:: CONMETR1 semantic presentation
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_minor_Scherungssatz means :
Def1:
:: CONMETR1:def 1
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1for b
9, b
10 being
Subset of a
1 holds
( b
9 // b
10 & b
1 in b
9 & b
3 in b
9 & b
5 in b
9 & b
7 in b
9 & b
2 in b
10 & b
4 in b
10 & b
6 in b
10 & b
8 in b
10 & not b
4 in b
9 & not b
2 in b
9 & not b
6 in b
9 & not b
8 in b
9 & not b
1 in b
10 & not b
3 in b
10 & not b
5 in b
10 & not b
7 in b
10 & b
3,b
2 // b
7,b
6 & b
2,b
1 // b
6,b
5 & b
1,b
4 // b
5,b
8 implies b
3,b
4 // b
7,b
8 );
end;
:: deftheorem Def1 defines satisfying_minor_Scherungssatz CONMETR1:def 1 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_minor_Scherungssatz iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1for b
10, b
11 being
Subset of b
1 holds
( b
10 // b
11 & b
2 in b
10 & b
4 in b
10 & b
6 in b
10 & b
8 in b
10 & b
3 in b
11 & b
5 in b
11 & b
7 in b
11 & b
9 in b
11 & not b
5 in b
10 & not b
3 in b
10 & not b
7 in b
10 & not b
9 in b
10 & not b
2 in b
11 & not b
4 in b
11 & not b
6 in b
11 & not b
8 in b
11 & b
4,b
3 // b
8,b
7 & b
3,b
2 // b
7,b
6 & b
2,b
5 // b
6,b
9 implies b
4,b
5 // b
8,b
9 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_major_Scherungssatz means :
Def2:
:: CONMETR1:def 2
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of a
1for b
10, b
11 being
Subset of a
1 holds
( b
10 is
being_line & b
11 is
being_line & b
1 in b
10 & b
1 in b
11 & b
2 in b
10 & b
4 in b
10 & b
6 in b
10 & b
8 in b
10 & b
3 in b
11 & b
5 in b
11 & b
7 in b
11 & b
9 in b
11 & not b
5 in b
10 & not b
3 in b
10 & not b
7 in b
10 & not b
9 in b
10 & not b
2 in b
11 & not b
4 in b
11 & not b
6 in b
11 & not b
8 in b
11 & b
4,b
3 // b
8,b
7 & b
3,b
2 // b
7,b
6 & b
2,b
5 // b
6,b
9 implies b
4,b
5 // b
8,b
9 );
end;
:: deftheorem Def2 defines satisfying_major_Scherungssatz CONMETR1:def 2 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_major_Scherungssatz iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11, b
12 being
Subset of b
1 holds
( b
11 is
being_line & b
12 is
being_line & b
2 in b
11 & b
2 in b
12 & b
3 in b
11 & b
5 in b
11 & b
7 in b
11 & b
9 in b
11 & b
4 in b
12 & b
6 in b
12 & b
8 in b
12 & b
10 in b
12 & not b
6 in b
11 & not b
4 in b
11 & not b
8 in b
11 & not b
10 in b
11 & not b
3 in b
12 & not b
5 in b
12 & not b
7 in b
12 & not b
9 in b
12 & b
5,b
4 // b
9,b
8 & b
4,b
3 // b
8,b
7 & b
3,b
6 // b
7,b
10 implies b
5,b
6 // b
9,b
10 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_Scherungssatz means :
Def3:
:: CONMETR1:def 3
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1for b
9, b
10 being
Subset of a
1 holds
( b
9 is
being_line & b
10 is
being_line & b
1 in b
9 & b
3 in b
9 & b
5 in b
9 & b
7 in b
9 & b
2 in b
10 & b
4 in b
10 & b
6 in b
10 & b
8 in b
10 & not b
4 in b
9 & not b
2 in b
9 & not b
6 in b
9 & not b
8 in b
9 & not b
1 in b
10 & not b
3 in b
10 & not b
5 in b
10 & not b
7 in b
10 & b
3,b
2 // b
7,b
6 & b
2,b
1 // b
6,b
5 & b
1,b
4 // b
5,b
8 implies b
3,b
4 // b
7,b
8 );
end;
:: deftheorem Def3 defines satisfying_Scherungssatz CONMETR1:def 3 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_Scherungssatz iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1for b
10, b
11 being
Subset of b
1 holds
( b
10 is
being_line & b
11 is
being_line & b
2 in b
10 & b
4 in b
10 & b
6 in b
10 & b
8 in b
10 & b
3 in b
11 & b
5 in b
11 & b
7 in b
11 & b
9 in b
11 & not b
5 in b
10 & not b
3 in b
10 & not b
7 in b
10 & not b
9 in b
10 & not b
2 in b
11 & not b
4 in b
11 & not b
6 in b
11 & not b
8 in b
11 & b
4,b
3 // b
8,b
7 & b
3,b
2 // b
7,b
6 & b
2,b
5 // b
6,b
9 implies b
4,b
5 // b
8,b
9 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_indirect_Scherungssatz means :
Def4:
:: CONMETR1:def 4
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1for b
9, b
10 being
Subset of a
1 holds
( b
9 is
being_line & b
10 is
being_line & b
1 in b
9 & b
3 in b
9 & b
6 in b
9 & b
8 in b
9 & b
2 in b
10 & b
4 in b
10 & b
5 in b
10 & b
7 in b
10 & not b
4 in b
9 & not b
2 in b
9 & not b
5 in b
9 & not b
7 in b
9 & not b
1 in b
10 & not b
3 in b
10 & not b
6 in b
10 & not b
8 in b
10 & b
3,b
2 // b
7,b
6 & b
2,b
1 // b
6,b
5 & b
1,b
4 // b
5,b
8 implies b
3,b
4 // b
7,b
8 );
end;
:: deftheorem Def4 defines satisfying_indirect_Scherungssatz CONMETR1:def 4 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_indirect_Scherungssatz iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1for b
10, b
11 being
Subset of b
1 holds
( b
10 is
being_line & b
11 is
being_line & b
2 in b
10 & b
4 in b
10 & b
7 in b
10 & b
9 in b
10 & b
3 in b
11 & b
5 in b
11 & b
6 in b
11 & b
8 in b
11 & not b
5 in b
10 & not b
3 in b
10 & not b
6 in b
10 & not b
8 in b
10 & not b
2 in b
11 & not b
4 in b
11 & not b
7 in b
11 & not b
9 in b
11 & b
4,b
3 // b
8,b
7 & b
3,b
2 // b
7,b
6 & b
2,b
5 // b
6,b
9 implies b
4,b
5 // b
8,b
9 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_minor_indirect_Scherungssatz means :
Def5:
:: CONMETR1:def 5
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1for b
9, b
10 being
Subset of a
1 holds
( b
9 // b
10 & b
1 in b
9 & b
3 in b
9 & b
6 in b
9 & b
8 in b
9 & b
2 in b
10 & b
4 in b
10 & b
5 in b
10 & b
7 in b
10 & not b
4 in b
9 & not b
2 in b
9 & not b
5 in b
9 & not b
7 in b
9 & not b
1 in b
10 & not b
3 in b
10 & not b
6 in b
10 & not b
8 in b
10 & b
3,b
2 // b
7,b
6 & b
2,b
1 // b
6,b
5 & b
1,b
4 // b
5,b
8 implies b
3,b
4 // b
7,b
8 );
end;
:: deftheorem Def5 defines satisfying_minor_indirect_Scherungssatz CONMETR1:def 5 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_minor_indirect_Scherungssatz iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1for b
10, b
11 being
Subset of b
1 holds
( b
10 // b
11 & b
2 in b
10 & b
4 in b
10 & b
7 in b
10 & b
9 in b
10 & b
3 in b
11 & b
5 in b
11 & b
6 in b
11 & b
8 in b
11 & not b
5 in b
10 & not b
3 in b
10 & not b
6 in b
10 & not b
8 in b
10 & not b
2 in b
11 & not b
4 in b
11 & not b
7 in b
11 & not b
9 in b
11 & b
4,b
3 // b
8,b
7 & b
3,b
2 // b
7,b
6 & b
2,b
5 // b
6,b
9 implies b
4,b
5 // b
8,b
9 ) );
definition
let c
1 be
AffinPlane;
attr a
1 is
satisfying_major_indirect_Scherungssatz means :
Def6:
:: CONMETR1:def 6
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of a
1for b
10, b
11 being
Subset of a
1 holds
( b
10 is
being_line & b
11 is
being_line & b
1 in b
10 & b
1 in b
11 & b
2 in b
10 & b
4 in b
10 & b
7 in b
10 & b
9 in b
10 & b
3 in b
11 & b
5 in b
11 & b
6 in b
11 & b
8 in b
11 & not b
5 in b
10 & not b
3 in b
10 & not b
6 in b
10 & not b
8 in b
10 & not b
2 in b
11 & not b
4 in b
11 & not b
7 in b
11 & not b
9 in b
11 & b
4,b
3 // b
8,b
7 & b
3,b
2 // b
7,b
6 & b
2,b
5 // b
6,b
9 implies b
4,b
5 // b
8,b
9 );
end;
:: deftheorem Def6 defines satisfying_major_indirect_Scherungssatz CONMETR1:def 6 :
for b
1 being
AffinPlane holds
( b
1 is
satisfying_major_indirect_Scherungssatz iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1for b
11, b
12 being
Subset of b
1 holds
( b
11 is
being_line & b
12 is
being_line & b
2 in b
11 & b
2 in b
12 & b
3 in b
11 & b
5 in b
11 & b
8 in b
11 & b
10 in b
11 & b
4 in b
12 & b
6 in b
12 & b
7 in b
12 & b
9 in b
12 & not b
6 in b
11 & not b
4 in b
11 & not b
7 in b
11 & not b
9 in b
11 & not b
3 in b
12 & not b
5 in b
12 & not b
8 in b
12 & not b
10 in b
12 & b
5,b
4 // b
9,b
8 & b
4,b
3 // b
8,b
7 & b
3,b
6 // b
7,b
10 implies b
5,b
6 // b
9,b
10 ) );
theorem Th1: :: CONMETR1:1
theorem Th2: :: CONMETR1:2
theorem Th3: :: CONMETR1:3
theorem Th4: :: CONMETR1:4
theorem Th5: :: CONMETR1:5
theorem Th6: :: CONMETR1:6
theorem Th7: :: CONMETR1:7
theorem Th8: :: CONMETR1:8
theorem Th9: :: CONMETR1:9
theorem Th10: :: CONMETR1:10
theorem Th11: :: CONMETR1:11
theorem Th12: :: CONMETR1:12
theorem Th13: :: CONMETR1:13
theorem Th14: :: CONMETR1:14
theorem Th15: :: CONMETR1:15
theorem Th16: :: CONMETR1:16
theorem Th17: :: CONMETR1:17