End T11_1.
Section T11_2.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
Lemma l11_16 : forall A B C A' B' C',
Per A B C    -> A <> B  -> C <> B ->
Per A' B' C' -> A'<> B' -> C'<> B'->
CongA A B C A' B' C'.
Lemma l11_17 : forall A B C A' B' C',
Per A B C -> CongA A B C A' B' C' -> Per A' B' C'.
Lemma l11_18_1 : forall A B C D,
Bet C B D -> B <> C -> B <> D -> A <> B -> Per A B C -> CongA A B C A B D.
Lemma l11_18_2 : forall A B C D,
Bet C B D -> CongA A B C A B D -> Per A B C.
Lemma cong3_preserves_out : forall A B C A' B' C',
Out A B C -> Cong_3 A B C A' B' C' -> Out A' B' C'.
Lemma l11_21_a : forall A B C A' B' C', Out B A C -> CongA A B C A' B' C' -> Out B' A' C'.
Lemma l11_21_b : forall A B C A' B' C',
Out B A C -> Out B' A' C' -> CongA A B C A' B' C'.
Lemma conga_cop__or_out_ts : forall A B C C', Coplanar A B C C' -> CongA A B C A B C' ->
Out B C C' \/ TS A B C C'.
Lemma cong2_conga_cong : forall A B C A' B' C',
CongA A B C A' B' C' -> Cong A B A' B' -> Cong B C B' C' ->
Cong A C A' C'.
Lemma angle_construction_1 : forall A B C A' B' P,
~ Col A B C -> ~ Col A' B' P ->
exists C', CongA A B C A' B' C' /\ OS A' B' C' P.
Lemma angle_construction_2 : forall A B C A' B' P,
A <> B -> A <> C -> B <> C -> A' <> B' -> ~ Col A' B' P ->
exists C', CongA A B C A' B' C' /\ (OS A' B' C' P \/ Col A' B' C').
Lemma ex_conga_ts : forall A B C A' B' P,
~ Col A B C -> ~ Col A' B' P ->
exists C' : Tpoint, CongA A B C A' B' C' /\ TS A' B' C' P.
Lemma l11_15 : forall A B C D E P, ~ Col A B C -> ~ Col D E P ->
exists F, CongA A B C D E F /\ OS E D F P /\
(forall F1 F2, ((CongA A B C D E F1 /\ OS E D F1 P) /\
(CongA A B C D E F2 /\ OS E D F2 P)) -> Out E F1 F2).
Lemma l11_19 : forall A B P1 P2,
Per A B P1 -> Per A B P2 -> OS A B P1 P2 ->
Out B P1 P2.
Lemma l11_22_bet :
forall A B C P A' B' C' P',
Bet A B C ->
TS P' B' A' C' ->
CongA A B P A' B' P' /\ CongA P B C  P' B' C' ->
Bet A' B' C'.
Lemma l11_22a :
forall A B C P A' B' C' P',
TS B P A C /\ TS B' P' A' C' /\
CongA A B P A' B' P' /\ CongA P B C  P' B' C' ->
CongA A B C A' B' C'.
Lemma l11_22b :
forall A B C P A' B' C' P',
OS B P A C /\ OS B' P' A' C' /\
CongA A B P A' B' P' /\ CongA P B C  P' B' C' ->
CongA A B C A' B' C'.
Lemma l11_22 :
forall A B C P A' B' C' P',
((TS B P A C /\ TS B' P' A' C')\/
(OS B P A C /\ OS B' P' A' C')) /\
CongA A B P A' B' P' /\ CongA P B C  P' B' C' ->
CongA A B C A' B' C'.
Lemma l11_24 :
forall P A B C,
InAngle P A B C -> InAngle P C B A.
Lemma out_in_angle :
forall A B C P,
Out B A C -> Out B P A ->
InAngle P A B C.
Lemma col_in_angle :
forall A B C P,
A <> B -> C <> B -> P <> B ->
Out B A P \/ Out B C P ->
InAngle P A B C.
Lemma out321__inangle :
forall A B C P,
C <> B -> Out B A P ->
InAngle P A B C.
Lemma inangle1123 :
forall A B C,
A <> B -> C <> B ->
InAngle A A B C.
Lemma out341__inangle :
forall A B C P,
A <> B -> Out B C P ->
InAngle P A B C.
Lemma inangle3123 :
forall A B C,
A <> B -> C <> B ->
InAngle C A B C.
Lemma in_angle_two_sides :
forall A B C P,
~ Col B A P -> ~ Col B C P ->
InAngle P A B C ->
TS P B A C.
Lemma in_angle_out :
forall A B C P,
Out B A C ->
InAngle P A B C ->
Out B A P.
Lemma col_in_angle_out :
forall A B C P,
Col B A P ->
~ Bet A B C ->
InAngle P A B C ->
Out B A P.
Lemma l11_25_aux : forall P A B C A',
InAngle P A B C ->
~ Bet A B C ->
Out B A' A ->
InAngle P A' B C.
Lemma l11_25 : forall P A B C A' C' P',
InAngle P A B C ->
Out B A' A ->
Out B C' C ->
Out B P' P ->
InAngle P' A' B C'.
Lemma inangle_distincts : forall A B C P, InAngle P A B C ->
A <> B /\ C <> B /\ P <> B.
Lemma segment_construction_0 : forall A B A', exists B', Cong A' B' A B.
Lemma angle_construction_3 :
forall A B C A' B',
A <> B -> C <> B -> A' <> B' ->
exists C', CongA A B C A' B' C'.
Lemma l11_28 : forall A B C D A' B' C',
Cong_3 A B C A' B' C' -> Col A C D ->
exists D', Cong A D A' D' /\ Cong B D B' D' /\ Cong C D C' D'.
Lemma bet_conga__bet :
forall A B C A' B' C',
Bet A B C ->
CongA A B C A' B' C' ->
Bet A' B' C'.
Lemma out_in_angle_out :
forall A B C P,
Out B A C ->
InAngle P A B C ->
Out B A P.
Lemma in_angle_one_side :
forall A B C P,
~ Col A B C ->
~ Col B A P ->
InAngle P A B C ->
OS A B P C.
Lemma inangle_one_side : forall A B C P Q , ~ Col A B C -> ~ Col A B P -> ~ Col A B Q ->
InAngle P A B C -> InAngle Q A B C ->
OS A B P Q.
Lemma inangle_one_side2 : forall A B C P Q , ~ Col A B C -> ~ Col A B P -> ~ Col A B Q ->
~ Col C B P -> ~ Col C B Q ->
InAngle P A B C -> InAngle Q A B C ->
OS A B P Q /\ OS C B P Q.
Lemma col_conga_col : forall A B C D E F, Col A B C -> CongA A B C D E F -> Col D E F.
Lemma ncol_conga_ncol : forall A B C D E F, ~ Col A B C -> CongA A B C D E F -> ~ Col D E F.
Lemma angle_construction_4 :
forall A B C A' B' P,
A <> B -> C <> B -> A' <> B' ->
exists C', CongA A B C A' B' C' /\ Coplanar A' B' C' P.
Lemma lea_distincts : forall A B C D E F, LeA A B C D E F ->
A<>B /\ C<>B /\ D<>E /\ F<>E.
Lemma gea_distincts : forall A B C D E F, GeA A B C D E F ->
A<>B /\ C<>B /\ D<>E /\ F<>E.
Lemma l11_29_a : forall A B C D E F, LeA A B C D E F ->
exists Q, InAngle C A B Q /\ CongA A B Q D E F.
Lemma in_angle_line : forall A B C P, P <> B -> A <> B -> C <> B -> Bet A B C -> InAngle P A B C.
Lemma l11_29_b : forall A B C D E F, (exists Q, InAngle C A B Q /\ CongA A B Q D E F) ->
LeA A B C D E F.
Lemma bet_in_angle_bet : forall A B C P, Bet A B P -> InAngle P A B C -> Bet A B C.
Lemma lea_line : forall A B C P, Bet A B P -> LeA A B P A B C -> Bet A B C.
Lemma eq_conga_out : forall A B D E F, CongA A B A D E F -> Out E D F.
Lemma conga_ex_cong3 : forall A B C A' B' C',
CongA A B C A' B' C' -> exists AA, exists CC, Out B A AA -> Out B C CC -> Cong_3 AA B CC A' B' C'.
Lemma conga_preserves_in_angle : forall A B C I A' B' C' I',
CongA A B C A' B' C' -> CongA A B I A' B' I' ->
InAngle I A B C -> OS A' B' I' C' ->
InAngle I' A' B' C'.
Lemma l11_30 : forall A B C D E F A' B' C' D' E' F',
LeA A B C D E F ->
CongA A B C A' B' C' ->
CongA D E F D' E' F' ->
LeA A' B' C' D' E' F'.
Lemma l11_31_1 : forall A B C D E F,
Out B A C -> D <> E -> F <> E ->
LeA A B C D E F.
Lemma l11_31_2 : forall A B C D E F,
A <> B -> C <> B -> D <> E -> F <> E ->
Bet D E F ->
LeA A B C D E F.
Lemma lea_refl : forall A B C,
A <> B -> C <> B -> LeA A B C A B C.
Lemma conga__lea : forall A B C D E F,
CongA A B C D E F -> LeA A B C D E F.
Lemma conga__lea456123 : forall A B C D E F,
CongA A B C D E F -> LeA D E F A B C.
Lemma lea_out4__lea : forall A B C D E F A' C' D' F',
LeA A B C D E F -> Out B A A' -> Out B C C' -> Out E D D' -> Out E F F' ->
LeA A' B C' D' E F'.
Lemma lea121345 : forall A B C D E, A<>B -> C<>D -> D<>E -> LeA A B A C D E.
Lemma inangle__lea : forall A B C P, InAngle P A B C -> LeA A B P A B C.
Lemma in_angle_trans : forall A B C D E,
InAngle C A B D -> InAngle D A B E -> InAngle C A B E.
Lemma lea_trans : forall A B C A1 B1 C1 A2 B2 C2,
LeA A B C A1 B1 C1 ->
LeA A1 B1 C1 A2 B2 C2 ->
LeA A B C A2 B2 C2.
Lemma in_angle_asym : forall A B C D,
InAngle D A B C -> InAngle C A B D -> CongA A B C A B D.
Lemma lea_asym : forall A B C D E F,
LeA A B C D E F -> LeA D E F A B C -> CongA A B C D E F.
Lemma col_lta__bet : forall A B C X Y Z, Col X Y Z -> LtA A B C X Y Z -> Bet X Y Z.
Lemma col_lta__out : forall A B C X Y Z, Col A B C -> LtA A B C X Y Z -> Out B A C.
Lemma lta_distincts : forall A B C D E F, LtA A B C D E F ->
A<>B /\ C<>B /\ D<>E /\ F<>E /\ D <> F.
Lemma gta_distincts : forall A B C D E F, GtA A B C D E F ->
A<>B /\ C<>B /\ D<>E /\ F<>E /\ A <> C.
Lemma acute_distincts : forall A B C, Acute A B C -> A<>B /\ C<>B.
Lemma obtuse_distincts : forall A B C, Obtuse A B C -> A<>B /\ C<>B /\ A <> C.
Lemma two_sides_in_angle : forall A B C P P',
B <> P' ->
TS B P A C ->
Bet P B P' ->
InAngle P A B C \/ InAngle P' A B C.
Lemma in_angle_reverse :
forall A B A' C D,
A' <> B ->
Bet A B A' ->
InAngle C A B D ->
InAngle D A' B C.
Lemma in_angle_trans2 : forall A B C D E, InAngle C A B D -> InAngle D A B E -> InAngle D C B E.
Lemma l11_36 : forall A B C D E F A' D',
A <> B -> A' <> B -> D <> E -> D' <> E ->
Bet A B A' -> Bet D E D' ->
(LeA A B C D E F <-> LeA D' E F A' B C).
Lemma l11_41_aux : forall A B C D,
~ Col A B C ->
Bet B A D ->
A <> D ->
LtA A C B C A D.