:: AFF_3 semantic presentation
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES1 means :
Def1:
:: AFF_3:def 1
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, b
11, b
12 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 is_line & b
2 <> b
1 & b
2 <> b
3 & b
1 <> b
3 & b
4 in b
1 & b
5 in b
1 & b
6 in b
1 & b
4 in b
2 & b
7 in b
2 & b
8 in b
2 & b
4 in b
3 & b
9 in b
3 & b
10 in b
3 & b
4 <> b
5 & b
4 <> b
7 & b
4 <> b
9 & b
11 <> b
12 & not
LIN b
7,b
5,b
9 & not
LIN b
8,b
6,b
10 & b
5 <> b
6 &
LIN b
7,b
5,b
11 &
LIN b
8,b
6,b
11 &
LIN b
7,b
9,b
12 &
LIN b
8,b
10,b
12 & b
5,b
9 // b
6,b
10 implies b
5,b
9 // b
11,b
12 );
end;
:: deftheorem Def1 defines satisfying_DES1 AFF_3:def 1 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES1 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, b
12, b
13 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 is_line & b
3 <> b
2 & b
3 <> b
4 & b
2 <> b
4 & b
5 in b
2 & b
6 in b
2 & b
7 in b
2 & b
5 in b
3 & b
8 in b
3 & b
9 in b
3 & b
5 in b
4 & b
10 in b
4 & b
11 in b
4 & b
5 <> b
6 & b
5 <> b
8 & b
5 <> b
10 & b
12 <> b
13 & not
LIN b
8,b
6,b
10 & not
LIN b
9,b
7,b
11 & b
6 <> b
7 &
LIN b
8,b
6,b
12 &
LIN b
9,b
7,b
12 &
LIN b
8,b
10,b
13 &
LIN b
9,b
11,b
13 & b
6,b
10 // b
7,b
11 implies b
6,b
10 // b
12,b
13 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES1_1 means :
Def2:
:: AFF_3:def 2
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, b
11, b
12 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 is_line & b
2 <> b
1 & b
2 <> b
3 & b
1 <> b
3 & b
4 in b
1 & b
5 in b
1 & b
6 in b
1 & b
4 in b
2 & b
7 in b
2 & b
8 in b
2 & b
4 in b
3 & b
9 in b
3 & b
10 in b
3 & b
4 <> b
5 & b
4 <> b
7 & b
4 <> b
9 & b
11 <> b
12 & b
9 <> b
12 & not
LIN b
7,b
5,b
9 & not
LIN b
8,b
6,b
10 &
LIN b
7,b
5,b
11 &
LIN b
8,b
6,b
11 &
LIN b
7,b
9,b
12 &
LIN b
8,b
10,b
12 & b
5,b
9 // b
11,b
12 implies b
5,b
9 // b
6,b
10 );
end;
:: deftheorem Def2 defines satisfying_DES1_1 AFF_3:def 2 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES1_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, b
12, b
13 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 is_line & b
3 <> b
2 & b
3 <> b
4 & b
2 <> b
4 & b
5 in b
2 & b
6 in b
2 & b
7 in b
2 & b
5 in b
3 & b
8 in b
3 & b
9 in b
3 & b
5 in b
4 & b
10 in b
4 & b
11 in b
4 & b
5 <> b
6 & b
5 <> b
8 & b
5 <> b
10 & b
12 <> b
13 & b
10 <> b
13 & not
LIN b
8,b
6,b
10 & not
LIN b
9,b
7,b
11 &
LIN b
8,b
6,b
12 &
LIN b
9,b
7,b
12 &
LIN b
8,b
10,b
13 &
LIN b
9,b
11,b
13 & b
6,b
10 // b
12,b
13 implies b
6,b
10 // b
7,b
11 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES1_2 means :
Def3:
:: AFF_3:def 3
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, b
11, b
12 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 is_line & b
2 <> b
1 & b
2 <> b
3 & b
1 <> b
3 & b
4 in b
1 & b
5 in b
1 & b
6 in b
1 & b
4 in b
2 & b
7 in b
2 & b
8 in b
2 & b
9 in b
3 & b
10 in b
3 & b
4 <> b
5 & b
4 <> b
7 & b
4 <> b
9 & b
11 <> b
12 & not
LIN b
7,b
5,b
9 & not
LIN b
8,b
6,b
10 & b
9 <> b
10 &
LIN b
7,b
5,b
11 &
LIN b
8,b
6,b
11 &
LIN b
7,b
9,b
12 &
LIN b
8,b
10,b
12 & b
5,b
9 // b
6,b
10 & b
5,b
9 // b
11,b
12 implies b
4 in b
3 );
end;
:: deftheorem Def3 defines satisfying_DES1_2 AFF_3:def 3 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES1_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, b
12, b
13 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 is_line & b
3 <> b
2 & b
3 <> b
4 & b
2 <> b
4 & b
5 in b
2 & b
6 in b
2 & b
7 in b
2 & b
5 in b
3 & b
8 in b
3 & b
9 in b
3 & b
10 in b
4 & b
11 in b
4 & b
5 <> b
6 & b
5 <> b
8 & b
5 <> b
10 & b
12 <> b
13 & not
LIN b
8,b
6,b
10 & not
LIN b
9,b
7,b
11 & b
10 <> b
11 &
LIN b
8,b
6,b
12 &
LIN b
9,b
7,b
12 &
LIN b
8,b
10,b
13 &
LIN b
9,b
11,b
13 & b
6,b
10 // b
7,b
11 & b
6,b
10 // b
12,b
13 implies b
5 in b
4 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES1_3 means :
Def4:
:: AFF_3: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, b
11, b
12 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 is_line & b
2 <> b
1 & b
2 <> b
3 & b
1 <> b
3 & 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
4 in b
3 & b
9 in b
3 & b
10 in b
3 & b
4 <> b
5 & b
4 <> b
7 & b
4 <> b
9 & b
11 <> b
12 & not
LIN b
7,b
5,b
9 & not
LIN b
8,b
6,b
10 & b
7 <> b
8 & b
5 <> b
6 &
LIN b
7,b
5,b
11 &
LIN b
8,b
6,b
11 &
LIN b
7,b
9,b
12 &
LIN b
8,b
10,b
12 & b
5,b
9 // b
6,b
10 & b
5,b
9 // b
11,b
12 implies b
4 in b
2 );
end;
:: deftheorem Def4 defines satisfying_DES1_3 AFF_3:def 4 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES1_3 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, b
12, b
13 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 is_line & b
3 <> b
2 & b
3 <> b
4 & b
2 <> b
4 & 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
5 in b
4 & b
10 in b
4 & b
11 in b
4 & b
5 <> b
6 & b
5 <> b
8 & b
5 <> b
10 & b
12 <> b
13 & not
LIN b
8,b
6,b
10 & not
LIN b
9,b
7,b
11 & b
8 <> b
9 & b
6 <> b
7 &
LIN b
8,b
6,b
12 &
LIN b
9,b
7,b
12 &
LIN b
8,b
10,b
13 &
LIN b
9,b
11,b
13 & b
6,b
10 // b
7,b
11 & b
6,b
10 // b
12,b
13 implies b
5 in b
3 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES2 means :
Def5:
:: AFF_3: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, b
11 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 is_line & b
1 <> b
2 & b
1 <> b
3 & b
2 <> b
3 & b
4 in b
1 & b
5 in b
1 & b
6 in b
2 & b
7 in b
2 & b
8 in b
3 & b
9 in b
3 & b
1 // b
2 & b
1 // b
3 & not
LIN b
6,b
4,b
8 & not
LIN b
7,b
5,b
9 & b
10 <> b
11 & b
4 <> b
5 &
LIN b
6,b
4,b
10 &
LIN b
7,b
5,b
10 &
LIN b
6,b
8,b
11 &
LIN b
7,b
9,b
11 & b
4,b
8 // b
5,b
9 implies b
4,b
8 // b
10,b
11 );
end;
:: deftheorem Def5 defines satisfying_DES2 AFF_3:def 5 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES2 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, b
12 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 & b
5 in b
2 & b
6 in b
2 & b
7 in b
3 & b
8 in b
3 & b
9 in b
4 & b
10 in b
4 & b
2 // b
3 & b
2 // b
4 & not
LIN b
7,b
5,b
9 & not
LIN b
8,b
6,b
10 & b
11 <> b
12 & b
5 <> b
6 &
LIN b
7,b
5,b
11 &
LIN b
8,b
6,b
11 &
LIN b
7,b
9,b
12 &
LIN b
8,b
10,b
12 & b
5,b
9 // b
6,b
10 implies b
5,b
9 // b
11,b
12 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES2_1 means :
Def6:
:: AFF_3: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, b
11 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 is_line & b
1 <> b
2 & b
1 <> b
3 & b
2 <> b
3 & b
4 in b
1 & b
5 in b
1 & b
6 in b
2 & b
7 in b
2 & b
8 in b
3 & b
9 in b
3 & b
1 // b
2 & b
1 // b
3 & not
LIN b
6,b
4,b
8 & not
LIN b
7,b
5,b
9 & b
10 <> b
11 &
LIN b
6,b
4,b
10 &
LIN b
7,b
5,b
10 &
LIN b
6,b
8,b
11 &
LIN b
7,b
9,b
11 & b
4,b
8 // b
10,b
11 implies b
4,b
8 // b
5,b
9 );
end;
:: deftheorem Def6 defines satisfying_DES2_1 AFF_3:def 6 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES2_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, b
12 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 & b
5 in b
2 & b
6 in b
2 & b
7 in b
3 & b
8 in b
3 & b
9 in b
4 & b
10 in b
4 & b
2 // b
3 & b
2 // b
4 & not
LIN b
7,b
5,b
9 & not
LIN b
8,b
6,b
10 & b
11 <> b
12 &
LIN b
7,b
5,b
11 &
LIN b
8,b
6,b
11 &
LIN b
7,b
9,b
12 &
LIN b
8,b
10,b
12 & b
5,b
9 // b
11,b
12 implies b
5,b
9 // b
6,b
10 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES2_2 means :
Def7:
:: AFF_3:def 7
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, b
11 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 is_line & b
1 <> b
2 & b
1 <> b
3 & b
2 <> b
3 & b
4 in b
1 & b
5 in b
1 & b
6 in b
2 & b
7 in b
2 & b
8 in b
3 & b
9 in b
3 & b
1 // b
3 & not
LIN b
6,b
4,b
8 & not
LIN b
7,b
5,b
9 & b
10 <> b
11 & b
4 <> b
5 &
LIN b
6,b
4,b
10 &
LIN b
7,b
5,b
10 &
LIN b
6,b
8,b
11 &
LIN b
7,b
9,b
11 & b
4,b
8 // b
5,b
9 & b
4,b
8 // b
10,b
11 implies b
1 // b
2 );
end;
:: deftheorem Def7 defines satisfying_DES2_2 AFF_3:def 7 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES2_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, b
12 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 & b
5 in b
2 & b
6 in b
2 & b
7 in b
3 & b
8 in b
3 & b
9 in b
4 & b
10 in b
4 & b
2 // b
4 & not
LIN b
7,b
5,b
9 & not
LIN b
8,b
6,b
10 & b
11 <> b
12 & b
5 <> b
6 &
LIN b
7,b
5,b
11 &
LIN b
8,b
6,b
11 &
LIN b
7,b
9,b
12 &
LIN b
8,b
10,b
12 & b
5,b
9 // b
6,b
10 & b
5,b
9 // b
11,b
12 implies b
2 // b
3 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_DES2_3 means :
Def8:
:: AFF_3:def 8
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, b
11 being
Element of a
1 holds
( b
1 is_line & b
2 is_line & b
3 is_line & b
1 <> b
2 & b
1 <> b
3 & b
2 <> b
3 & b
4 in b
1 & b
5 in b
1 & b
6 in b
2 & b
7 in b
2 & b
8 in b
3 & b
9 in b
3 & b
1 // b
2 & not
LIN b
6,b
4,b
8 & not
LIN b
7,b
5,b
9 & b
10 <> b
11 & b
8 <> b
9 &
LIN b
6,b
4,b
10 &
LIN b
7,b
5,b
10 &
LIN b
6,b
8,b
11 &
LIN b
7,b
9,b
11 & b
4,b
8 // b
5,b
9 & b
4,b
8 // b
10,b
11 implies b
1 // b
3 );
end;
:: deftheorem Def8 defines satisfying_DES2_3 AFF_3:def 8 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_DES2_3 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, b
12 being
Element of b
1 holds
( b
2 is_line & b
3 is_line & b
4 is_line & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 & b
5 in b
2 & b
6 in b
2 & b
7 in b
3 & b
8 in b
3 & b
9 in b
4 & b
10 in b
4 & b
2 // b
3 & not
LIN b
7,b
5,b
9 & not
LIN b
8,b
6,b
10 & b
11 <> b
12 & b
9 <> b
10 &
LIN b
7,b
5,b
11 &
LIN b
8,b
6,b
11 &
LIN b
7,b
9,b
12 &
LIN b
8,b
10,b
12 & b
5,b
9 // b
6,b
10 & b
5,b
9 // b
11,b
12 implies b
2 // b
4 ) );
theorem Th1: :: AFF_3:1
canceled;
theorem Th2: :: AFF_3:2
canceled;
theorem Th3: :: AFF_3:3
canceled;
theorem Th4: :: AFF_3:4
canceled;
theorem Th5: :: AFF_3:5
canceled;
theorem Th6: :: AFF_3:6
canceled;
theorem Th7: :: AFF_3:7
canceled;
theorem Th8: :: AFF_3:8
canceled;
theorem Th9: :: AFF_3:9
theorem Th10: :: AFF_3:10
theorem Th11: :: AFF_3:11
theorem Th12: :: AFF_3:12
theorem Th13: :: AFF_3:13
theorem Th14: :: AFF_3:14
theorem Th15: :: AFF_3:15
theorem Th16: :: AFF_3:16
theorem Th17: :: AFF_3:17
theorem Th18: :: AFF_3:18