:: PARDEPAP semantic presentation
theorem Th1: :: PARDEPAP:1
for b
1 being
AffinPlane holds
( b
1 satisfies_PAP implies for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
2,b
4 & b
5,b
6 // b
5,b
7 & b
2,b
6 // b
3,b
5 & b
3,b
7 // b
4,b
6 implies b
4,b
5 // b
2,b
7 ) )
theorem Th2: :: PARDEPAP:2
for b
1 being
AffinPlane holds
( b
1 satisfies_DES implies for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( not b
2,b
3 // b
2,b
5 & not b
2,b
3 // b
2,b
7 & 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
3,b
5 // b
4,b
6 & b
3,b
7 // b
4,b
8 implies b
5,b
7 // b
6,b
8 ) )
theorem Th3: :: PARDEPAP:3
for b
1 being
AffinPlane holds
( b
1 satisfies_des implies for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not b
2,b
3 // b
2,b
4 & not b
2,b
3 // b
2,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 Th4: :: PARDEPAP:4
canceled;
theorem Th5: :: PARDEPAP:5
ex b
1 being
AffinPlane st
( ( for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( not b
2,b
3 // b
2,b
5 & not b
2,b
3 // b
2,b
7 & 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
3,b
5 // b
4,b
6 & b
3,b
7 // b
4,b
8 implies b
5,b
7 // b
6,b
8 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not b
2,b
3 // b
2,b
4 & not b
2,b
3 // b
2,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 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
2,b
4 & b
5,b
6 // b
5,b
7 & b
2,b
6 // b
3,b
5 & b
3,b
7 // b
4,b
6 implies b
4,b
5 // b
2,b
7 ) ) & ( for b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( not b
2,b
3 // b
2,b
4 & b
2,b
3 // b
4,b
5 & b
2,b
4 // b
3,b
5 & b
2,b
5 // b
3,b
4 ) ) )
theorem Th6: :: PARDEPAP:6
for b
1 being
AffinPlanefor b
2, b
3 being
Element of b
1 holds
ex b
4 being
Element of b
1 st
for b
5, b
6 being
Element of b
1 holds
( b
2,b
3 // b
2,b
4 & not for b
7 being
Element of b
1 holds
( b
2,b
4 // b
2,b
5 & not ( b
2,b
6 // b
2,b
7 & b
4,b
6 // b
5,b
7 ) ) )