(** * Group II Order *)

(** Definition of the Between predicate of Hilbert.
    Note that it is different from the Between of Tarski.
    The Between of Hilbert is strict. *)

Definition Between_H := fun A B C =>
  Bet A B C /\ A <> B /\ B <> C /\ A <> C.

(* Specifications: *)
Lemma axiom_between_col : forall A B C, Between_H A B C -> Col_H A B C.
Lemma axiom_between_diff : forall A B C, Between_H A B C -> A<>C.
Lemma axiom_between_comm : forall A B C, Between_H A B C -> Between_H C B A.
Lemma axiom_between_out : forall A B, A <> B -> exists C, Between_H A B C.
Lemma axiom_between_only_one : forall A B C, Between_H A B C -> ~ Between_H B C A.
Lemma between_one : forall A B C, A<>B -> A<>C -> B<>C -> Col A B C -> Between_H A B C \/ Between_H B C A \/ Between_H B A C.
Lemma axiom_between_one : forall A B C, A<>B -> A<>C -> B<>C -> Col_H A B C -> Between_H A B C \/ Between_H B C A \/ Between_H B A C.