Article HESSENBE, MML version 4.99.1005
:: HESSENBE:th 3
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4 being Element of the carrier of b1
st b2,b3,b4 is_collinear
holds b3,b4,b2 is_collinear & b4,b2,b3 is_collinear & b3,b2,b4 is_collinear & b2,b4,b3 is_collinear & b4,b3,b2 is_collinear;
:: HESSENBE:th 4
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <> b3 & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear
holds b2,b4,b5 is_collinear;
:: HESSENBE:th 5
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b2 <> b3 & b4,b5,b2 is_collinear & b4,b5,b3 is_collinear & b2,b3,b6 is_collinear
holds b4,b5,b6 is_collinear;
:: HESSENBE:th 6
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds ex b4 being Element of the carrier of b1 st
not b2,b3,b4 is_collinear;
:: HESSENBE:th 7
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2 being Element of the carrier of b1 holds
ex b3, b4 being Element of the carrier of b1 st
not b2,b3,b4 is_collinear;
:: HESSENBE:th 8
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5 being Element of the carrier of b1
st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2 <> b5
holds not b2,b5,b4 is_collinear;
:: HESSENBE:th 9
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5 being Element of the carrier of b1
st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b4,b5 is_collinear
holds b2 = b5;
:: HESSENBE:th 10
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st not b2,b3,b4 is_collinear & b2,b4,b5 is_collinear & b3,b4,b6 is_collinear & b4 <> b5 & b7,b5,b6 is_collinear & b2,b3,b7 is_collinear & b2 <> b7
holds b6 <> b4;
:: HESSENBE:th 12
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b3,b4,b6 is_collinear & b2,b4,b7 is_collinear & b6,b5,b7 is_collinear & b5 <> b2 & b5 <> b3 & b6 <> b3 & b6 <> b4
holds b2 <> b7 & b4 <> b7;
:: HESSENBE:th 13
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b4,b6,b5 is_collinear & b6 <> b4 & b5 <> b2
holds not b6,b2,b4 is_collinear;
:: HESSENBE:th 14
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b4,b5,b6 is_collinear & b2 <> b5 & b5 <> b6
holds not b3,b2,b6 is_collinear;
:: HESSENBE:th 15
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b4,b6,b5 is_collinear & b5 <> b6 & b3 <> b5
holds not b5,b3,b6 is_collinear;
:: HESSENBE:th 16
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b4,b6,b2 is_collinear & b2 <> b5 & b2 <> b6
holds not b5,b2,b6 is_collinear;
:: HESSENBE:th 17
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st b2 <> b3 & b4 <> b5 & b4,b5,b6 is_collinear & b4,b5,b7 is_collinear & b2,b3,b6 is_collinear & b2,b3,b7 is_collinear & not b2,b3,b4 is_collinear
holds b6 = b7;
:: HESSENBE:th 19
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b4,b6 is_collinear & b2 <> b5 & b2 <> b6
holds not b2,b5,b6 is_collinear;
:: HESSENBE:th 20
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank Pappian 2-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
st b2 <> b3 & b4 <> b3 & b5 <> b6 & b7 <> b5 & b7 <> b6 & not b4,b2,b7 is_collinear & b4,b2,b3 is_collinear & b7,b5,b6 is_collinear & b4,b5,b8 is_collinear & b7,b2,b8 is_collinear & b4,b6,b9 is_collinear & b3,b7,b9 is_collinear & b2,b6,b10 is_collinear & b3,b5,b10 is_collinear
holds b10,b9,b8 is_collinear;
:: HESSENBE:th 21
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank Pappian 2-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b2 <> b3 & b4 <> b3 & b2 <> b5 & b6 <> b5 & b2 <> b7 & b8 <> b7 & not b2,b4,b6 is_collinear & not b2,b4,b8 is_collinear & not b2,b6,b8 is_collinear & b4,b6,b9 is_collinear & b3,b5,b9 is_collinear & b6,b8,b10 is_collinear & b5,b7,b10 is_collinear & b4,b8,b11 is_collinear & b3,b7,b11 is_collinear & b2,b4,b3 is_collinear & b2,b6,b5 is_collinear & b2,b8,b7 is_collinear
holds b10,b11,b9 is_collinear;
:: HESSENBE:condreg 1
registration
cluster non empty reflexive transitive proper Vebleian at_least_3rank Pappian -> Desarguesian (CollStr);
end;