Section Intuitionistic_Tarski_to_Tarski.
Context `{BTn:intuitionistic_Tarski_neutral_dimensionless}.

(* Specifications: *)
Lemma col_dec : forall A B C, ICol A B C \/ ~ ICol A B C.
Lemma eq_dec : forall A B :ITpoint, A=B \/ A<>B.
Definition BetT A B C := IBet A B C \/ A=B \/ B=C.
Lemma bet_id : forall A B : ITpoint, BetT A B A -> A = B.
Lemma IT_chara : forall A B C,
IT A B C <-> A=B \/ B=C \/ IBet A B C.
Lemma BetT_symmetry : forall A B C, BetT A B C -> BetT C B A.
Lemma BetT_id : forall A B, BetT A B A -> A=B.
(* Lemma 4.1 page 22 *)
Lemma pasch_col_case : forall A B C P Q : ITpoint,
BetT A P C ->
BetT B Q C -> ICol A B C -> exists x : ITpoint, BetT P x B /\ BetT Q x A.
Lemma pasch : forall A B C P Q : ITpoint,
BetT A P C ->
BetT B Q C -> exists x : ITpoint, BetT P x B /\ BetT Q x A.
Lemma five_segment:
forall A A' B B' C C' D D' : ITpoint,
ICong A B A' B' ->
ICong B C B' C' ->
ICong A D A' D' ->
ICong B D B' D' ->
BetT A B C -> BetT A' B' C' -> A <> B -> ICong C D C' D'.
Lemma IT_trivial : forall A B, IT A A B.
Lemma IT_trivial2 : forall A B, IT A B B.
Lemma another_point : forall A, exists B:ITpoint, A<>B.
Lemma segment_construction :
forall A B C D : ITpoint,
exists E : ITpoint, BetT A B E /\ ICong B E C D.
Lemma lower_dim :
~ (BetT beeson_s_axioms.PA beeson_s_axioms.PB beeson_s_axioms.PC \/
BetT beeson_s_axioms.PB beeson_s_axioms.PC beeson_s_axioms.PA \/
BetT beeson_s_axioms.PC beeson_s_axioms.PA beeson_s_axioms.PB).
Lemma eq_dec_points_from_classic : forall A B : ITpoint, A = B \/ A <> B.
Instance IT_to_T : Tarski_neutral_dimensionless.