(* Julien Narboux *)
(* Formalization of *)
(* Tarski's axiomatic *)
Require Export Ch02_congruence_properties.
(** Axiom 12 of Givant. *)
Proof with eTarski.
intros.
assert (exists x : Point, Bet A B x /\ Cong B x B B)...
DecompExAnd H x.
assert (B = x).
assert (Cong B B B x)...
congruence.
Qed.
Hint Resolve beetween_trivial : Tarski.
(** Axiom 14 of Givant. *)
Proof with eTarski.
intros.
assert (Bet B C C)...
assert (exists x, Bet B x B /\ Bet C x A)...
DecompExAnd H1 x.
assert (B=x)...
congruence.
Qed.
Hint Resolve between_symmetry : sTarski.
Hint Resolve beetween_trivial beetween_trivial2 : Tarski.

Lemma between_egality : forall A B C : Point, Bet A B C -> Bet B A C -> A = B.