Ltac Between := eauto with Between.(*
Axiom 20.1 of Givant
Lemma unique_triangle : forall A B C C' D,
A <> B -> Cong A C A C' -> Cong B C B C' -> 
Bet B D C' -> (Bet A D C \/ Bet A C D) -> C=C'.
Axiom 21 of Givant
Lemma existence_triangle : forall A B A' B' C',
 Cong A B A' B' -> exists C X, Cong A C A' C' /\ Cong B C B' C' /\ Bet C X P /\ Col A B X.
*)

(** We formalize $Bet_n$ only for n=4. *)

Definition Bet_4 := fun A1 A2 A3 A4 =>
   Bet A1 A2 A3 /\ Bet A2 A3 A4 /\ Bet A1 A3 A4 /\ Bet A1 A2 A4.

(* Specifications: *)
Lemma l_3_9_4 : forall A1 A2 A3 A4, Bet_4 A1 A2 A3 A4 -> Bet_4 A4 A3 A2 A1.
Lemma two_distinct_points : exists x, exists y: Point, x <> y.
Lemma point_construction_different : forall A B, exists C, Bet A B C /\ B <> C.
Lemma another_point : forall A:Point, exists B, A<>B.
Lemma l3_17 : forall A B C A' B' P, Bet A B C -> Bet A' B' C -> Bet A P A' -> exists Q, Bet P Q C /\ Bet B Q B'.