Proof with Tarski.
intros.
unfold out.
assert (exists X : Point, (Bet A R X \/ Bet A X R) /\ Cong A X B C).
eapply (segment_construction_2 R A).
auto.
DecompExAnd H1 X.
exists X.
intuition;treat_equalities...
Qed.

(* Specifications: *)
Lemma l6_13_1 : forall P A B,
out P A B -> le P A P B -> Bet P A B.