Article PARDEPAP, MML version 4.99.1005
:: PARDEPAP:th 1
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is Pappian
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st b2,b3 // b2,b4 & b5,b6 // b5,b7 & b2,b6 // b3,b5 & b3,b7 // b4,b6
holds b4,b5 // b2,b7;
:: PARDEPAP:th 2
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is Desarguesian
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st 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
holds b5,b7 // b6,b8;
:: PARDEPAP:th 3
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is translational
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st 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
holds b4,b6 // b5,b7;
:: PARDEPAP:th 5
theorem
ex b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct st
(for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st 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
holds b5,b7 // b6,b8) &
(for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st 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
holds b4,b6 // b5,b7) &
(for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st b2,b3 // b2,b4 & b5,b6 // b5,b7 & b2,b6 // b3,b5 & b3,b7 // b4,b6
holds b4,b5 // b2,b7) &
(for b2, b3, b4, b5 being Element of the carrier of b1
st not b2,b3 // b2,b4 & b2,b3 // b4,b5 & b2,b4 // b3,b5
holds not b2,b5 // b3,b4);
:: PARDEPAP:th 6
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
for b5, b6 being Element of the carrier of b1 holds
b2,b3 // b2,b4 &
(ex b7 being Element of the carrier of b1 st
(b2,b4 // b2,b5 implies b2,b6 // b2,b7 & b4,b6 // b5,b7));