S A B Y) = (S A B W) + W**Y/U**V * (S4 A U B V)).
apply elim_area_on_parallel.
unfold on_parallel.
repeat split...
unfold not;intro;subst U...
assert (P<>Q).
unfold not;intro;subst P...
assert (U<>V).
unfold not;intro;subst U...
cases_equality Y W.
subst Y.
assert (parallel P Q W R)...
unfold parallel in H7.
rewrite H7.
field...
assert (~ Col W R Y).
unfold not;intro.
assert (parallel U V Y R).
eapply col_par_par.
apply H7.
Geometry.
Geometry.
assert (parallel U V P Q).
eapply parallel_transitivity with (C:=Y) (D:=R)...
intuition.
assert (W**Y/U**V = (S4 W P R Q) / (S4 U P V Q)).
eapply elim_length_ratio_on_parallel_d_2...
unfold on_parallel_d;repeat split...
assert (R**Y = R**Y / P**Q * P**Q).
field...
eauto.
rewrite H.
rewrite H9.
replace (S4 P U Q V) with (- S4 U P V Q)...
replace (S4 W P R Q) with (- S4 P W Q R)...
field;split...
Qed.

Lemma elim_area_on_inter_parallel_parallel_RY :
forall P Q R U V W Y A B : Point,
on_inter_parallel_parallel Y R P Q W U V ->
R = Y ->
S A B Y = S A B W +
(W ** R / P ** Q + R ** Y / P ** Q) / (U ** V / P ** Q) * S4 A U B V.