Proof with Tarski.
intros.
unfold Col in *;intuition;assert (P<>Q);auto;clear H;intuition;Between.
(* *)
assert (Bet Q A B \/ Bet Q B A).
apply (l5_2 P Q A B);intuition;Between.
intuition;Between.
(* *)
assert (Bet Q A B \/ Bet Q B A).
apply (l5_3 Q A B P);intuition;Between.
intuition;Between.
(* *)
assert (Bet Q A B \/ Bet Q B A).
apply (l5_1 Q P A B);intuition;Between.
intuition;Between.
Qed.

Theorem l6_21 : forall A B C D P Q,
~ Col  A B C -> C<>D ->
Col A B P -> Col A B Q ->
Col C D P -> Col C D Q ->
P=Q.