:: Elementary Variants of Affine Configurational Theorems
:: by Krzysztof Pra\.zmowski and Krzysztof Radziszewski
::
:: Copyright (c) 1990-2021 Association of Mizar Users

theorem Th1: :: PARDEPAP:1
for SAS being AffinPlane st SAS is Pappian holds
for a1, a2, a3, b1, b2, b3 being Element of SAS st a1,a2 // a1,a3 & b1,b2 // b1,b3 & a1,b2 // a2,b1 & a2,b3 // a3,b2 holds
a3,b1 // a1,b3
proof end;

theorem Th2: :: PARDEPAP:2
for SAS being AffinPlane st SAS is Desarguesian holds
for o, a, a9, b, b9, c, c9 being Element of SAS st not o,a // o,b & not o,a // o,c & o,a // o,a9 & o,b // o,b9 & o,c // o,c9 & a,b // a9,b9 & a,c // a9,c9 holds
b,c // b9,c9
proof end;

theorem Th3: :: PARDEPAP:3
for SAS being AffinPlane st SAS is translational holds
for a, a9, b, b9, c, c9 being Element of SAS st not a,a9 // a,b & not a,a9 // a,c & a,a9 // b,b9 & a,a9 // c,c9 & a,b // a9,b9 & a,c // a9,c9 holds
b,c // b9,c9
proof end;

theorem :: PARDEPAP:4
ex SAS being AffinPlane st
( ( for o, a, a9, b, b9, c, c9 being Element of SAS st not o,a // o,b & not o,a // o,c & o,a // o,a9 & o,b // o,b9 & o,c // o,c9 & a,b // a9,b9 & a,c // a9,c9 holds
b,c // b9,c9 ) & ( for a, a9, b, b9, c, c9 being Element of SAS st not a,a9 // a,b & not a,a9 // a,c & a,a9 // b,b9 & a,a9 // c,c9 & a,b // a9,b9 & a,c // a9,c9 holds
b,c // b9,c9 ) & ( for a1, a2, a3, b1, b2, b3 being Element of SAS st a1,a2 // a1,a3 & b1,b2 // b1,b3 & a1,b2 // a2,b1 & a2,b3 // a3,b2 holds
a3,b1 // a1,b3 ) & ( for a, b, c, d being Element of SAS st not a,b // a,c & a,b // c,d & a,c // b,d holds
not a,d // b,c ) )
proof end;

theorem :: PARDEPAP:5
for SAS being AffinPlane
for o, a being Element of SAS ex p being Element of SAS st
for b, c being Element of SAS holds
( o,a // o,p & ex d being Element of SAS st
( o,p // o,b implies ( o,c // o,d & p,c // b,d ) ) )
proof end;