Require Export GeoCoq.Tarski_dev.Ch02_cong.
Section T2_1.
Context `{Tn:Tarski_neutral_dimensionless}.

(* Specifications: *)
Lemma bet_col : forall A B C, Bet A B C -> Col A B C.
Lemma between_trivial : forall A B : Tpoint, Bet A B B.
Lemma between_symmetry : forall A B C : Tpoint, Bet A B C -> Bet C B A.
(** This lemma is used by tactics for trying several permutations. *)
Lemma Bet_cases :
forall A B C,
Bet A B C \/ Bet C B A ->
Bet A B C.
Lemma Bet_perm :
forall A B C,
Bet A B C ->
Bet A B C /\ Bet C B A.
Lemma between_trivial2 : forall A B : Tpoint, Bet A A B.
Lemma between_equality : forall A B C : Tpoint, Bet A B C -> Bet B A C -> A = B.
Lemma between_equality_2 : forall A B C : Tpoint, Bet A B C -> Bet A C B -> B = C.
Lemma between_exchange3 : forall A B C D, Bet A B C -> Bet A C D -> Bet B C D.
Lemma bet_neq12__neq : forall A B C, Bet A B C -> A <> B -> A <> C.
Lemma bet_neq21__neq : forall A B C, Bet A B C -> B <> A -> A <> C.
Lemma bet_neq23__neq : forall A B C, Bet A B C -> B <> C -> A <> C.
Lemma bet_neq32__neq : forall A B C, Bet A B C -> C <> B -> A <> C.
Lemma not_bet_distincts : forall A B C, ~ Bet A B C -> A <> B /\ B <> C.
End T2_1.