Proof with Tarski.
intros.
unfold Col in *;intuition;assert (P<>Q);auto;clear H;intuition;Between.
(* *)
assert (P<>Q);auto;clear H1.
assert (Bet P A B \/ Bet P B A).
apply (l5_1 P Q A B)...
intuition.
(* *)
assert (Bet P A B \/ Bet P B A).
apply (l5_3 P A B Q);intuition;Between.
intuition;Between.
(* *)
assert (Bet P A B \/ Bet P B A).
apply (l5_2 Q P A B);intuition;Between.
intuition;Between.
(* *)
Qed.

Lemma col_transitivity_2 : forall P Q A B,
P<>Q -> Col P Q A -> Col P Q B -> Col Q A B.