:: AFF_2 semantic presentation
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_PPAP means :
Def1:
:: AFF_2:def 1
for b
1, b
2 being
Subset of a
1for b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 in b
1 & b
4 in b
1 & b
5 in b
1 & b
6 in b
2 & b
7 in b
2 & b
8 in b
2 & b
3,b
7 // b
4,b
6 & b
4,b
8 // b
5,b
7 implies b
3,b
8 // b
5,b
6 );
end;
:: deftheorem Def1 defines satisfying_PPAP AFF_2:def 1 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_PPAP iff for b
2, b
3 being
Subset of b
1for b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 in b
2 & b
5 in b
2 & b
6 in b
2 & b
7 in b
3 & b
8 in b
3 & b
9 in b
3 & b
4,b
8 // b
5,b
7 & b
5,b
9 // b
6,b
8 implies b
4,b
9 // b
6,b
7 ) );
definition
let c
1 be
AffinSpace;
attr a
1 is
Pappian means :
Def2:
:: AFF_2:def 2
for b
1, b
2 being
Subset of a
1for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
1 <> b
2 & b
3 in b
1 & b
3 in b
2 & b
3 <> b
4 & b
3 <> b
7 & b
3 <> b
5 & b
3 <> b
8 & b
3 <> b
6 & b
3 <> b
9 & b
4 in b
1 & b
5 in b
1 & b
6 in b
1 & b
7 in b
2 & b
8 in b
2 & b
9 in b
2 & b
4,b
8 // b
5,b
7 & b
5,b
9 // b
6,b
8 implies b
4,b
9 // b
6,b
7 );
end;
:: deftheorem Def2 defines Pappian AFF_2:def 2 :
for b
1 being
AffinSpace holds
( b
1 is
Pappian iff for b
2, b
3 being
Subset of b
1for b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
2 <> b
3 & b
4 in b
2 & b
4 in b
3 & b
4 <> b
5 & b
4 <> b
8 & b
4 <> b
6 & b
4 <> b
9 & b
4 <> b
7 & b
4 <> b
10 & b
5 in b
2 & b
6 in b
2 & b
7 in b
2 & b
8 in b
3 & b
9 in b
3 & b
10 in b
3 & b
5,b
9 // b
6,b
8 & b
6,b
10 // b
7,b
9 implies b
5,b
10 // b
7,b
8 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_PAP_1 means :
Def3:
:: AFF_2:def 3
for b
1, b
2 being
Subset of a
1for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
1 <> b
2 & b
3 in b
1 & b
3 in b
2 & b
3 <> b
4 & b
3 <> b
7 & b
3 <> b
5 & b
3 <> b
8 & b
3 <> b
6 & b
3 <> b
9 & b
4 in b
1 & b
5 in b
1 & b
6 in b
1 & b
8 in b
2 & b
9 in b
2 & b
4,b
8 // b
5,b
7 & b
5,b
9 // b
6,b
8 & b
4,b
9 // b
6,b
7 & b
5 <> b
6 implies b
7 in b
2 );
end;
:: deftheorem Def3 defines satisfying_PAP_1 AFF_2:def 3 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_PAP_1 iff for b
2, b
3 being
Subset of b
1for b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
2 <> b
3 & b
4 in b
2 & b
4 in b
3 & b
4 <> b
5 & b
4 <> b
8 & b
4 <> b
6 & b
4 <> b
9 & b
4 <> b
7 & b
4 <> b
10 & b
5 in b
2 & b
6 in b
2 & b
7 in b
2 & b
9 in b
3 & b
10 in b
3 & b
5,b
9 // b
6,b
8 & b
6,b
10 // b
7,b
9 & b
5,b
10 // b
7,b
8 & b
6 <> b
7 implies b
8 in b
3 ) );
definition
let c
1 be
AffinSpace;
attr a
1 is
Desarguesian means :
Def4:
:: AFF_2:def 4
for b
1, b
2, b
3 being
Subset of a
1for b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of a
1 holds
( b
4 in b
1 & b
4 in b
2 & b
4 in b
3 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
5 in b
1 & b
8 in b
1 & b
6 in b
2 & b
9 in b
2 & b
7 in b
3 & b
10 in b
3 & b
1 is_line & b
2 is_line & b
3 is_line & b
1 <> b
2 & b
1 <> b
3 & b
5,b
6 // b
8,b
9 & b
5,b
7 // b
8,b
10 implies b
6,b
7 // b
9,b
10 );
end;
:: deftheorem Def4 defines Desarguesian AFF_2:def 4 :
for b
1 being
AffinSpace holds
( b
1 is
Desarguesian iff for b
2, b
3, b
4 being
Subset of b
1for b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of b
1 holds
( b
5 in b
2 & b
5 in b
3 & b
5 in b
4 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 in b
2 & b
9 in b
2 & b
7 in b
3 & b
10 in b
3 & b
8 in b
4 & b
11 in b
4 & b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
6,b
7 // b
9,b
10 & b
6,b
8 // b
9,b
11 implies b
7,b
8 // b
10,b
11 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES_1 means :
Def5:
:: AFF_2:def 5
for b
1, b
2, b
3 being
Subset of a
1for b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of a
1 holds
( b
4 in b
1 & b
4 in b
2 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
5 in b
1 & b
8 in b
1 & b
6 in b
2 & b
9 in b
2 & b
7 in b
3 & b
10 in b
3 & b
1 is_line & b
2 is_line & b
3 is_line & b
1 <> b
2 & b
1 <> b
3 & b
5,b
6 // b
8,b
9 & b
5,b
7 // b
8,b
10 & b
6,b
7 // b
9,b
10 & not
LIN b
5,b
6,b
7 & b
7 <> b
10 implies b
4 in b
3 );
end;
:: deftheorem Def5 defines satisfying_DES_1 AFF_2:def 5 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES_1 iff for b
2, b
3, b
4 being
Subset of b
1for b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of b
1 holds
( b
5 in b
2 & b
5 in b
3 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 in b
2 & b
9 in b
2 & b
7 in b
3 & b
10 in b
3 & b
8 in b
4 & b
11 in b
4 & b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
6,b
7 // b
9,b
10 & b
6,b
8 // b
9,b
11 & b
7,b
8 // b
10,b
11 & not
LIN b
6,b
7,b
8 & b
8 <> b
11 implies b
5 in b
4 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES_2 means :: AFF_2:def 6
for b
1, b
2, b
3 being
Subset of a
1for b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of a
1 holds
( b
4 in b
1 & b
4 in b
2 & b
4 in b
3 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
5 in b
1 & b
8 in b
1 & b
6 in b
2 & b
9 in b
2 & b
7 in b
3 & b
1 is_line & b
2 is_line & b
3 is_line & b
1 <> b
2 & b
1 <> b
3 & b
5,b
6 // b
8,b
9 & b
5,b
7 // b
8,b
10 & b
6,b
7 // b
9,b
10 implies b
10 in b
3 );
end;
:: deftheorem Def6 defines satisfying_DES_2 AFF_2:def 6 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES_2 iff for b
2, b
3, b
4 being
Subset of b
1for b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of b
1 holds
( b
5 in b
2 & b
5 in b
3 & b
5 in b
4 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 in b
2 & b
9 in b
2 & b
7 in b
3 & b
10 in b
3 & b
8 in b
4 & b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
6,b
7 // b
9,b
10 & b
6,b
8 // b
9,b
11 & b
7,b
8 // b
10,b
11 implies b
11 in b
4 ) );
definition
let c
1 be
AffinSpace;
attr a
1 is
Moufangian means :
Def7:
:: AFF_2:def 7
for b
1 being
Subset of a
1for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( b
1 is_line & b
2 in b
1 & b
5 in b
1 & b
8 in b
1 & not b
3 in b
1 & b
2 <> b
5 & b
3 <> b
4 &
LIN b
2,b
3,b
6 &
LIN b
2,b
4,b
7 & b
3,b
4 // b
6,b
7 & b
3,b
5 // b
6,b
8 & b
3,b
4 // b
1 implies b
4,b
5 // b
7,b
8 );
end;
:: deftheorem Def7 defines Moufangian AFF_2:def 7 :
for b
1 being
AffinSpace holds
( b
1 is
Moufangian iff for b
2 being
Subset of b
1for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
2 is_line & b
3 in b
2 & b
6 in b
2 & b
9 in b
2 & not b
4 in b
2 & b
3 <> b
6 & b
4 <> b
5 &
LIN b
3,b
4,b
7 &
LIN b
3,b
5,b
8 & b
4,b
5 // b
7,b
8 & b
4,b
6 // b
7,b
9 & b
4,b
5 // b
2 implies b
5,b
6 // b
8,b
9 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_TDES_1 means :
Def8:
:: AFF_2:def 8
for b
1 being
Subset of a
1for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( b
1 is_line & b
2 in b
1 & b
5 in b
1 & b
8 in b
1 & not b
3 in b
1 & b
2 <> b
5 & b
3 <> b
4 &
LIN b
2,b
3,b
6 & b
3,b
4 // b
6,b
7 & b
4,b
5 // b
7,b
8 & b
3,b
5 // b
6,b
8 & b
3,b
4 // b
1 implies
LIN b
2,b
4,b
7 );
end;
:: deftheorem Def8 defines satisfying_TDES_1 AFF_2:def 8 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_TDES_1 iff for b
2 being
Subset of b
1for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
2 is_line & b
3 in b
2 & b
6 in b
2 & b
9 in b
2 & not b
4 in b
2 & b
3 <> b
6 & b
4 <> b
5 &
LIN b
3,b
4,b
7 & b
4,b
5 // b
7,b
8 & b
5,b
6 // b
8,b
9 & b
4,b
6 // b
7,b
9 & b
4,b
5 // b
2 implies
LIN b
3,b
5,b
8 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_TDES_2 means :
Def9:
:: AFF_2:def 9
for b
1 being
Subset of a
1for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( b
1 is_line & b
2 in b
1 & b
5 in b
1 & b
8 in b
1 & not b
3 in b
1 & b
2 <> b
5 & b
3 <> b
4 &
LIN b
2,b
3,b
6 &
LIN b
2,b
4,b
7 & b
4,b
5 // b
7,b
8 & b
3,b
5 // b
6,b
8 & b
3,b
4 // b
1 implies b
3,b
4 // b
6,b
7 );
end;
:: deftheorem Def9 defines satisfying_TDES_2 AFF_2:def 9 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_TDES_2 iff for b
2 being
Subset of b
1for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
2 is_line & b
3 in b
2 & b
6 in b
2 & b
9 in b
2 & not b
4 in b
2 & b
3 <> b
6 & b
4 <> b
5 &
LIN b
3,b
4,b
7 &
LIN b
3,b
5,b
8 & b
5,b
6 // b
8,b
9 & b
4,b
6 // b
7,b
9 & b
4,b
5 // b
2 implies b
4,b
5 // b
7,b
8 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_TDES_3 means :
Def10:
:: AFF_2:def 10
for b
1 being
Subset of a
1for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( b
1 is_line & b
2 in b
1 & b
5 in b
1 & not b
3 in b
1 & b
2 <> b
5 & b
3 <> b
4 &
LIN b
2,b
3,b
6 &
LIN b
2,b
4,b
7 & b
3,b
4 // b
6,b
7 & b
3,b
5 // b
6,b
8 & b
4,b
5 // b
7,b
8 & b
3,b
4 // b
1 implies b
8 in b
1 );
end;
:: deftheorem Def10 defines satisfying_TDES_3 AFF_2:def 10 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_TDES_3 iff for b
2 being
Subset of b
1for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
2 is_line & b
3 in b
2 & b
6 in b
2 & not b
4 in b
2 & b
3 <> b
6 & b
4 <> b
5 &
LIN b
3,b
4,b
7 &
LIN b
3,b
5,b
8 & b
4,b
5 // b
7,b
8 & b
4,b
6 // b
7,b
9 & b
5,b
6 // b
8,b
9 & b
4,b
5 // b
2 implies b
9 in b
2 ) );
definition
let c
1 be
AffinSpace;
attr a
1 is
translational means :
Def11:
:: AFF_2:def 11
for b
1, b
2, b
3 being
Subset of a
1for b
4, b
5, b
6, b
7, b
8, b
9 being
Element of a
1 holds
( b
1 // b
2 & b
1 // b
3 & b
4 in b
1 & b
7 in b
1 & b
5 in b
2 & b
8 in b
2 & b
6 in b
3 & b
9 in b
3 & b
1 is_line & b
2 is_line & b
3 is_line & b
1 <> b
2 & b
1 <> b
3 & b
4,b
5 // b
7,b
8 & b
4,b
6 // b
7,b
9 implies b
5,b
6 // b
8,b
9 );
end;
:: deftheorem Def11 defines translational AFF_2:def 11 :
for b
1 being
AffinSpace holds
( b
1 is
translational iff for b
2, b
3, b
4 being
Subset of b
1for b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1 holds
( b
2 // b
3 & b
2 // b
4 & b
5 in b
2 & b
8 in b
2 & b
6 in b
3 & b
9 in b
3 & b
7 in b
4 & b
10 in b
4 & b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
5,b
6 // b
8,b
9 & b
5,b
7 // b
8,b
10 implies b
6,b
7 // b
9,b
10 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_des_1 means :
Def12:
:: AFF_2:def 12
for b
1, b
2, b
3 being
Subset of a
1for b
4, b
5, b
6, b
7, b
8, b
9 being
Element of a
1 holds
( b
1 // b
2 & b
4 in b
1 & b
7 in b
1 & b
5 in b
2 & b
8 in b
2 & b
6 in b
3 & b
9 in b
3 & b
1 is_line & b
2 is_line & b
3 is_line & b
1 <> b
2 & b
1 <> b
3 & b
4,b
5 // b
7,b
8 & b
4,b
6 // b
7,b
9 & b
5,b
6 // b
8,b
9 & not
LIN b
4,b
5,b
6 & b
6 <> b
9 implies b
1 // b
3 );
end;
:: deftheorem Def12 defines satisfying_des_1 AFF_2:def 12 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_des_1 iff for b
2, b
3, b
4 being
Subset of b
1for b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1 holds
( b
2 // b
3 & b
5 in b
2 & b
8 in b
2 & b
6 in b
3 & b
9 in b
3 & b
7 in b
4 & b
10 in b
4 & b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
5,b
6 // b
8,b
9 & b
5,b
7 // b
8,b
10 & b
6,b
7 // b
9,b
10 & not
LIN b
5,b
6,b
7 & b
7 <> b
10 implies b
2 // b
4 ) );
definition
let c
1 be
AffinSpace;
attr a
1 is
satisfying_pap means :
Def13:
:: AFF_2:def 13
for b
1, b
2 being
Subset of a
1for b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 in b
1 & b
4 in b
1 & b
5 in b
1 & b
1 // b
2 & b
1 <> b
2 & b
6 in b
2 & b
7 in b
2 & b
8 in b
2 & b
3,b
7 // b
4,b
6 & b
4,b
8 // b
5,b
7 implies b
3,b
8 // b
5,b
6 );
end;
:: deftheorem Def13 defines satisfying_pap AFF_2:def 13 :
for b
1 being
AffinSpace holds
( b
1 is
satisfying_pap iff for b
2, b
3 being
Subset of b
1for b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 in b
2 & b
5 in b
2 & b
6 in b
2 & b
2 // b
3 & b
2 <> b
3 & b
7 in b
3 & b
8 in b
3 & b
9 in b
3 & b
4,b
8 // b
5,b
7 & b
5,b
9 // b
6,b
8 implies b
4,b
9 // b
6,b
7 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_pap_1 means :
Def14:
:: AFF_2:def 14
for b
1, b
2 being
Subset of a
1for b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 in b
1 & b
4 in b
1 & b
5 in b
1 & b
1 // b
2 & b
1 <> b
2 & b
6 in b
2 & b
7 in b
2 & b
3,b
7 // b
4,b
6 & b
4,b
8 // b
5,b
7 & b
3,b
8 // b
5,b
6 & b
6 <> b
7 implies b
8 in b
2 );
end;
:: deftheorem Def14 defines satisfying_pap_1 AFF_2:def 14 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_pap_1 iff for b
2, b
3 being
Subset of b
1for b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 in b
2 & b
5 in b
2 & b
6 in b
2 & b
2 // b
3 & b
2 <> b
3 & b
7 in b
3 & b
8 in b
3 & b
4,b
8 // b
5,b
7 & b
5,b
9 // b
6,b
8 & b
4,b
9 // b
6,b
7 & b
7 <> b
8 implies b
9 in b
3 ) );
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
theorem Th16: :: AFF_2:16
theorem Th17: :: AFF_2:17
theorem Th18: :: AFF_2:18
theorem Th19: :: AFF_2:19
theorem Th20: :: AFF_2:20
theorem Th21: :: AFF_2:21
theorem Th22: :: AFF_2:22
theorem Th23: :: AFF_2:23
theorem Th24: :: AFF_2:24
theorem Th25: :: AFF_2:25
theorem Th26: :: AFF_2:26
theorem Th27: :: AFF_2:27
theorem Th28: :: AFF_2:28
theorem Th29: :: AFF_2:29