Proof with Tarski.
unfold is_midpoint.
intros.
intuition.
cases_equality A P.
treat_equalities...
eapply construction_unicity; try apply H0...
esTarski.
Qed.

(* Specifications: *)
Lemma l7_9 : forall P Q A X,
is_midpoint A P X -> is_midpoint A Q X -> P=Q.