Proof with eTarski.
intros.
assert (exists x,Bet B x B /\ Bet C x A)...
DecompExAnd H1 x.
apply between_symmetry.
assert (B=x)...
congruence.
Qed.

Lemma between_exchange3 : forall A B C D,
Bet A B C -> Bet A C D -> Bet B C D.