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

Lemma between_egality : forall A B C : Point, Bet A B C -> Bet B A C -> A = B.
Proof with Tarski.