Article AFF_2, MML version 4.99.1005
:: AFF_2:attrnot 1 => AFF_2:attr 1
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_PPAP means
for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b3 in b1 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b2 & b3,b7 // b4,b6 & b4,b8 // b5,b7
holds b3,b8 // b5,b6;
end;
:: AFF_2:dfs 1
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_PPAP
it is sufficient to prove
thus for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b3 in b1 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b2 & b3,b7 // b4,b6 & b4,b8 // b5,b7
holds b3,b8 // b5,b6;
:: AFF_2:def 1
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_PPAP
iff
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & b4 in b2 & b5 in b2 & b6 in b2 & b7 in b3 & b8 in b3 & b9 in b3 & b4,b8 // b5,b7 & b5,b9 // b6,b8
holds b4,b9 // b6,b7;
:: AFF_2:prednot 1 => AFF_2:attr 1
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_PPAP for satisfying_PPAP;
end;
:: AFF_2:attrnot 2 => AFF_2:attr 2
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
attr a1 is Pappian means
for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b1 <> b2 & b3 in b1 & b3 in b2 & b3 <> b4 & b3 <> b7 & b3 <> b5 & b3 <> b8 & b3 <> b6 & b3 <> b9 & b4 in b1 & b5 in b1 & b6 in b1 & b7 in b2 & b8 in b2 & b9 in b2 & b4,b8 // b5,b7 & b5,b9 // b6,b8
holds b4,b9 // b6,b7;
end;
:: AFF_2:dfs 2
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
To prove
a1 is Pappian
it is sufficient to prove
thus for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b1 <> b2 & b3 in b1 & b3 in b2 & b3 <> b4 & b3 <> b7 & b3 <> b5 & b3 <> b8 & b3 <> b6 & b3 <> b9 & b4 in b1 & b5 in b1 & b6 in b1 & b7 in b2 & b8 in b2 & b9 in b2 & b4,b8 // b5,b7 & b5,b9 // b6,b8
holds b4,b9 // b6,b7;
:: AFF_2:def 2
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
b1 is Pappian
iff
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & b2 <> b3 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b8 & b4 <> b6 & b4 <> b9 & b4 <> b7 & b4 <> b10 & b5 in b2 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b10 in b3 & b5,b9 // b6,b8 & b6,b10 // b7,b9
holds b5,b10 // b7,b8;
:: AFF_2:prednot 2 => AFF_2:attr 2
notation
let a1 be non empty non trivial AffinSpace-like AffinStruct;
synonym a1 satisfies_PAP for Pappian;
end;
:: AFF_2:attrnot 3 => AFF_2:attr 3
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_PAP_1 means
for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b1 <> b2 & b3 in b1 & b3 in b2 & b3 <> b4 & b3 <> b7 & b3 <> b5 & b3 <> b8 & b3 <> b6 & b3 <> b9 & b4 in b1 & b5 in b1 & b6 in b1 & b8 in b2 & b9 in b2 & b4,b8 // b5,b7 & b5,b9 // b6,b8 & b4,b9 // b6,b7 & b5 <> b6
holds b7 in b2;
end;
:: AFF_2:dfs 3
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_PAP_1
it is sufficient to prove
thus for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b1 <> b2 & b3 in b1 & b3 in b2 & b3 <> b4 & b3 <> b7 & b3 <> b5 & b3 <> b8 & b3 <> b6 & b3 <> b9 & b4 in b1 & b5 in b1 & b6 in b1 & b8 in b2 & b9 in b2 & b4,b8 // b5,b7 & b5,b9 // b6,b8 & b4,b9 // b6,b7 & b5 <> b6
holds b7 in b2;
:: AFF_2:def 3
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_PAP_1
iff
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & b2 <> b3 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b8 & b4 <> b6 & b4 <> b9 & b4 <> b7 & b4 <> b10 & b5 in b2 & b6 in b2 & b7 in b2 & b9 in b3 & b10 in b3 & b5,b9 // b6,b8 & b6,b10 // b7,b9 & b5,b10 // b7,b8 & b6 <> b7
holds b8 in b3;
:: AFF_2:prednot 3 => AFF_2:attr 3
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_PAP_1 for satisfying_PAP_1;
end;
:: AFF_2:attrnot 4 => AFF_2:attr 4
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
attr a1 is Desarguesian means
for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b4 in b1 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 in b1 & b8 in b1 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b5,b6 // b8,b9 & b5,b7 // b8,b10
holds b6,b7 // b9,b10;
end;
:: AFF_2:dfs 4
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
To prove
a1 is Desarguesian
it is sufficient to prove
thus for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b4 in b1 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 in b1 & b8 in b1 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b5,b6 // b8,b9 & b5,b7 // b8,b10
holds b6,b7 // b9,b10;
:: AFF_2:def 4
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
b1 is Desarguesian
iff
for b2, b3, b4 being Element of bool the carrier of b1
for b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b5 in b2 & b5 in b3 & b5 in b4 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b8 in b4 & b11 in b4 & b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b2 <> b3 & b2 <> b4 & b6,b7 // b9,b10 & b6,b8 // b9,b11
holds b7,b8 // b10,b11;
:: AFF_2:prednot 4 => AFF_2:attr 4
notation
let a1 be non empty non trivial AffinSpace-like AffinStruct;
synonym a1 satisfies_DES for Desarguesian;
end;
:: AFF_2:attrnot 5 => AFF_2:attr 5
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_DES_1 means
for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b4 in b1 & b4 in b2 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 in b1 & b8 in b1 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b5,b6 // b8,b9 & b5,b7 // b8,b10 & b6,b7 // b9,b10 & not LIN b5,b6,b7 & b7 <> b10
holds b4 in b3;
end;
:: AFF_2:dfs 5
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_DES_1
it is sufficient to prove
thus for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b4 in b1 & b4 in b2 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 in b1 & b8 in b1 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b5,b6 // b8,b9 & b5,b7 // b8,b10 & b6,b7 // b9,b10 & not LIN b5,b6,b7 & b7 <> b10
holds b4 in b3;
:: AFF_2:def 5
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_DES_1
iff
for b2, b3, b4 being Element of bool the carrier of b1
for b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b5 in b2 & b5 in b3 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b8 in b4 & b11 in b4 & b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b2 <> b3 & b2 <> b4 & b6,b7 // b9,b10 & b6,b8 // b9,b11 & b7,b8 // b10,b11 & not LIN b6,b7,b8 & b8 <> b11
holds b5 in b4;
:: AFF_2:prednot 5 => AFF_2:attr 5
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_DES_1 for satisfying_DES_1;
end;
:: AFF_2:attrnot 6 => AFF_2:attr 6
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_DES_2 means
for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b4 in b1 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 in b1 & b8 in b1 & b6 in b2 & b9 in b2 & b7 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b5,b6 // b8,b9 & b5,b7 // b8,b10 & b6,b7 // b9,b10
holds b10 in b3;
end;
:: AFF_2:dfs 6
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_DES_2
it is sufficient to prove
thus for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b4 in b1 & b4 in b2 & b4 in b3 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 in b1 & b8 in b1 & b6 in b2 & b9 in b2 & b7 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b5,b6 // b8,b9 & b5,b7 // b8,b10 & b6,b7 // b9,b10
holds b10 in b3;
:: AFF_2:def 6
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_DES_2
iff
for b2, b3, b4 being Element of bool the carrier of b1
for b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b5 in b2 & b5 in b3 & b5 in b4 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 in b2 & b9 in b2 & b7 in b3 & b10 in b3 & b8 in b4 & b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b2 <> b3 & b2 <> b4 & b6,b7 // b9,b10 & b6,b8 // b9,b11 & b7,b8 // b10,b11
holds b11 in b4;
:: AFF_2:prednot 6 => AFF_2:attr 6
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_DES_2 for satisfying_DES_2;
end;
:: AFF_2:attrnot 7 => AFF_2:attr 7
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
attr a1 is Moufangian means
for b1 being Element of bool the carrier of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 in b1 & b5 in b1 & b8 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & LIN b2,b4,b7 & b3,b4 // b6,b7 & b3,b5 // b6,b8 & b3,b4 // b1
holds b4,b5 // b7,b8;
end;
:: AFF_2:dfs 7
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
To prove
a1 is Moufangian
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 in b1 & b5 in b1 & b8 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & LIN b2,b4,b7 & b3,b4 // b6,b7 & b3,b5 // b6,b8 & b3,b4 // b1
holds b4,b5 // b7,b8;
:: AFF_2:def 7
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
b1 is Moufangian
iff
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
st b2 is being_line(b1) & b3 in b2 & b6 in b2 & b9 in b2 & not b4 in b2 & b3 <> b6 & b4 <> b5 & LIN b3,b4,b7 & LIN b3,b5,b8 & b4,b5 // b7,b8 & b4,b6 // b7,b9 & b4,b5 // b2
holds b5,b6 // b8,b9;
:: AFF_2:prednot 7 => AFF_2:attr 7
notation
let a1 be non empty non trivial AffinSpace-like AffinStruct;
synonym a1 satisfies_TDES for Moufangian;
end;
:: AFF_2:attrnot 8 => AFF_2:attr 8
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_TDES_1 means
for b1 being Element of bool the carrier of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 in b1 & b5 in b1 & b8 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & b3,b4 // b6,b7 & b4,b5 // b7,b8 & b3,b5 // b6,b8 & b3,b4 // b1
holds LIN b2,b4,b7;
end;
:: AFF_2:dfs 8
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_TDES_1
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 in b1 & b5 in b1 & b8 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & b3,b4 // b6,b7 & b4,b5 // b7,b8 & b3,b5 // b6,b8 & b3,b4 // b1
holds LIN b2,b4,b7;
:: AFF_2:def 8
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_TDES_1
iff
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
st b2 is being_line(b1) & b3 in b2 & b6 in b2 & b9 in b2 & not b4 in b2 & b3 <> b6 & b4 <> b5 & LIN b3,b4,b7 & b4,b5 // b7,b8 & b5,b6 // b8,b9 & b4,b6 // b7,b9 & b4,b5 // b2
holds LIN b3,b5,b8;
:: AFF_2:prednot 8 => AFF_2:attr 8
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_TDES_1 for satisfying_TDES_1;
end;
:: AFF_2:attrnot 9 => AFF_2:attr 9
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_TDES_2 means
for b1 being Element of bool the carrier of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 in b1 & b5 in b1 & b8 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & LIN b2,b4,b7 & b4,b5 // b7,b8 & b3,b5 // b6,b8 & b3,b4 // b1
holds b3,b4 // b6,b7;
end;
:: AFF_2:dfs 9
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_TDES_2
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 in b1 & b5 in b1 & b8 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & LIN b2,b4,b7 & b4,b5 // b7,b8 & b3,b5 // b6,b8 & b3,b4 // b1
holds b3,b4 // b6,b7;
:: AFF_2:def 9
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_TDES_2
iff
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
st b2 is being_line(b1) & b3 in b2 & b6 in b2 & b9 in b2 & not b4 in b2 & b3 <> b6 & b4 <> b5 & LIN b3,b4,b7 & LIN b3,b5,b8 & b5,b6 // b8,b9 & b4,b6 // b7,b9 & b4,b5 // b2
holds b4,b5 // b7,b8;
:: AFF_2:prednot 9 => AFF_2:attr 9
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_TDES_2 for satisfying_TDES_2;
end;
:: AFF_2:attrnot 10 => AFF_2:attr 10
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_TDES_3 means
for b1 being Element of bool the carrier of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 in b1 & b5 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & LIN b2,b4,b7 & b3,b4 // b6,b7 & b3,b5 // b6,b8 & b4,b5 // b7,b8 & b3,b4 // b1
holds b8 in b1;
end;
:: AFF_2:dfs 10
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_TDES_3
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 in b1 & b5 in b1 & not b3 in b1 & b2 <> b5 & b3 <> b4 & LIN b2,b3,b6 & LIN b2,b4,b7 & b3,b4 // b6,b7 & b3,b5 // b6,b8 & b4,b5 // b7,b8 & b3,b4 // b1
holds b8 in b1;
:: AFF_2:def 10
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_TDES_3
iff
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
st b2 is being_line(b1) & b3 in b2 & b6 in b2 & not b4 in b2 & b3 <> b6 & b4 <> b5 & LIN b3,b4,b7 & LIN b3,b5,b8 & b4,b5 // b7,b8 & b4,b6 // b7,b9 & b5,b6 // b8,b9 & b4,b5 // b2
holds b9 in b2;
:: AFF_2:prednot 10 => AFF_2:attr 10
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_TDES_3 for satisfying_TDES_3;
end;
:: AFF_2:attrnot 11 => AFF_2:attr 11
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
attr a1 is translational means
for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
st b1 // b2 & b1 // b3 & b4 in b1 & b7 in b1 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b4,b5 // b7,b8 & b4,b6 // b7,b9
holds b5,b6 // b8,b9;
end;
:: AFF_2:dfs 11
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
To prove
a1 is translational
it is sufficient to prove
thus for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
st b1 // b2 & b1 // b3 & b4 in b1 & b7 in b1 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b4,b5 // b7,b8 & b4,b6 // b7,b9
holds b5,b6 // b8,b9;
:: AFF_2:def 11
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
b1 is translational
iff
for b2, b3, b4 being Element of bool the carrier of b1
for b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
st b2 // b3 & b2 // b4 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b7 in b4 & b10 in b4 & b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b2 <> b3 & b2 <> b4 & b5,b6 // b8,b9 & b5,b7 // b8,b10
holds b6,b7 // b9,b10;
:: AFF_2:prednot 11 => AFF_2:attr 11
notation
let a1 be non empty non trivial AffinSpace-like AffinStruct;
synonym a1 satisfies_des for translational;
end;
:: AFF_2:attrnot 12 => AFF_2:attr 12
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_des_1 means
for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
st b1 // b2 & b4 in b1 & b7 in b1 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b4,b5 // b7,b8 & b4,b6 // b7,b9 & b5,b6 // b8,b9 & not LIN b4,b5,b6 & b6 <> b9
holds b1 // b3;
end;
:: AFF_2:dfs 12
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_des_1
it is sufficient to prove
thus for b1, b2, b3 being Element of bool the carrier of a1
for b4, b5, b6, b7, b8, b9 being Element of the carrier of a1
st b1 // b2 & b4 in b1 & b7 in b1 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b4,b5 // b7,b8 & b4,b6 // b7,b9 & b5,b6 // b8,b9 & not LIN b4,b5,b6 & b6 <> b9
holds b1 // b3;
:: AFF_2:def 12
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_des_1
iff
for b2, b3, b4 being Element of bool the carrier of b1
for b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
st b2 // b3 & b5 in b2 & b8 in b2 & b6 in b3 & b9 in b3 & b7 in b4 & b10 in b4 & b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b2 <> b3 & b2 <> b4 & b5,b6 // b8,b9 & b5,b7 // b8,b10 & b6,b7 // b9,b10 & not LIN b5,b6,b7 & b7 <> b10
holds b2 // b4;
:: AFF_2:prednot 12 => AFF_2:attr 12
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_des_1 for satisfying_des_1;
end;
:: AFF_2:attrnot 13 => AFF_2:attr 13
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
attr a1 is satisfying_pap means
for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b3 in b1 & b4 in b1 & b5 in b1 & b1 // b2 & b1 <> b2 & b6 in b2 & b7 in b2 & b8 in b2 & b3,b7 // b4,b6 & b4,b8 // b5,b7
holds b3,b8 // b5,b6;
end;
:: AFF_2:dfs 13
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
To prove
a1 is satisfying_pap
it is sufficient to prove
thus for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b3 in b1 & b4 in b1 & b5 in b1 & b1 // b2 & b1 <> b2 & b6 in b2 & b7 in b2 & b8 in b2 & b3,b7 // b4,b6 & b4,b8 // b5,b7
holds b3,b8 // b5,b6;
:: AFF_2:def 13
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
b1 is satisfying_pap
iff
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & b4 in b2 & b5 in b2 & b6 in b2 & b2 // b3 & b2 <> b3 & b7 in b3 & b8 in b3 & b9 in b3 & b4,b8 // b5,b7 & b5,b9 // b6,b8
holds b4,b9 // b6,b7;
:: AFF_2:prednot 13 => AFF_2:attr 13
notation
let a1 be non empty non trivial AffinSpace-like AffinStruct;
synonym a1 satisfies_pap for satisfying_pap;
end;
:: AFF_2:attrnot 14 => AFF_2:attr 14
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
attr a1 is satisfying_pap_1 means
for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b3 in b1 & b4 in b1 & b5 in b1 & b1 // b2 & b1 <> b2 & b6 in b2 & b7 in b2 & b3,b7 // b4,b6 & b4,b8 // b5,b7 & b3,b8 // b5,b6 & b6 <> b7
holds b8 in b2;
end;
:: AFF_2:dfs 14
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
a1 is satisfying_pap_1
it is sufficient to prove
thus for b1, b2 being Element of bool the carrier of a1
for b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
st b1 is being_line(a1) & b2 is being_line(a1) & b3 in b1 & b4 in b1 & b5 in b1 & b1 // b2 & b1 <> b2 & b6 in b2 & b7 in b2 & b3,b7 // b4,b6 & b4,b8 // b5,b7 & b3,b8 // b5,b6 & b6 <> b7
holds b8 in b2;
:: AFF_2:def 14
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_pap_1
iff
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
st b2 is being_line(b1) & b3 is being_line(b1) & b4 in b2 & b5 in b2 & b6 in b2 & b2 // b3 & b2 <> b3 & b7 in b3 & b8 in b3 & b4,b8 // b5,b7 & b5,b9 // b6,b8 & b4,b9 // b6,b7 & b7 <> b8
holds b9 in b3;
:: AFF_2:prednot 14 => AFF_2:attr 14
notation
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
synonym a1 satisfies_pap_1 for satisfying_pap_1;
end;
:: AFF_2:th 15
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is Pappian
iff
b1 is satisfying_PAP_1;
:: AFF_2:th 16
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is Desarguesian
iff
b1 is satisfying_DES_1;
:: AFF_2:th 17
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is Moufangian
holds b1 is satisfying_TDES_1;
:: AFF_2:th 18
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is satisfying_TDES_1
holds b1 is satisfying_TDES_2;
:: AFF_2:th 19
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is satisfying_TDES_2
holds b1 is satisfying_TDES_3;
:: AFF_2:th 20
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is satisfying_TDES_3
holds b1 is Moufangian;
:: AFF_2:th 21
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is translational
iff
b1 is satisfying_des_1;
:: AFF_2:th 22
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_pap
iff
b1 is satisfying_pap_1;
:: AFF_2:th 23
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is Pappian
holds b1 is satisfying_pap;
:: AFF_2:th 24
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is satisfying_PPAP
iff
b1 is Pappian & b1 is satisfying_pap;
:: AFF_2:th 25
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is Pappian
holds b1 is Desarguesian;
:: AFF_2:th 26
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is Desarguesian
holds b1 is Moufangian;
:: AFF_2:th 27
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is satisfying_TDES_1
holds b1 is satisfying_des_1;
:: AFF_2:th 28
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is Moufangian
holds b1 is translational;
:: AFF_2:th 29
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is translational
holds b1 is satisfying_pap;