Proof with Tarski.
unfold is_midpoint.
intros.
assert (exists E : Point, Bet A P E /\ Cong P E A P)...
DecompExAnd H E.
exists E.
intuition.
Qed.
Hint Resolve symmetric_point_construction : Midpoint.

Lemma symmetric_point_unicity : forall A P P1 P2,
is_midpoint P A P1 -> is_midpoint P A P2 -> P1=P2.