(** 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).

(* Specifications: *)
Lemma l6_2 : forall A B C P, A<>P -> B<>P -> C<>P -> Bet A P C -> (Bet B P C <-> out P A B).
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).
Lemma l6_3_2 : forall A B P, (A<>P /\ B<>P /\ exists C, C<>P /\ Bet A P C /\ Bet B P C) -> out P A B.
Lemma l6_4_1 : forall A B P, out P A B -> Col A P B /\ ~ Bet A P B.
Lemma l6_4_2 : forall A B P, Col A P B /\ ~ Bet A P B -> out P A B.
Lemma l6_5 : forall P A, A<>P -> out P A A.
Lemma l6_6 : forall P A B, out P A B -> out P B A.
Lemma l6_7 : forall P A B C, out P A B -> out P B C -> out P A C.
Lemma l6_11_unicity : forall A B C R X Y, R<>A -> B<>C -> out A X R -> Cong A X B C -> out A Y R -> Cong A Y B C -> X=Y.
Lemma l6_11_existence : forall A B C R, R<>A -> B<>C -> exists X, out A X R /\ Cong A X B C.
Lemma l6_13_1 : forall P A B, out P A B -> le P A P B -> Bet P A B.
Lemma l6_13_2 : forall P A B, out P A B -> Bet P A B -> le P A P B.
Lemma l6_16_1 : forall P Q S X, P<>Q -> S<>P -> Col S P Q -> (Col X P Q -> Col X P S).
Lemma l6_16_2 : forall P Q S X, P<>Q -> S<>P -> Col S P Q -> (Col X P S -> Col X P Q).