:: CONMETR semantic presentation
definition
let c
1 be
OrtAfPl;
attr a
1 is
satisfying_OPAP means :
Def1:
:: CONMETR:def 1
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of a
1for b
8, b
9 being
Subset of a
1 holds
( b
1 in b
8 & b
2 in b
8 & b
3 in b
8 & b
4 in b
8 & b
1 in b
9 & b
5 in b
9 & b
6 in b
9 & b
7 in b
9 & not b
6 in b
8 & not b
4 in b
9 & b
8 _|_ b
9 & b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
1 <> b
5 & b
1 <> b
6 & b
1 <> b
7 & b
4,b
6 // b
3,b
5 & b
4,b
7 // b
2,b
5 implies b
2,b
6 // b
3,b
7 );
attr a
1 is
satisfying_PAP means :: CONMETR:def 2
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of a
1for b
8, b
9 being
Subset of a
1 holds
( b
8 is_line & b
9 is_line & b
1 in b
8 & b
2 in b
8 & b
3 in b
8 & b
4 in b
8 & b
1 in b
9 & b
5 in b
9 & b
6 in b
9 & b
7 in b
9 & not b
6 in b
8 & not b
4 in b
9 & b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
1 <> b
5 & b
1 <> b
6 & b
1 <> b
7 & b
4,b
6 // b
3,b
5 & b
4,b
7 // b
2,b
5 implies b
2,b
6 // b
3,b
7 );
attr a
1 is
satisfying_MH1 means :
Def3:
:: CONMETR: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 _|_ 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
2 in b
9 & not b
4 in b
9 & b
1,b
2 _|_ b
5,b
6 & b
2,b
3 _|_ b
6,b
7 & b
3,b
4 _|_ b
7,b
8 implies b
1,b
4 _|_ b
5,b
8 );
attr a
1 is
satisfying_MH2 means :
Def4:
:: CONMETR: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 _|_ 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
2 in b
9 & not b
4 in b
9 & b
1,b
2 _|_ b
5,b
6 & b
2,b
3 _|_ b
6,b
7 & b
3,b
4 _|_ b
7,b
8 implies b
1,b
4 _|_ b
5,b
8 );
attr a
1 is
satisfying_TDES means :
Def5:
:: CONMETR:def 5
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of a
1 holds
( b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
1 <> b
5 & b
1 <> b
6 & b
1 <> b
7 & not
LIN b
4,b
5,b
2 & not
LIN b
4,b
5,b
6 &
LIN b
1,b
2,b
3 &
LIN b
1,b
4,b
5 &
LIN b
1,b
6,b
7 & b
2,b
4 // b
3,b
5 & b
2,b
4 // b
1,b
6 & b
4,b
6 // b
5,b
7 implies b
2,b
6 // b
3,b
7 );
attr a
1 is
satisfying_SCH means :: CONMETR:def 6
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_line & b
10 is_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 );
attr a
1 is
satisfying_OSCH means :
Def7:
:: CONMETR:def 7
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 );
attr a
1 is
satisfying_des means :
Def8:
:: CONMETR:def 8
for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( not
LIN b
1,b
2,b
3 & not
LIN b
1,b
2,b
5 & b
1,b
2 // b
3,b
4 & b
1,b
2 // b
5,b
6 & b
1,b
3 // b
2,b
4 & b
1,b
5 // b
2,b
6 implies b
3,b
5 // b
4,b
6 );
end;
:: deftheorem Def1 defines satisfying_OPAP CONMETR:def 1 :
for b
1 being
OrtAfPl holds
( b
1 is
satisfying_OPAP iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1for b
9, b
10 being
Subset of b
1 holds
( b
2 in b
9 & b
3 in b
9 & b
4 in b
9 & b
5 in b
9 & b
2 in b
10 & b
6 in b
10 & b
7 in b
10 & b
8 in b
10 & not b
7 in b
9 & not b
5 in b
10 & b
9 _|_ b
10 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
2 <> b
7 & b
2 <> b
8 & b
5,b
7 // b
4,b
6 & b
5,b
8 // b
3,b
6 implies b
3,b
7 // b
4,b
8 ) );
:: deftheorem Def2 defines satisfying_PAP CONMETR:def 2 :
for b
1 being
OrtAfPl holds
( b
1 is
satisfying_PAP iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1for b
9, b
10 being
Subset of b
1 holds
( b
9 is_line & b
10 is_line & b
2 in b
9 & b
3 in b
9 & b
4 in b
9 & b
5 in b
9 & b
2 in b
10 & b
6 in b
10 & b
7 in b
10 & b
8 in b
10 & not b
7 in b
9 & not b
5 in b
10 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
2 <> b
7 & b
2 <> b
8 & b
5,b
7 // b
4,b
6 & b
5,b
8 // b
3,b
6 implies b
3,b
7 // b
4,b
8 ) );
:: deftheorem Def3 defines satisfying_MH1 CONMETR:def 3 :
for b
1 being
OrtAfPl holds
( b
1 is
satisfying_MH1 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
3 in b
10 & not b
5 in b
10 & b
2,b
3 _|_ b
6,b
7 & b
3,b
4 _|_ b
7,b
8 & b
4,b
5 _|_ b
8,b
9 implies b
2,b
5 _|_ b
6,b
9 ) );
:: deftheorem Def4 defines satisfying_MH2 CONMETR:def 4 :
for b
1 being
OrtAfPl holds
( b
1 is
satisfying_MH2 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
3 in b
10 & not b
5 in b
10 & b
2,b
3 _|_ b
6,b
7 & b
3,b
4 _|_ b
7,b
8 & b
4,b
5 _|_ b
8,b
9 implies b
2,b
5 _|_ b
6,b
9 ) );
:: deftheorem Def5 defines satisfying_TDES CONMETR:def 5 :
for b
1 being
OrtAfPl holds
( b
1 is
satisfying_TDES iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
2 <> b
7 & b
2 <> b
8 & not
LIN b
5,b
6,b
3 & not
LIN b
5,b
6,b
7 &
LIN b
2,b
3,b
4 &
LIN b
2,b
5,b
6 &
LIN b
2,b
7,b
8 & b
3,b
5 // b
4,b
6 & b
3,b
5 // b
2,b
7 & b
5,b
7 // b
6,b
8 implies b
3,b
7 // b
4,b
8 ) );
:: deftheorem Def6 defines satisfying_SCH CONMETR:def 6 :
for b
1 being
OrtAfPl holds
( b
1 is
satisfying_SCH 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_line & b
11 is_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 ) );
:: deftheorem Def7 defines satisfying_OSCH CONMETR:def 7 :
for b
1 being
OrtAfPl holds
( b
1 is
satisfying_OSCH 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 ) );
:: deftheorem Def8 defines satisfying_des CONMETR:def 8 :
for b
1 being
OrtAfPl holds
( b
1 is
satisfying_des iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not
LIN b
2,b
3,b
4 & not
LIN b
2,b
3,b
6 & b
2,b
3 // b
4,b
5 & b
2,b
3 // b
6,b
7 & b
2,b
4 // b
3,b
5 & b
2,b
6 // b
3,b
7 implies b
4,b
6 // b
5,b
7 ) );
theorem Th1: :: CONMETR:1
theorem Th2: :: CONMETR:2
for b
1 being
OrtAfPlfor b
2, b
3 being
Element of b
1 holds
not ( b
2 <> b
3 & ( for b
4 being
Element of b
1 holds
not (
LIN b
2,b
3,b
4 & b
2 <> b
4 & b
3 <> b
4 ) ) )
theorem Th3: :: CONMETR:3
theorem Th4: :: CONMETR:4
theorem Th5: :: CONMETR:5
theorem Th6: :: CONMETR:6
for b
1 being
OrtAfPlfor b
2, b
3, b
4, b
5 being
Element of b
1for b
6 being
Subset of b
1for b
7 being
Subset of
(Af b1)for b
8, b
9 being
Element of
(Af b1) holds
( b
4 = b
8 & b
5 = b
9 & b
6 = b
7 & b
2 in b
6 & b
3 in b
6 & b
8,b
9 // b
7 implies b
4,b
5 // b
2,b
3 )
theorem Th7: :: CONMETR:7
theorem Th8: :: CONMETR:8
theorem Th9: :: CONMETR:9
theorem Th10: :: CONMETR:10
theorem Th11: :: CONMETR:11
theorem Th12: :: CONMETR:12
theorem Th13: :: CONMETR:13
theorem Th14: :: CONMETR:14
theorem Th15: :: CONMETR:15