Proof with Tarski.
intros.
assert (exists x, exists y:Point, x <> y)...
DecompEx H U.
DecompEx H0 V.
assert (exists C, Bet A B C /\ Cong B C U V)...
DecompEx H0 C.
exists C.
intuition.
rewrite H1 in H2.
assert (U=V).
eapply cong_identity.
apply cong_symmetry.
eauto.
auto.
Qed.
Hint Resolve point_construction_different : Tarski.
(** Given a point, we can build another one which is different from the first one. *)

Lemma another_point : forall A:Point, exists B, A<>B.