(* Julien Narboux *)
(* Formalization of *)
(* Tarski's axiomatic *)
Require Export Ch05_between_transitivity_le.
(** The predicate out P A B informally means that P is on the line A B outside the segment A B. *)
Definition out := fun P A B => A<>P /\ B<>P /\ (Bet P A B \/ Bet P B A).
Proof with Tarski.
intros.
unfold iff;split;intro.
unfold out;repeat split;auto.
eapply l5_2;eauto;sTarski.
unfold out in H3.
decompose [and] H3.
elim H7;intro;Between.
Qed.

Lemma l6_3_1 : forall A B P,
out P A B -> (A<>P /\ B<>P /\ exists C, C<>P /\ Bet A P C /\ Bet B P C).