Require Export GeoCoq.Tarski_dev.Ch05_bet_le.
Ltac eCol := eauto with col.
Section T6_1.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
Lemma bet_out : forall A B C, B <> A -> Bet A B C -> Out A B C.
Lemma out_dec : forall P A B, Out P A B \/ ~ Out P A B.
Lemma out_diff1 : forall A B C, Out A B C -> B <> A.
Lemma out_diff2 : forall A B C, Out A B C -> C <> A.
Lemma out_distinct : forall A B C, Out A B C -> B <> A /\ C <> A.
Lemma out_col : forall A B C, Out A B C -> Col A B C.
Lemma l6_2 : forall A B C P,  A<>P -> B<>P -> C<>P -> Bet A P C -> (Bet B P C <-> Out P A B).
Lemma bet_out__bet : forall A B C P, Bet A P C -> Out P A B -> Bet B P C.
Lemma l6_3_1 : forall A B P, Out P A B -> (A<>P /\ B<>P /\ exists C, C<>P /\ Bet A P C /\ Bet B P C).
Lemma l6_3_2 : forall A B P,
(A<>P /\ B<>P /\ exists C, C<>P /\ Bet A P C /\ Bet B P C) -> Out P A B.
Lemma l6_4_1 : forall A B P, Out P A B -> Col A P B /\ ~ Bet A P B.
Lemma l6_4_2 : forall A B P, Col A P B /\ ~ Bet A P B -> Out P A B.
(** out reflexivity. l6_5 *)
Lemma out_trivial : forall P A, A<>P -> Out P A A.
(** out symmetry. *)
Lemma l6_6 : forall P A B, Out P A B -> Out P B A.
(** out transitivity. *)
Lemma l6_7 : forall P A B C, Out P A B -> Out P B C -> Out P A C.
Lemma bet_out_out_bet : forall A B C A' C',
Bet A B C -> Out B A A' -> Out B C C' -> Bet A' B C'.
Lemma out2_bet_out : forall A B C X P,
Out B A C -> Out B X P -> Bet A X C -> Out B A P /\ Out B C P.
Lemma l6_11_uniqueness : forall A B C R X Y,
R<>A -> B<>C ->
Out A X R -> Cong A X B C ->
Out A Y R -> Cong A Y B C ->
X=Y.
Lemma l6_11_existence : forall A B C R,
R<>A -> B<>C -> exists X, Out A X R /\ Cong A X B C.
Lemma segment_construction_3 : forall A B X Y, A <> B -> X <> Y -> exists C, Out A B C /\ Cong A C X Y.
Lemma l6_13_1 : forall P A B, Out P A B -> Le
P A P B -> Bet P A B.
Lemma l6_13_2 : forall P A B, Out P A B -> Bet P A B -> Le
P A P B.
Lemma l6_16_1 : forall P Q S X, P<>Q -> Col S P Q -> Col X P Q -> Col X P S.
Lemma col_transitivity_1 : forall P Q A B,
P<>Q -> Col P Q A -> Col P Q B -> Col P A B.
Lemma col_transitivity_2 : forall P Q A B,
P<>Q -> Col P Q A -> Col P Q B -> Col Q A B.