Proof with Geometry.
intros.
assert (P<>Q).
unfold not;intro.
assert (parallel P Q U V).
subst P.
Geometry.
intuition.
assert (U<>V).
unfold not;intro.
assert (parallel P Q U V).
subst U.
Geometry.
intuition.
assert ({R':Point | (parallel P Q R R') /\ R<>R'}).
apply euclid_parallel_existence_strong...
DecompExAnd H2 R'.
assert ({W':Point | (parallel U V W W') /\ W<>W'}).
apply euclid_parallel_existence_strong...
DecompExAnd H2 W'.
assert {Y : Point | Col Y R R' /\ Col Y W W'}.
apply inter_llex...
unfold not;intro.
assert (parallel R R' U V).
eapply parallel_transitivity.
apply H7.
Geometry.
Geometry.
assert (parallel P Q U V).
eapply parallel_transitivity;apply H5 || Geometry .
intuition.
DecompExAnd H2 Y.
exists Y.
split.
cut (parallel P Q R Y).
Geometry.
eapply col_par_par.
apply H5.
Geometry.
Geometry.
cut (parallel U V W Y).
Geometry.
eapply col_par_par.
apply H7.
Geometry.
Geometry.
Qed.

Lemma on_inter_parallel_parallel_ex : forall P Q R U V W:Point,
~ parallel P Q U V -> ~ Col R U V ->
{Y :Point | (on_inter_parallel_parallel Y R P Q W U V)}.