Proof with eTarski.
intros.
eapply between_symmetry.
eapply between_exchange2;apply between_symmetry.
2:apply H.
eapply outer_transitivity_between2...
Qed.

(* Specifications: *)
Lemma between_exchange4 : forall A B C D,
Bet A B C -> Bet A C D -> Bet A B D.