(* Lemma t2_8 : forall A B C D E : Tpoint, Bet A B C -> Bet D B E -> Cong A B D B -> Cong B C B E -> Cong A E C D. *)

(* Specifications: *)
Lemma col3 : forall X Y A B C,
X <> Y ->
Col X Y A -> Col X Y B -> Col X Y C ->
Col A B C.
Lemma colx : forall A B C X Y, A <> B -> Col X Y A -> Col X Y B -> Col A B C -> Col X Y C.
Lemma out2__bet : forall A B C, Out A B C -> Out C A B -> Bet A B C.
Lemma bet2_le2__le1346 : forall A B C A' B' C', Bet A B C -> Bet A' B' C' -> Le
A B A' B' -> Le
B C B' C' ->
Le
A C A' C'.
Lemma bet2_le2__le2356 : forall A B C A' B' C', Bet A B C -> Bet A' B' C' ->
Le A B A' B' -> Le A' C' A C -> Le B' C' B C.
Lemma bet2_le2__le1245 : forall A B C A' B' C', Bet A B C -> Bet A' B' C' ->
Le B C B' C' -> Le A' C' A C -> Le A' B' A B.
Lemma cong_preserves_bet : forall B A' A0 E D' D0,
Bet B A' A0 -> Cong B A' E D' -> Cong B A0 E D0 -> Out E D' D0 ->
Bet E D' D0.
Lemma out_cong_cong : forall B A A0 E D D0,
Out B A A0 -> Out E D D0 ->
Cong B A E D -> Cong B A0 E D0 ->
Cong A A0 D D0.
Lemma not_out_bet : forall A B C, Col A B C -> ~ Out B A C -> Bet A B C.
Lemma or_bet_out : forall A B C, Bet A B C \/ Out B A C \/ ~Col A B C.
Lemma not_bet_out : forall A B C,
Col A B C -> ~Bet A B C -> Out B A C.
Lemma not_bet_and_out :
forall A B C,
~ (Bet A B C /\ Out B A C).
Lemma out_to_bet :
forall A B C A' B' C',
Col A' B' C' ->
(Out B A C <-> Out B' A' C') ->
Bet A B C ->
Bet A' B' C'.
Lemma col_out2_col  : forall A B C AA CC, Col A B C -> Out B A AA -> Out B C CC -> Col AA B CC.
Lemma bet2_out_out : forall A B C B' C', B <> A -> B' <> A -> Out A C C' -> Bet A B C -> Bet A B' C' -> Out A B B'.
Lemma bet2__out : forall A B C B', A <> B -> A <> B' -> Bet A B C -> Bet A B' C -> Out A B B'.
Lemma out2_out_1 : forall B C D X,
Out B X C -> Out B X D -> Out B C D.
Lemma out2_out_2 : forall B C D X,
Out B C X -> Out B D X -> Out B C D.
Lemma out_bet_out_1 : forall A B C P,
Out P A C -> Bet A B C -> Out P A B.
Lemma out_bet_out_2 : forall A B C P,
Out P A C -> Bet A B C -> Out P B C.
Lemma out_bet__out : forall A B P Q,
Bet P Q A -> Out Q A B -> Out P A B.
Lemma segment_reverse : forall A B C, Bet A B C -> exists B', Bet A B' C /\ Cong C B' A B.
Lemma diff_col_ex : forall A B, exists C, A <> C /\ B <> C /\ Col A B C.
Lemma diff_bet_ex3 : forall A B C,
Bet A B C ->
exists D, A <> D /\ B <> D /\ C <> D /\ Col A B D.
Lemma diff_col_ex3 : forall A B C,
Col A B C -> exists D, A <> D /\ B <> D /\ C <> D /\ Col A B D.