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.

Lemma point_construction_different : forall A B, exists C, Bet A B C /\ B <> C.