Proof with Tarski.
intros.
unfold Col in *.
elim H1;intro;elim H2;intro.
right.
assert (Bet P Q X \/ Bet P X Q).
apply (l5_2 S P Q X);Between;intuition;Between...
intuition;Between...
elim H4;intro.
left;Between.
left;Between.
elim H3;intro.
left;Between.
left;Between.
elim H3;intro.
elim H4;intro.
right;left;Between.
right.
assert (Bet P Q X \/ Bet P X Q).
apply (l5_3 P Q X S);Between.
intuition;Between.
elim H4;intro.
right.
assert (Bet P Q X \/ Bet P X Q).
apply (l5_1 P S Q X);Between.
intuition;Between.
right;right;Between.
Qed.
Definition Inter := fun X P Q R S =>
P<>Q /\ R <> S /\ Col X P Q /\ Col X R S /\
((exists X, Col X P Q /\ ~ Col X R S) \/ (exists X, ~ Col X P Q /\ Col X R S)).

Lemma col_transitivity_1 : forall P Q A B,
P<>Q -> Col P Q A -> Col P Q B -> Col P A B.