Proof with Tarski.
unfold out.
intros.
decompose [and] H1;clear H1.
decompose [and] H3;clear H3.
assert (Cong A X A Y).
eapply cong_transitivity with (C:=B) (D:=C);sTarski.
(* *)
intuition.
(* *)
eapply l4_19;try apply H6;auto.
eapply l4_3.
apply between_symmetry.
apply H6.
apply between_symmetry.
apply H8.
Tarski.
esTarski.
(* *)
assert (Bet A X Y);Between.
eapply between_cong.
apply H10.
assumption.
(* *)
assert (Bet A Y X);Between.
symmetry.
eapply between_cong.
apply H10.
sTarski.
(* *)
assert (Bet A X Y \/ Bet A Y X).
eapply l5_1.
2:apply H6.
auto.
auto.
elim H10;intro.
(* *)
eapply between_cong;eauto.
symmetry;eapply between_cong;eauto;sTarski.
Qed.

Lemma l6_11_existence : forall A B C R,
R<>A -> B<>C ->
exists X, out A X R /\ Cong A X B C.