(** * 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.

Lemma axiom_between_out : forall A B, A <> B -> exists C, Between_H A B C.