(* 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.

Lemma beetween_trivial2 : forall A B : Point, Bet A A B.