Proof with intuition.
intros.
assert  {Y : Point | Col Y P Q /\ Col Y U V}.
apply inter_llex...
DecompExAnd H0 Y.
exists Y.
unfold inter_ll...
Qed.

(* Specifications: *)
Theorem on_parallelex : forall P Q R : Point,
Q<>R ->
~ Col P Q R ->
exists Y, parallel Q R P Y.
Theorem on_parallel_ex : forall P Q R : Point,
Q<>R ->
~ Col P Q R ->
exists Y, on_parallel Y P Q R.
Lemma  on_parallel_d_ex :
forall (P Q R : Point) (lambda : F),
P<>Q ->
~ Col P Q R ->
exists Y, on_parallel_d Y R P Q lambda.