:: PARDEPAP semantic presentation

theorem Th1: :: PARDEPAP:1
for b1 being AffinPlane holds
( b1 satisfies_PAP implies for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2,b3 // b2,b4 & b5,b6 // b5,b7 & b2,b6 // b3,b5 & b3,b7 // b4,b6 implies b4,b5 // b2,b7 ) )
proof end;

theorem Th2: :: PARDEPAP:2
for b1 being AffinPlane holds
( b1 satisfies_DES implies for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
( not b2,b3 // b2,b5 & not b2,b3 // b2,b7 & b2,b3 // b2,b4 & b2,b5 // b2,b6 & b2,b7 // b2,b8 & b3,b5 // b4,b6 & b3,b7 // b4,b8 implies b5,b7 // b6,b8 ) )
proof end;

theorem Th3: :: PARDEPAP:3
for b1 being AffinPlane holds
( b1 satisfies_des implies for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( not b2,b3 // b2,b4 & not b2,b3 // b2,b6 & b2,b3 // b4,b5 & b2,b3 // b6,b7 & b2,b4 // b3,b5 & b2,b6 // b3,b7 implies b4,b6 // b5,b7 ) )
proof end;

theorem Th4: :: PARDEPAP:4
canceled;

theorem Th5: :: PARDEPAP:5
ex b1 being AffinPlane st
( ( for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
( not b2,b3 // b2,b5 & not b2,b3 // b2,b7 & b2,b3 // b2,b4 & b2,b5 // b2,b6 & b2,b7 // b2,b8 & b3,b5 // b4,b6 & b3,b7 // b4,b8 implies b5,b7 // b6,b8 ) ) & ( for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( not b2,b3 // b2,b4 & not b2,b3 // b2,b6 & b2,b3 // b4,b5 & b2,b3 // b6,b7 & b2,b4 // b3,b5 & b2,b6 // b3,b7 implies b4,b6 // b5,b7 ) ) & ( for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2,b3 // b2,b4 & b5,b6 // b5,b7 & b2,b6 // b3,b5 & b3,b7 // b4,b6 implies b4,b5 // b2,b7 ) ) & ( for b2, b3, b4, b5 being Element of b1 holds
not ( not b2,b3 // b2,b4 & b2,b3 // b4,b5 & b2,b4 // b3,b5 & b2,b5 // b3,b4 ) ) )
proof end;

theorem Th6: :: PARDEPAP:6
for b1 being AffinPlane
for b2, b3 being Element of b1 holds
ex b4 being Element of b1 st
for b5, b6 being Element of b1 holds
( b2,b3 // b2,b4 & not for b7 being Element of b1 holds
( b2,b4 // b2,b5 & not ( b2,b6 // b2,b7 & b4,b6 // b5,b7 ) ) )
proof end;