:: AFF_2 semantic presentation

definition
let c1 be AffinPlane;
attr a1 is satisfying_PPAP means :Def1: :: AFF_2:def 1
for b1, b2 being Subset of a1
for b3, b4, b5, b6, b7, b8 being Element of a1 holds
( b1 is_line & b2 is_line & b3 in b1 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b2 & b3,b7 // b4,b6 & b4,b8 // b5,b7 implies b3,b8 // b5,b6 );
end;

:: deftheorem Def1 defines satisfying_PPAP AFF_2:def 1 :
for b1 being AffinPlane holds
( b1 is satisfying_PPAP iff for b2, b3 being Subset of b1
for b4, b5, b6, b7, b8, b9 being Element of b1 holds
( b2 is_line & b3 is_line & b4 in b2 & b5 in b2 & b6 in b2 & b7 in b3 & b8 in b3 & b9 in b3 & b4,b8 // b5,b7 & b5,b9 // b6,b8 implies b4,b9 // b6,b7 ) );

notation
let c1 be AffinPlane;
synonym c1 satisfies_PPAP for satisfying_PPAP c1;
end;

definition
let c1 be AffinSpace;
attr a1 is Pappian means :Def2: :: AFF_2:def 2
for b1, b2 being Subset of a1
for b3, b4, b5, b6, b7, b8, b9 being Element of a1 holds
( b1 is_line & b2 is_line & b1 <> b2 & b3 in b1 & b3 in b2 & b3 <> b4 & b3 <> b7 & b3 <> b5 & b3 <> b8 & b3 <> b6 & b3 <> b9 & b4 in b1 & b5 in b1 & b6 in b1 & b7 in b2 & b8 in b2 & b9 in b2 & b4,b8 // b5,b7 & b5,b9 // b6,b8 implies b4,b9 // b6,b7 );
end;

:: deftheorem Def2 defines Pappian AFF_2:def 2 :
for b1 being AffinSpace holds
( b1 is Pappian iff for b2, b3 being Subset of b1
for b4, b5, b6, b7, b8, b9, b10 being Element of b1 holds
( b2 is_line & b3 is_line & b2 <> b3 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b8 & b4 <> b6 & b4 <> b9 & b4 <> b7 & b4 <> b10 & b5 in b2 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b10 in b3 & b5,b9 // b6,b8 & b6,b10 // b7,b9 implies b5,b10 // b7,b8 ) );

notation
let c1 be AffinSpace;
synonym c1 satisfies_PAP for Pappian c1;
end;

definition
let c1 be AffinPlane;
attr a1 is satisfying_PAP_1 means :Def3: :: AFF_2:def 3
for b1, b2 being Subset of a1
for b3, b4, b5, b6, b7, b8, b9 being Element of a1 holds
( b1 is_line & b2 is_line & b1 <> b2 & b3 in b1 & b3 in b2 & b3 <> b4 & b3 <> b7 & b3 <> b5 & b3 <> b8 & b3 <> b6 & b3 <> b9 & b4 in b1 & b5 in b1 & b6 in b1 & b8 in b2 & b9 in b2 & b4,b8 // b5,b7 & b5,b9 // b6,b8 & b4,b9 // b6,b7 & b5 <> b6 implies b7 in b2 );
end;

:: deftheorem Def3 defines satisfying_PAP_1 AFF_2:def 3 :
for b1 being AffinPlane holds
( b1 is satisfying_PAP_1 iff for b2, b3 being Subset of b1
for b4, b5, b6, b7, b8, b9, b10 being Element of b1 holds
( b2 is_line & b3 is_line & b2 <> b3 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b8 & b4 <> b6 & b4 <> b9 & b4 <> b7 & b4 <> b10 & b5 in b2 & b6 in b2 & b7 in b2 & b9 in b3 & b10 in b3 & b5,b9 // b6,b8 & b6,b10 // b7,b9 & b5,b10 // b7,b8 & b6 <> b7 implies b8 in b3 ) );

notation
let c1 be AffinPlane;
synonym c1 satisfies_PAP_1 for satisfying_PAP_1 c1;
end;

definition
let c1 be AffinSpace;
attr a1 is Desarguesian means :Def4: :: AFF_2:def 4
for b1, b2, b3 being Subset of a1
for b4, b5, b6, b7, b8, b9, b10 being Element of a1 holds
( b4 in b1 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 in b1 & b8 in b1 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b1 is_line & b2 is_line & b3 is_line & b1 <> b2 & b1 <> b3 & b5,b6 // b8,b9 & b5,b7 // b8,b10 implies b6,b7 // b9,b10 );
end;

:: deftheorem Def4 defines Desarguesian AFF_2:def 4 :
for b1 being AffinSpace holds
( b1 is Desarguesian iff for b2, b3, b4 being Subset of b1
for b5, b6, b7, b8, b9, b10, b11 being Element of b1 holds
( b5 in b2 & b5 in b3 & b5 in b4 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b8 in b4 & b11 in b4 & b2 is_line & b3 is_line & b4 is_line & b2 <> b3 & b2 <> b4 & b6,b7 // b9,b10 & b6,b8 // b9,b11 implies b7,b8 // b10,b11 ) );

notation
let c1 be AffinSpace;
synonym c1 satisfies_DES for Desarguesian c1;
end;

definition
let c1 be AffinPlane;
attr a1 is satisfying_DES_1 means :Def5: :: AFF_2:def 5
for b1, b2, b3 being Subset of a1
for b4, b5, b6, b7, b8, b9, b10 being Element of a1 holds
( b4 in b1 & b4 in b2 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 in b1 & b8 in b1 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b1 is_line & b2 is_line & b3 is_line & b1 <> b2 & b1 <> b3 & b5,b6 // b8,b9 & b5,b7 // b8,b10 & b6,b7 // b9,b10 & not LIN b5,b6,b7 & b7 <> b10 implies b4 in b3 );
end;

:: deftheorem Def5 defines satisfying_DES_1 AFF_2:def 5 :
for b1 being AffinPlane holds
( b1 is satisfying_DES_1 iff for b2, b3, b4 being Subset of b1
for b5, b6, b7, b8, b9, b10, b11 being Element of b1 holds
( b5 in b2 & b5 in b3 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b8 in b4 & b11 in b4 & b2 is_line & b3 is_line & b4 is_line & b2 <> b3 & b2 <> b4 & b6,b7 // b9,b10 & b6,b8 // b9,b11 & b7,b8 // b10,b11 & not LIN b6,b7,b8 & b8 <> b11 implies b5 in b4 ) );

notation
let c1 be AffinPlane;
synonym c1 satisfies_DES_1 for satisfying_DES_1 c1;
end;

definition
let c1 be AffinPlane;
attr a1 is satisfying_DES_2 means :: AFF_2:def 6
for b1, b2, b3 being Subset of a1
for b4, b5, b6, b7, b8, b9, b10 being Element of a1 holds
( b4 in b1 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 in b1 & b8 in b1 & b6 in b2 & b9 in b2 & b7 in b3 & b1 is_line & b2 is_line & b3 is_line & b1 <> b2 & b1 <> b3 & b5,b6 // b8,b9 & b5,b7 // b8,b10 & b6,b7 // b9,b10 implies b10 in b3 );
end;

:: deftheorem Def6 defines satisfying_DES_2 AFF_2:def 6 :
for b1 being AffinPlane holds
( b1 is satisfying_DES_2 iff for b2, b3, b4 being Subset of b1
for b5, b6, b7, b8, b9, b10, b11 being Element of b1 holds
( b5 in b2 & b5 in b3 & b5 in b4 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b8 in b4 & b2 is_line & b3 is_line & b4 is_line & b2 <> b3 & b2 <> b4 & b6,b7 // b9,b10 & b6,b8 // b9,b11 & b7,b8 // b10,b11 implies b11 in b4 ) );

notation
let c1 be AffinPlane;
synonym c1 satisfies_DES_2 for satisfying_DES_2 c1;
end;

definition
let c1 be AffinSpace;
attr a1 is Moufangian means :Def7: :: AFF_2:def 7
for b1 being Subset of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of a1 holds
( b1 is_line & b2 in b1 & b5 in b1 & b8 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & LIN b2,b4,b7 & b3,b4 // b6,b7 & b3,b5 // b6,b8 & b3,b4 // b1 implies b4,b5 // b7,b8 );
end;

:: deftheorem Def7 defines Moufangian AFF_2:def 7 :
for b1 being AffinSpace holds
( b1 is Moufangian iff for b2 being Subset of b1
for b3, b4, b5, b6, b7, b8, b9 being Element of b1 holds
( b2 is_line & b3 in b2 & b6 in b2 & b9 in b2 & not b4 in b2 & b3 <> b6 & b4 <> b5 & LIN b3,b4,b7 & LIN b3,b5,b8 & b4,b5 // b7,b8 & b4,b6 // b7,b9 & b4,b5 // b2 implies b5,b6 // b8,b9 ) );

notation
let c1 be AffinSpace;
synonym c1 satisfies_TDES for Moufangian c1;
end;

definition
let c1 be AffinPlane;
attr a1 is satisfying_TDES_1 means :Def8: :: AFF_2:def 8
for b1 being Subset of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of a1 holds
( b1 is_line & b2 in b1 & b5 in b1 & b8 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & b3,b4 // b6,b7 & b4,b5 // b7,b8 & b3,b5 // b6,b8 & b3,b4 // b1 implies LIN b2,b4,b7 );
end;

:: deftheorem Def8 defines satisfying_TDES_1 AFF_2:def 8 :
for b1 being AffinPlane holds
( b1 is satisfying_TDES_1 iff for b2 being Subset of b1
for b3, b4, b5, b6, b7, b8, b9 being Element of b1 holds
( b2 is_line & b3 in b2 & b6 in b2 & b9 in b2 & not b4 in b2 & b3 <> b6 & b4 <> b5 & LIN b3,b4,b7 & b4,b5 // b7,b8 & b5,b6 // b8,b9 & b4,b6 // b7,b9 & b4,b5 // b2 implies LIN b3,b5,b8 ) );

notation
let c1 be AffinPlane;
synonym c1 satisfies_TDES_1 for satisfying_TDES_1 c1;
end;

definition
let c1 be AffinPlane;
attr a1 is satisfying_TDES_2 means :Def9: :: AFF_2:def 9
for b1 being Subset of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of a1 holds
( b1 is_line & b2 in b1 & b5 in b1 & b8 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & LIN b2,b4,b7 & b4,b5 // b7,b8 & b3,b5 // b6,b8 & b3,b4 // b1 implies b3,b4 // b6,b7 );
end;

:: deftheorem Def9 defines satisfying_TDES_2 AFF_2:def 9 :
for b1 being AffinPlane holds
( b1 is satisfying_TDES_2 iff for b2 being Subset of b1
for b3, b4, b5, b6, b7, b8, b9 being Element of b1 holds
( b2 is_line & b3 in b2 & b6 in b2 & b9 in b2 & not b4 in b2 & b3 <> b6 & b4 <> b5 & LIN b3,b4,b7 & LIN b3,b5,b8 & b5,b6 // b8,b9 & b4,b6 // b7,b9 & b4,b5 // b2 implies b4,b5 // b7,b8 ) );

notation
let c1 be AffinPlane;
synonym c1 satisfies_TDES_2 for satisfying_TDES_2 c1;
end;

definition
let c1 be AffinPlane;
attr a1 is satisfying_TDES_3 means :Def10: :: AFF_2:def 10
for b1 being Subset of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of a1 holds
( b1 is_line & b2 in b1 & b5 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & LIN b2,b4,b7 & b3,b4 // b6,b7 & b3,b5 // b6,b8 & b4,b5 // b7,b8 & b3,b4 // b1 implies b8 in b1 );
end;

:: deftheorem Def10 defines satisfying_TDES_3 AFF_2:def 10 :
for b1 being AffinPlane holds
( b1 is satisfying_TDES_3 iff for b2 being Subset of b1
for b3, b4, b5, b6, b7, b8, b9 being Element of b1 holds
( b2 is_line & b3 in b2 & b6 in b2 & not b4 in b2 & b3 <> b6 & b4 <> b5 & LIN b3,b4,b7 & LIN b3,b5,b8 & b4,b5 // b7,b8 & b4,b6 // b7,b9 & b5,b6 // b8,b9 & b4,b5 // b2 implies b9 in b2 ) );

notation
let c1 be AffinPlane;
synonym c1 satisfies_TDES_3 for satisfying_TDES_3 c1;
end;

definition
let c1 be AffinSpace;
attr a1 is translational means :Def11: :: AFF_2:def 11
for b1, b2, b3 being Subset of a1
for b4, b5, b6, b7, b8, b9 being Element of a1 holds
( b1 // b2 & b1 // b3 & b4 in b1 & b7 in b1 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b1 is_line & b2 is_line & b3 is_line & b1 <> b2 & b1 <> b3 & b4,b5 // b7,b8 & b4,b6 // b7,b9 implies b5,b6 // b8,b9 );
end;

:: deftheorem Def11 defines translational AFF_2:def 11 :
for b1 being AffinSpace holds
( b1 is translational iff for b2, b3, b4 being Subset of b1
for b5, b6, b7, b8, b9, b10 being Element of b1 holds
( b2 // b3 & b2 // b4 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b7 in b4 & b10 in b4 & b2 is_line & b3 is_line & b4 is_line & b2 <> b3 & b2 <> b4 & b5,b6 // b8,b9 & b5,b7 // b8,b10 implies b6,b7 // b9,b10 ) );

notation
let c1 be AffinSpace;
synonym c1 satisfies_des for translational c1;
end;

definition
let c1 be AffinPlane;
attr a1 is satisfying_des_1 means :Def12: :: AFF_2:def 12
for b1, b2, b3 being Subset of a1
for b4, b5, b6, b7, b8, b9 being Element of a1 holds
( b1 // b2 & b4 in b1 & b7 in b1 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b1 is_line & b2 is_line & b3 is_line & b1 <> b2 & b1 <> b3 & b4,b5 // b7,b8 & b4,b6 // b7,b9 & b5,b6 // b8,b9 & not LIN b4,b5,b6 & b6 <> b9 implies b1 // b3 );
end;

:: deftheorem Def12 defines satisfying_des_1 AFF_2:def 12 :
for b1 being AffinPlane holds
( b1 is satisfying_des_1 iff for b2, b3, b4 being Subset of b1
for b5, b6, b7, b8, b9, b10 being Element of b1 holds
( b2 // b3 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b7 in b4 & b10 in b4 & b2 is_line & b3 is_line & b4 is_line & b2 <> b3 & b2 <> b4 & b5,b6 // b8,b9 & b5,b7 // b8,b10 & b6,b7 // b9,b10 & not LIN b5,b6,b7 & b7 <> b10 implies b2 // b4 ) );

notation
let c1 be AffinPlane;
synonym c1 satisfies_des_1 for satisfying_des_1 c1;
end;

definition
let c1 be AffinSpace;
attr a1 is satisfying_pap means :Def13: :: AFF_2:def 13
for b1, b2 being Subset of a1
for b3, b4, b5, b6, b7, b8 being Element of a1 holds
( b1 is_line & b2 is_line & b3 in b1 & b4 in b1 & b5 in b1 & b1 // b2 & b1 <> b2 & b6 in b2 & b7 in b2 & b8 in b2 & b3,b7 // b4,b6 & b4,b8 // b5,b7 implies b3,b8 // b5,b6 );
end;

:: deftheorem Def13 defines satisfying_pap AFF_2:def 13 :
for b1 being AffinSpace holds
( b1 is satisfying_pap iff for b2, b3 being Subset of b1
for b4, b5, b6, b7, b8, b9 being Element of b1 holds
( b2 is_line & b3 is_line & b4 in b2 & b5 in b2 & b6 in b2 & b2 // b3 & b2 <> b3 & b7 in b3 & b8 in b3 & b9 in b3 & b4,b8 // b5,b7 & b5,b9 // b6,b8 implies b4,b9 // b6,b7 ) );

notation
let c1 be AffinSpace;
synonym c1 satisfies_pap for satisfying_pap c1;
end;

definition
let c1 be AffinPlane;
attr a1 is satisfying_pap_1 means :Def14: :: AFF_2:def 14
for b1, b2 being Subset of a1
for b3, b4, b5, b6, b7, b8 being Element of a1 holds
( b1 is_line & b2 is_line & b3 in b1 & b4 in b1 & b5 in b1 & b1 // b2 & b1 <> b2 & b6 in b2 & b7 in b2 & b3,b7 // b4,b6 & b4,b8 // b5,b7 & b3,b8 // b5,b6 & b6 <> b7 implies b8 in b2 );
end;

:: deftheorem Def14 defines satisfying_pap_1 AFF_2:def 14 :
for b1 being AffinPlane holds
( b1 is satisfying_pap_1 iff for b2, b3 being Subset of b1
for b4, b5, b6, b7, b8, b9 being Element of b1 holds
( b2 is_line & b3 is_line & b4 in b2 & b5 in b2 & b6 in b2 & b2 // b3 & b2 <> b3 & b7 in b3 & b8 in b3 & b4,b8 // b5,b7 & b5,b9 // b6,b8 & b4,b9 // b6,b7 & b7 <> b8 implies b9 in b3 ) );

notation
let c1 be AffinPlane;
synonym c1 satisfies_pap_1 for satisfying_pap_1 c1;
end;

theorem Th1: :: AFF_2:1
canceled;

theorem Th2: :: AFF_2:2
canceled;

theorem Th3: :: AFF_2:3
canceled;

theorem Th4: :: AFF_2:4
canceled;

theorem Th5: :: AFF_2:5
canceled;

theorem Th6: :: AFF_2:6
canceled;

theorem Th7: :: AFF_2:7
canceled;

theorem Th8: :: AFF_2:8
canceled;

theorem Th9: :: AFF_2:9
canceled;

theorem Th10: :: AFF_2:10
canceled;

theorem Th11: :: AFF_2:11
canceled;

theorem Th12: :: AFF_2:12
canceled;

theorem Th13: :: AFF_2:13
canceled;

theorem Th14: :: AFF_2:14
canceled;

theorem Th15: :: AFF_2:15
for b1 being AffinPlane holds
( b1 satisfies_PAP iff b1 satisfies_PAP_1 )
proof end;

theorem Th16: :: AFF_2:16
for b1 being AffinPlane holds
( b1 satisfies_DES iff b1 satisfies_DES_1 )
proof end;

theorem Th17: :: AFF_2:17
for b1 being AffinPlane holds
( b1 satisfies_TDES implies b1 satisfies_TDES_1 )
proof end;

theorem Th18: :: AFF_2:18
for b1 being AffinPlane holds
( b1 satisfies_TDES_1 implies b1 satisfies_TDES_2 )
proof end;

theorem Th19: :: AFF_2:19
for b1 being AffinPlane holds
( b1 satisfies_TDES_2 implies b1 satisfies_TDES_3 )
proof end;

theorem Th20: :: AFF_2:20
for b1 being AffinPlane holds
( b1 satisfies_TDES_3 implies b1 satisfies_TDES )
proof end;

theorem Th21: :: AFF_2:21
for b1 being AffinPlane holds
( b1 satisfies_des iff b1 satisfies_des_1 )
proof end;

theorem Th22: :: AFF_2:22
for b1 being AffinPlane holds
( b1 satisfies_pap iff b1 satisfies_pap_1 )
proof end;

theorem Th23: :: AFF_2:23
for b1 being AffinPlane holds
( b1 satisfies_PAP implies b1 satisfies_pap )
proof end;

theorem Th24: :: AFF_2:24
for b1 being AffinPlane holds
( b1 satisfies_PPAP iff ( b1 satisfies_PAP & b1 satisfies_pap ) )
proof end;

theorem Th25: :: AFF_2:25
for b1 being AffinPlane holds
( b1 satisfies_PAP implies b1 satisfies_DES )
proof end;

theorem Th26: :: AFF_2:26
for b1 being AffinPlane holds
( b1 satisfies_DES implies b1 satisfies_TDES )
proof end;

theorem Th27: :: AFF_2:27
for b1 being AffinPlane holds
( b1 satisfies_TDES_1 implies b1 satisfies_des_1 )
proof end;

theorem Th28: :: AFF_2:28
for b1 being AffinPlane holds
( b1 satisfies_TDES implies b1 satisfies_des )
proof end;

theorem Th29: :: AFF_2:29
for b1 being AffinPlane holds
( b1 satisfies_des implies b1 satisfies_pap )
proof end;