Definition T A B C : Prop := ~ (A<>B /\ B<>C /\ ~ BetH A B C).Definition ColB A B C := ~ (~ T C A B /\ ~ T A C B /\ ~ T A B C).

(* Specifications: *)
Lemma between_identity_B : forall A B, ~ BetH A B A.
Lemma Bet_T : forall A B C, Bet A B C -> T A B C.
Lemma BetH_Bet : forall A B C, BetH A B C -> Bet A B C.
Lemma T_Bet : forall A B C, T A B C -> Bet A B C.
Lemma NT_NBet : forall A B C, ~ T A B C -> ~ Bet A B C.
Lemma T_dec : forall A B C, T A B C \/ ~ T A B C.
Lemma between_inner_transitivity_B : forall A B C D : Tpoint, BetH A B D -> BetH B C D -> BetH A B C.
Lemma ColB_Col : forall A B C, ColB A B C -> Col A B C.
Lemma Diff_Col_ColB : forall A B C, Col A B C -> ColB A B C.
Lemma NColB_NDiffCol : forall A B C, ~ ColB A B C -> ~ Col A B C.
Lemma NColB_NColOrEq : forall A B C, ~ ColB A B C -> ~ Col A B C.
Lemma inner_pasch_B : forall A B C P Q, BetH A P C -> BetH B Q C -> ~ ColB A B C -> exists x, BetH P x B /\ BetH Q x A.
Lemma between_symmetry_B : forall A B C, BetH A B C -> BetH C B A.
Lemma five_segment_B : forall A A' B B' C C' D D' : Tpoint, Cong A B A' B' -> Cong B C B' C' -> Cong A D A' D' -> Cong B D B' D' -> ~ (A <> B /\ B <> C /\ ~ BetH A B C) -> ~ (A' <> B' /\ B' <> C' /\ ~ BetH A' B' C') -> A <> B -> Cong C D C' D'.
Lemma segment_construction_B : forall A B C D, A<>B -> exists E, T A B E /\ Cong B E C D.
Lemma lower_dim_B : ~ T PC PA PB /\ ~ T PA PC PB /\ ~ T PA PB PC.