Article CONMETR1, MML version 4.99.1005
:: CONMETR1:attrnot 1 => CONMETR1:attr 1
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_minor_Scherungssatz means
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
for b9, b10 being Element of bool the carrier of a1
st b9 // b10 & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b4 in b9 & not b2 in b9 & not b6 in b9 & not b8 in b9 & not b1 in b10 & not b3 in b10 & not b5 in b10 & not b7 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
holds b3,b4 // b7,b8;
end;
:: CONMETR1:dfs 1
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_minor_Scherungssatz
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
for b9, b10 being Element of bool the carrier of a1
st b9 // b10 & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b4 in b9 & not b2 in b9 & not b6 in b9 & not b8 in b9 & not b1 in b10 & not b3 in b10 & not b5 in b10 & not b7 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
holds b3,b4 // b7,b8;
:: CONMETR1:def 1
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_minor_Scherungssatz
iff
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
for b10, b11 being Element of bool the carrier of b1
st b10 // b11 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & b3 in b11 & b5 in b11 & b7 in b11 & b9 in b11 & not b5 in b10 & not b3 in b10 & not b7 in b10 & not b9 in b10 & not b2 in b11 & not b4 in b11 & not b6 in b11 & not b8 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
holds b4,b5 // b8,b9;
:: CONMETR1:prednot 1 => CONMETR1:attr 1
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_minor_SCH for satisfying_minor_Scherungssatz;
end;
:: CONMETR1:attrnot 2 => CONMETR1:attr 2
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_major_Scherungssatz means
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
for b10, b11 being Element of bool the carrier of a1
st b10 is being_line(a1) & b11 is being_line(a1) & b1 in b10 & b1 in b11 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & b3 in b11 & b5 in b11 & b7 in b11 & b9 in b11 & not b5 in b10 & not b3 in b10 & not b7 in b10 & not b9 in b10 & not b2 in b11 & not b4 in b11 & not b6 in b11 & not b8 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
holds b4,b5 // b8,b9;
end;
:: CONMETR1:dfs 2
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_major_Scherungssatz
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
for b10, b11 being Element of bool the carrier of a1
st b10 is being_line(a1) & b11 is being_line(a1) & b1 in b10 & b1 in b11 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & b3 in b11 & b5 in b11 & b7 in b11 & b9 in b11 & not b5 in b10 & not b3 in b10 & not b7 in b10 & not b9 in b10 & not b2 in b11 & not b4 in b11 & not b6 in b11 & not b8 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
holds b4,b5 // b8,b9;
:: CONMETR1:def 2
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_major_Scherungssatz
iff
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11, b12 being Element of bool the carrier of b1
st b11 is being_line(b1) & b12 is being_line(b1) & b2 in b11 & b2 in b12 & b3 in b11 & b5 in b11 & b7 in b11 & b9 in b11 & b4 in b12 & b6 in b12 & b8 in b12 & b10 in b12 & not b6 in b11 & not b4 in b11 & not b8 in b11 & not b10 in b11 & not b3 in b12 & not b5 in b12 & not b7 in b12 & not b9 in b12 & b5,b4 // b9,b8 & b4,b3 // b8,b7 & b3,b6 // b7,b10
holds b5,b6 // b9,b10;
:: CONMETR1:prednot 2 => CONMETR1:attr 2
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_major_SCH for satisfying_major_Scherungssatz;
end;
:: CONMETR1:attrnot 3 => CONMETR1:attr 3
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_Scherungssatz means
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
for b9, b10 being Element of bool the carrier of a1
st b9 is being_line(a1) & b10 is being_line(a1) & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b4 in b9 & not b2 in b9 & not b6 in b9 & not b8 in b9 & not b1 in b10 & not b3 in b10 & not b5 in b10 & not b7 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
holds b3,b4 // b7,b8;
end;
:: CONMETR1:dfs 3
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_Scherungssatz
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
for b9, b10 being Element of bool the carrier of a1
st b9 is being_line(a1) & b10 is being_line(a1) & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b4 in b9 & not b2 in b9 & not b6 in b9 & not b8 in b9 & not b1 in b10 & not b3 in b10 & not b5 in b10 & not b7 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
holds b3,b4 // b7,b8;
:: CONMETR1:def 3
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_Scherungssatz
iff
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
for b10, b11 being Element of bool the carrier of b1
st b10 is being_line(b1) & b11 is being_line(b1) & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & b3 in b11 & b5 in b11 & b7 in b11 & b9 in b11 & not b5 in b10 & not b3 in b10 & not b7 in b10 & not b9 in b10 & not b2 in b11 & not b4 in b11 & not b6 in b11 & not b8 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
holds b4,b5 // b8,b9;
:: CONMETR1:prednot 3 => CONMETR1:attr 3
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_SCH for satisfying_Scherungssatz;
end;
:: CONMETR1:attrnot 4 => CONMETR1:attr 4
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_indirect_Scherungssatz means
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
for b9, b10 being Element of bool the carrier of a1
st b9 is being_line(a1) & b10 is being_line(a1) & b1 in b9 & b3 in b9 & b6 in b9 & b8 in b9 & b2 in b10 & b4 in b10 & b5 in b10 & b7 in b10 & not b4 in b9 & not b2 in b9 & not b5 in b9 & not b7 in b9 & not b1 in b10 & not b3 in b10 & not b6 in b10 & not b8 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
holds b3,b4 // b7,b8;
end;
:: CONMETR1:dfs 4
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_indirect_Scherungssatz
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
for b9, b10 being Element of bool the carrier of a1
st b9 is being_line(a1) & b10 is being_line(a1) & b1 in b9 & b3 in b9 & b6 in b9 & b8 in b9 & b2 in b10 & b4 in b10 & b5 in b10 & b7 in b10 & not b4 in b9 & not b2 in b9 & not b5 in b9 & not b7 in b9 & not b1 in b10 & not b3 in b10 & not b6 in b10 & not b8 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
holds b3,b4 // b7,b8;
:: CONMETR1:def 4
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_indirect_Scherungssatz
iff
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
for b10, b11 being Element of bool the carrier of b1
st b10 is being_line(b1) & b11 is being_line(b1) & b2 in b10 & b4 in b10 & b7 in b10 & b9 in b10 & b3 in b11 & b5 in b11 & b6 in b11 & b8 in b11 & not b5 in b10 & not b3 in b10 & not b6 in b10 & not b8 in b10 & not b2 in b11 & not b4 in b11 & not b7 in b11 & not b9 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
holds b4,b5 // b8,b9;
:: CONMETR1:prednot 4 => CONMETR1:attr 4
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_SCH* for satisfying_indirect_Scherungssatz;
end;
:: CONMETR1:attrnot 5 => CONMETR1:attr 5
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_minor_indirect_Scherungssatz means
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
for b9, b10 being Element of bool the carrier of a1
st b9 // b10 & b1 in b9 & b3 in b9 & b6 in b9 & b8 in b9 & b2 in b10 & b4 in b10 & b5 in b10 & b7 in b10 & not b4 in b9 & not b2 in b9 & not b5 in b9 & not b7 in b9 & not b1 in b10 & not b3 in b10 & not b6 in b10 & not b8 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
holds b3,b4 // b7,b8;
end;
:: CONMETR1:dfs 5
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_minor_indirect_Scherungssatz
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
for b9, b10 being Element of bool the carrier of a1
st b9 // b10 & b1 in b9 & b3 in b9 & b6 in b9 & b8 in b9 & b2 in b10 & b4 in b10 & b5 in b10 & b7 in b10 & not b4 in b9 & not b2 in b9 & not b5 in b9 & not b7 in b9 & not b1 in b10 & not b3 in b10 & not b6 in b10 & not b8 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
holds b3,b4 // b7,b8;
:: CONMETR1:def 5
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_minor_indirect_Scherungssatz
iff
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
for b10, b11 being Element of bool the carrier of b1
st b10 // b11 & b2 in b10 & b4 in b10 & b7 in b10 & b9 in b10 & b3 in b11 & b5 in b11 & b6 in b11 & b8 in b11 & not b5 in b10 & not b3 in b10 & not b6 in b10 & not b8 in b10 & not b2 in b11 & not b4 in b11 & not b7 in b11 & not b9 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
holds b4,b5 // b8,b9;
:: CONMETR1:prednot 5 => CONMETR1:attr 5
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_minor_SCH* for satisfying_minor_indirect_Scherungssatz;
end;
:: CONMETR1:attrnot 6 => CONMETR1:attr 6
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_major_indirect_Scherungssatz means
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
for b10, b11 being Element of bool the carrier of a1
st b10 is being_line(a1) & b11 is being_line(a1) & b1 in b10 & b1 in b11 & b2 in b10 & b4 in b10 & b7 in b10 & b9 in b10 & b3 in b11 & b5 in b11 & b6 in b11 & b8 in b11 & not b5 in b10 & not b3 in b10 & not b6 in b10 & not b8 in b10 & not b2 in b11 & not b4 in b11 & not b7 in b11 & not b9 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
holds b4,b5 // b8,b9;
end;
:: CONMETR1:dfs 6
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_major_indirect_Scherungssatz
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
for b10, b11 being Element of bool the carrier of a1
st b10 is being_line(a1) & b11 is being_line(a1) & b1 in b10 & b1 in b11 & b2 in b10 & b4 in b10 & b7 in b10 & b9 in b10 & b3 in b11 & b5 in b11 & b6 in b11 & b8 in b11 & not b5 in b10 & not b3 in b10 & not b6 in b10 & not b8 in b10 & not b2 in b11 & not b4 in b11 & not b7 in b11 & not b9 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
holds b4,b5 // b8,b9;
:: CONMETR1:def 6
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_major_indirect_Scherungssatz
iff
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
for b11, b12 being Element of bool the carrier of b1
st b11 is being_line(b1) & b12 is being_line(b1) & b2 in b11 & b2 in b12 & b3 in b11 & b5 in b11 & b8 in b11 & b10 in b11 & b4 in b12 & b6 in b12 & b7 in b12 & b9 in b12 & not b6 in b11 & not b4 in b11 & not b7 in b11 & not b9 in b11 & not b3 in b12 & not b5 in b12 & not b8 in b12 & not b10 in b12 & b5,b4 // b9,b8 & b4,b3 // b8,b7 & b3,b6 // b7,b10
holds b5,b6 // b9,b10;
:: CONMETR1:prednot 6 => CONMETR1:attr 6
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_major_SCH* for satisfying_major_indirect_Scherungssatz;
end;
:: CONMETR1:th 1
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_indirect_Scherungssatz
iff
b1 is satisfying_minor_indirect_Scherungssatz & b1 is satisfying_major_indirect_Scherungssatz;
:: CONMETR1:th 2
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_Scherungssatz
iff
b1 is satisfying_minor_Scherungssatz & b1 is satisfying_major_Scherungssatz;
:: CONMETR1:th 3
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is satisfying_minor_indirect_Scherungssatz
holds b1 is satisfying_minor_Scherungssatz;
:: CONMETR1:th 4
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is satisfying_major_indirect_Scherungssatz
holds b1 is satisfying_major_Scherungssatz;
:: CONMETR1:th 5
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is satisfying_indirect_Scherungssatz
holds b1 is satisfying_Scherungssatz;
:: CONMETR1:th 6
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is translational
holds b1 is satisfying_minor_Scherungssatz;
:: CONMETR1:th 7
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is Desarguesian
holds b1 is satisfying_major_Scherungssatz;
:: CONMETR1:th 8
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is Desarguesian
iff
b1 is satisfying_Scherungssatz;
:: CONMETR1:th 9
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_pap
iff
b1 is satisfying_minor_indirect_Scherungssatz;
:: CONMETR1:th 10
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is Pappian
iff
b1 is satisfying_major_indirect_Scherungssatz;
:: CONMETR1:th 11
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_PPAP
iff
b1 is satisfying_indirect_Scherungssatz;
:: CONMETR1:th 12
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is satisfying_major_indirect_Scherungssatz
holds b1 is satisfying_minor_indirect_Scherungssatz;
:: CONMETR1:th 13
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
Af b1 is satisfying_Scherungssatz
iff
b1 is satisfying_SCH;
:: CONMETR1:th 14
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is satisfying_TDES
iff
Af b1 is Moufangian;
:: CONMETR1:th 15
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
Af b1 is translational
iff
b1 is satisfying_des;
:: CONMETR1:th 16
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is satisfying_PAP
iff
Af b1 is Pappian;
:: CONMETR1:th 17
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is satisfying_DES
iff
Af b1 is Desarguesian;