Proof with eTarski.
intros.
cases_equality A B.
assert (A' = B').
assert (Cong A' B' A B).
sTarski.
eapply cong_identity.
rewrite H3 in H4.
apply H4.
rewrite <- H3 in H2.
rewrite <- H4 in H2.
assumption.
assert (Cong C A C' A').
eapply five_segments; try apply H1...
eTarski.
Qed.
(** Unicity of segment construction. *)

(* Specifications: *)
Lemma construction_unicity : forall Q A B C x y,
~(Q=A) ->
Bet Q A x -> Cong A x B C ->
Bet Q A y -> Cong A y B C ->
x=y.