Proof with Geometry.
intros.
unfold on_parallel_d in H.
DecompAndAll.
cases_equality R Y.
subst R;basic_simpl.
IsoleVar lambda H3...
assert (lambda = 0).
rewrite H3.
field...
rewrite H1.
ring.
assert (parallel R Y P Q)...
assert (Th := on_line_dex_spec_strong_f R Y P Q H1 H)...
DecompExAnd Th T.
assert (R <> T).
unfold not;intro;subst R;basic_simpl...
assert (S A B Y = lambda * S A B T + (1 - lambda) * S A B R).
apply elim_area_on_line_d.
rewrite <- H7 in H3.
unfold on_line_d in *...
assert (S R A B + S Q A B = S T A B + S P A B).
apply l2_11a_strong_strong_strong...
IsoleVar (S T A B) H9.
uniformize_signed_areas.
rewrite H9 in H6.
replace (S4 A P B Q) with (S Q A B - S P A B).
rewrite H6;ring.
unfold S4 in |- *;uniformize_signed_areas;ring.
Qed.

Theorem elim_area_on_parallel :
forall A B P Q R Y : Point,
on_parallel Y R P Q ->
S A B Y = S A B R + R ** Y / P ** Q * S4 A P B Q.