Proof with eTarski.
intros.
cases_equality B C.
subst B...
eapply outer_transitivity_between2.
3:apply H1.
eapply between_inner_transitivity...
assumption.
Qed.
(** Axiom 16 of Givant. *)

Lemma outer_transitivity_between : forall A B C D,
Bet A B C -> Bet B C D -> B<>C -> Bet A B D.