Article INCPROJ, MML version 4.99.1005
:: INCPROJ:modenot 1 => INCPROJ:mode 1
definition
let a1 be non empty reflexive transitive proper CollStr;
redefine mode LINE of a1 -> Element of bool the carrier of a1;
end;
:: INCPROJ:funcnot 1 => INCPROJ:func 1
definition
let a1 be non empty reflexive transitive proper CollStr;
func ProjectiveLines A1 -> set equals
{b1 where b1 is Element of bool the carrier of a1: b1 is LINE of a1};
end;
:: INCPROJ:def 1
theorem
for b1 being non empty reflexive transitive proper CollStr holds
ProjectiveLines b1 = {b2 where b2 is Element of bool the carrier of b1: b2 is LINE of b1};
:: INCPROJ:funcreg 1
registration
let a1 be non empty reflexive transitive proper CollStr;
cluster ProjectiveLines a1 -> non empty;
end;
:: INCPROJ:th 2
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2 being set holds
b2 is LINE of b1
iff
b2 is Element of ProjectiveLines b1;
:: INCPROJ:funcnot 2 => INCPROJ:func 2
definition
let a1 be non empty reflexive transitive proper CollStr;
func Proj_Inc A1 -> Relation of the carrier of a1,ProjectiveLines a1 means
for b1, b2 being set holds
[b1,b2] in it
iff
b1 in the carrier of a1 &
b2 in ProjectiveLines a1 &
(ex b3 being set st
b2 = b3 & b1 in b3);
end;
:: INCPROJ:def 2
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2 being Relation of the carrier of b1,ProjectiveLines b1 holds
b2 = Proj_Inc b1
iff
for b3, b4 being set holds
[b3,b4] in b2
iff
b3 in the carrier of b1 &
b4 in ProjectiveLines b1 &
(ex b5 being set st
b4 = b5 & b3 in b5);
:: INCPROJ:funcnot 3 => INCPROJ:func 3
definition
let a1 be non empty reflexive transitive proper CollStr;
func IncProjSp_of A1 -> IncProjStr equals
IncProjStr(#the carrier of a1,ProjectiveLines a1,Proj_Inc a1#);
end;
:: INCPROJ:def 3
theorem
for b1 being non empty reflexive transitive proper CollStr holds
IncProjSp_of b1 = IncProjStr(#the carrier of b1,ProjectiveLines b1,Proj_Inc b1#);
:: INCPROJ:funcreg 2
registration
let a1 be non empty reflexive transitive proper CollStr;
cluster IncProjSp_of a1 -> strict;
end;
:: INCPROJ:th 4
theorem
for b1 being non empty reflexive transitive proper CollStr holds
the Points of IncProjSp_of b1 = the carrier of b1 & the Lines of IncProjSp_of b1 = ProjectiveLines b1 & the Inc of IncProjSp_of b1 = Proj_Inc b1;
:: INCPROJ:th 5
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2 being set holds
b2 is Element of the carrier of b1
iff
b2 is Element of the Points of IncProjSp_of b1;
:: INCPROJ:th 6
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2 being set holds
b2 is LINE of b1
iff
b2 is Element of the Lines of IncProjSp_of b1;
:: INCPROJ:th 9
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2 being Element of the Points of IncProjSp_of b1
for b3 being Element of the Lines of IncProjSp_of b1
for b4 being Element of the carrier of b1
for b5 being LINE of b1
st b2 = b4 & b3 = b5
holds b2 on b3
iff
b4 in b5;
:: INCPROJ:th 10
theorem
for b1 being non empty reflexive transitive proper CollStr holds
ex b2, b3, b4 being Element of the carrier of b1 st
b2 <> b3 & b3 <> b4 & b4 <> b2;
:: INCPROJ:th 11
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2 being Element of the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
b2 <> b3;
:: INCPROJ:th 12
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being Element of the Points of IncProjSp_of b1
for b4, b5 being Element of the Lines of IncProjSp_of b1
st b2 on b4 & b3 on b4 & b2 on b5 & b3 on b5 & b2 <> b3
holds b4 = b5;
:: INCPROJ:th 13
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3 being Element of the Points of IncProjSp_of b1 holds
ex b4 being Element of the Lines of IncProjSp_of b1 st
b2 on b4 & b3 on b4;
:: INCPROJ:th 14
theorem
for b1 being non empty reflexive transitive proper CollStr
for b2, b3, b4 being Element of the Points of IncProjSp_of b1
for b5, b6, b7 being Element of the carrier of b1
st b2 = b5 & b3 = b6 & b4 = b7
holds b5,b6,b7 is_collinear
iff
ex b8 being Element of the Lines of IncProjSp_of b1 st
b2 on b8 & b3 on b8 & b4 on b8;
:: INCPROJ:th 15
theorem
for b1 being non empty reflexive transitive proper CollStr holds
ex b2 being Element of the Points of IncProjSp_of b1 st
ex b3 being Element of the Lines of IncProjSp_of b1 st
not b2 on b3;
:: INCPROJ:th 16
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2 being Element of the Lines of IncProjSp_of b1 holds
ex b3, b4, b5 being Element of the Points of IncProjSp_of b1 st
b3 <> b4 & b4 <> b5 & b5 <> b3 & b3 on b2 & b4 on b2 & b5 on b2;
:: INCPROJ:th 17
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
for b2, b3, b4, b5, b6 being Element of the Points of IncProjSp_of b1
for b7, b8, b9, b10 being Element of the Lines of IncProjSp_of b1
st b2 on b7 & b3 on b7 & b4 on b8 & b5 on b8 & b6 on b7 & b6 on b8 & b2 on b9 & b4 on b9 & b3 on b10 & b5 on b10 & not b6 on b9 & not b6 on b10 & b7 <> b8
holds ex b11 being Element of the Points of IncProjSp_of b1 st
b11 on b9 & b11 on b10;
:: INCPROJ:th 18
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
st for b2, b3, b4, b5 being Element of the carrier of b1 holds
ex b6 being Element of the carrier of b1 st
b2,b3,b6 is_collinear & b4,b5,b6 is_collinear
for b2, b3 being Element of the Lines of IncProjSp_of b1 holds
ex b4 being Element of the Points of IncProjSp_of b1 st
b4 on b2 & b4 on b3;
:: INCPROJ:th 19
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
st ex b2, b3, b4, b5 being Element of the carrier of b1 st
for b6 being Element of the carrier of b1
st b2,b3,b6 is_collinear
holds not b4,b5,b6 is_collinear
holds ex b2, b3 being Element of the Lines of IncProjSp_of b1 st
for b4 being Element of the Points of IncProjSp_of b1
st b4 on b2
holds not b4 on b3;
:: INCPROJ:th 20
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
st for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
ex b7, b8 being Element of the carrier of b1 st
b2,b4,b7 is_collinear & b3,b5,b8 is_collinear & b6,b7,b8 is_collinear
for b2 being Element of the Points of IncProjSp_of b1
for b3, b4 being Element of the Lines of IncProjSp_of b1 holds
ex b5, b6 being Element of the Points of IncProjSp_of b1 st
ex b7 being Element of the Lines of IncProjSp_of b1 st
b2 on b7 & b5 on b7 & b6 on b7 & b5 on b3 & b6 on b4;
:: INCPROJ:th 21
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
st for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st b2,b3,b4 is_collinear & b5,b6,b4 is_collinear & b2,b5,b7 is_collinear & b3,b6,b7 is_collinear & b2,b6,b8 is_collinear & b3,b5,b8 is_collinear & b7,b4,b8 is_collinear & not b2,b3,b6 is_collinear & not b2,b3,b5 is_collinear & not b2,b5,b6 is_collinear
holds b3,b5,b6 is_collinear
for b2, b3, b4, b5, b6, b7, b8 being Element of the Points of IncProjSp_of b1
for b9, b10, b11, b12, b13, b14, b15 being Element of the Lines of IncProjSp_of b1
st not b3 on b9 & not b4 on b9 & not b2 on b10 & not b5 on b10 & not b2 on b11 & not b4 on b11 & not b3 on b12 & not b5 on b12 & {b6,b2,b5} on b9 & {b6,b3,b4} on b10 & {b7,b3,b5} on b11 & {b7,b2,b4} on b12 & {b8,b2,b3} on b13 & {b8,b4,b5} on b14 & {b6,b7} on b15
holds not b8 on b15;
:: INCPROJ:th 22
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
st for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b2 <> b6 & b3 <> b6 & b2 <> b7 & b4 <> b7 & b2 <> b8 & b5 <> b8 & not b2,b3,b4 is_collinear & not b2,b3,b5 is_collinear & not b2,b4,b5 is_collinear & b3,b4,b11 is_collinear & b6,b7,b11 is_collinear & b4,b5,b9 is_collinear & b7,b8,b9 is_collinear & b3,b5,b10 is_collinear & b6,b8,b10 is_collinear & b2,b3,b6 is_collinear & b2,b4,b7 is_collinear & b2,b5,b8 is_collinear
holds b9,b10,b11 is_collinear
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the Points of IncProjSp_of b1
for b12, b13, b14, b15, b16, b17, b18, b19, b20 being Element of the Lines of IncProjSp_of b1
st {b2,b3,b4} on b12 & {b2,b6,b5} on b13 & {b2,b8,b7} on b14 & {b8,b6,b11} on b15 & {b8,b9,b4} on b16 & {b6,b10,b4} on b17 & {b11,b5,b7} on b18 & {b3,b9,b7} on b19 & {b3,b10,b5} on b20 & b12,b13,b14 are_mutually_different & b2 <> b4 & b2 <> b6 & b2 <> b8 & b2 <> b3 & b2 <> b5 & b2 <> b7 & b4 <> b3 & b6 <> b5 & b8 <> b7
holds ex b21 being Element of the Lines of IncProjSp_of b1 st
{b9,b10,b11} on b21;
:: INCPROJ:th 23
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr
st for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b2 <> b4 & b2 <> b5 & b4 <> b5 & b3 <> b4 & b3 <> b5 & b2 <> b7 & b2 <> b8 & b7 <> b8 & b6 <> b7 & b6 <> b8 & not b2,b3,b6 is_collinear & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b6,b7 is_collinear & b2,b6,b8 is_collinear & b3,b7,b11 is_collinear & b6,b4,b11 is_collinear & b3,b8,b10 is_collinear & b5,b6,b10 is_collinear & b4,b8,b9 is_collinear & b5,b7,b9 is_collinear
holds b9,b10,b11 is_collinear
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the Points of IncProjSp_of b1
for b12, b13, b14, b15, b16, b17, b18, b19, b20 being Element of the Lines of IncProjSp_of b1
st b2,b3,b4,b5 are_mutually_different & b2,b6,b7,b8 are_mutually_different & b14 <> b17 & b2 on b14 & b2 on b17 & {b4,b8,b9} on b12 & {b5,b6,b10} on b15 & {b3,b7,b11} on b18 & {b3,b8,b10} on b13 & {b5,b7,b9} on b16 & {b4,b6,b11} on b19 & {b6,b7,b8} on b14 & {b3,b4,b5} on b17 & {b9,b10} on b20
holds b11 on b20;
:: INCPROJ:attrnot 1 => INCPROJ:attr 1
definition
let a1 be IncProjStr;
attr a1 is partial means
for b1, b2 being Element of the Points of a1
for b3, b4 being Element of the Lines of a1
st b1 on b3 & b2 on b3 & b1 on b4 & b2 on b4 & b1 <> b2
holds b3 = b4;
end;
:: INCPROJ:dfs 4
definiens
let a1 be IncProjStr;
To prove
a1 is partial
it is sufficient to prove
thus for b1, b2 being Element of the Points of a1
for b3, b4 being Element of the Lines of a1
st b1 on b3 & b2 on b3 & b1 on b4 & b2 on b4 & b1 <> b2
holds b3 = b4;
:: INCPROJ:def 9
theorem
for b1 being IncProjStr holds
b1 is partial
iff
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
st b2 on b4 & b3 on b4 & b2 on b5 & b3 on b5 & b2 <> b3
holds b4 = b5;
:: INCPROJ:attrnot 2 => INCSP_1:attr 6
definition
let a1 be IncProjStr;
attr a1 is linear means
for b1, b2 being Element of the Points of a1 holds
ex b3 being Element of the Lines of a1 st
b1 on b3 & b2 on b3;
end;
:: INCPROJ:dfs 5
definiens
let a1 be IncProjStr;
To prove
a1 is linear
it is sufficient to prove
thus for b1, b2 being Element of the Points of a1 holds
ex b3 being Element of the Lines of a1 st
b1 on b3 & b2 on b3;
:: INCPROJ:def 10
theorem
for b1 being IncProjStr holds
b1 is linear
iff
for b2, b3 being Element of the Points of b1 holds
ex b4 being Element of the Lines of b1 st
b2 on b4 & b3 on b4;
:: INCPROJ:attrnot 3 => INCPROJ:attr 2
definition
let a1 be IncProjStr;
attr a1 is up-2-dimensional means
ex b1 being Element of the Points of a1 st
ex b2 being Element of the Lines of a1 st
not b1 on b2;
end;
:: INCPROJ:dfs 6
definiens
let a1 be IncProjStr;
To prove
a1 is up-2-dimensional
it is sufficient to prove
thus ex b1 being Element of the Points of a1 st
ex b2 being Element of the Lines of a1 st
not b1 on b2;
:: INCPROJ:def 11
theorem
for b1 being IncProjStr holds
b1 is up-2-dimensional
iff
ex b2 being Element of the Points of b1 st
ex b3 being Element of the Lines of b1 st
not b2 on b3;
:: INCPROJ:attrnot 4 => INCPROJ:attr 3
definition
let a1 be IncProjStr;
attr a1 is up-3-rank means
for b1 being Element of the Lines of a1 holds
ex b2, b3, b4 being Element of the Points of a1 st
b2 <> b3 & b3 <> b4 & b4 <> b2 & b2 on b1 & b3 on b1 & b4 on b1;
end;
:: INCPROJ:dfs 7
definiens
let a1 be IncProjStr;
To prove
a1 is up-3-rank
it is sufficient to prove
thus for b1 being Element of the Lines of a1 holds
ex b2, b3, b4 being Element of the Points of a1 st
b2 <> b3 & b3 <> b4 & b4 <> b2 & b2 on b1 & b3 on b1 & b4 on b1;
:: INCPROJ:def 12
theorem
for b1 being IncProjStr holds
b1 is up-3-rank
iff
for b2 being Element of the Lines of b1 holds
ex b3, b4, b5 being Element of the Points of b1 st
b3 <> b4 & b4 <> b5 & b5 <> b3 & b3 on b2 & b4 on b2 & b5 on b2;
:: INCPROJ:attrnot 5 => INCPROJ:attr 4
definition
let a1 be IncProjStr;
attr a1 is Vebleian means
for b1, b2, b3, b4, b5, b6 being Element of the Points of a1
for b7, b8, b9, b10 being Element of the Lines of a1
st b1 on b7 & b2 on b7 & b3 on b8 & b4 on b8 & b5 on b7 & b5 on b8 & b1 on b9 & b3 on b9 & b2 on b10 & b4 on b10 & not b5 on b9 & not b5 on b10 & b7 <> b8
holds ex b11 being Element of the Points of a1 st
b11 on b9 & b11 on b10;
end;
:: INCPROJ:dfs 8
definiens
let a1 be IncProjStr;
To prove
a1 is Vebleian
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6 being Element of the Points of a1
for b7, b8, b9, b10 being Element of the Lines of a1
st b1 on b7 & b2 on b7 & b3 on b8 & b4 on b8 & b5 on b7 & b5 on b8 & b1 on b9 & b3 on b9 & b2 on b10 & b4 on b10 & not b5 on b9 & not b5 on b10 & b7 <> b8
holds ex b11 being Element of the Points of a1 st
b11 on b9 & b11 on b10;
:: INCPROJ:def 13
theorem
for b1 being IncProjStr holds
b1 is Vebleian
iff
for b2, b3, b4, b5, b6, b7 being Element of the Points of b1
for b8, b9, b10, b11 being Element of the Lines of b1
st b2 on b8 & b3 on b8 & b4 on b9 & b5 on b9 & b6 on b8 & b6 on b9 & b2 on b10 & b4 on b10 & b3 on b11 & b5 on b11 & not b6 on b10 & not b6 on b11 & b8 <> b9
holds ex b12 being Element of the Points of b1 st
b12 on b10 & b12 on b11;
:: INCPROJ:funcreg 3
registration
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
cluster IncProjSp_of a1 -> linear partial up-2-dimensional up-3-rank Vebleian;
end;
:: INCPROJ:exreg 1
registration
cluster strict linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
end;
:: INCPROJ:modenot 2
definition
mode IncProjSp is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
end;
:: INCPROJ:funcreg 4
registration
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
cluster IncProjSp_of a1 -> linear partial up-2-dimensional up-3-rank Vebleian;
end;
:: INCPROJ:attrnot 6 => INCPROJ:attr 5
definition
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
attr a1 is 2-dimensional means
for b1, b2 being Element of the Lines of a1 holds
ex b3 being Element of the Points of a1 st
b3 on b1 & b3 on b2;
end;
:: INCPROJ:dfs 9
definiens
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
To prove
a1 is 2-dimensional
it is sufficient to prove
thus for b1, b2 being Element of the Lines of a1 holds
ex b3 being Element of the Points of a1 st
b3 on b1 & b3 on b2;
:: INCPROJ:def 14
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr holds
b1 is 2-dimensional
iff
for b2, b3 being Element of the Lines of b1 holds
ex b4 being Element of the Points of b1 st
b4 on b2 & b4 on b3;
:: INCPROJ:attrnot 7 => INCPROJ:attr 5
notation
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
antonym up-3-dimensional for 2-dimensional;
end;
:: INCPROJ:attrnot 8 => INCPROJ:attr 6
definition
let a1 be IncProjStr;
attr a1 is at_most-3-dimensional means
for b1 being Element of the Points of a1
for b2, b3 being Element of the Lines of a1 holds
ex b4, b5 being Element of the Points of a1 st
ex b6 being Element of the Lines of a1 st
b1 on b6 & b4 on b6 & b5 on b6 & b4 on b2 & b5 on b3;
end;
:: INCPROJ:dfs 10
definiens
let a1 be IncProjStr;
To prove
a1 is at_most-3-dimensional
it is sufficient to prove
thus for b1 being Element of the Points of a1
for b2, b3 being Element of the Lines of a1 holds
ex b4, b5 being Element of the Points of a1 st
ex b6 being Element of the Lines of a1 st
b1 on b6 & b4 on b6 & b5 on b6 & b4 on b2 & b5 on b3;
:: INCPROJ:def 16
theorem
for b1 being IncProjStr holds
b1 is at_most-3-dimensional
iff
for b2 being Element of the Points of b1
for b3, b4 being Element of the Lines of b1 holds
ex b5, b6 being Element of the Points of b1 st
ex b7 being Element of the Lines of b1 st
b2 on b7 & b5 on b7 & b6 on b7 & b5 on b3 & b6 on b4;
:: INCPROJ:attrnot 9 => INCPROJ:attr 7
definition
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
attr a1 is 3-dimensional means
a1 is at_most-3-dimensional & a1 is up-3-dimensional;
end;
:: INCPROJ:dfs 11
definiens
let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
To prove
a1 is 3-dimensional
it is sufficient to prove
thus a1 is at_most-3-dimensional & a1 is up-3-dimensional;
:: INCPROJ:def 17
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr holds
b1 is 3-dimensional
iff
b1 is at_most-3-dimensional & b1 is up-3-dimensional;
:: INCPROJ:attrnot 10 => INCPROJ:attr 8
definition
let a1 be IncProjStr;
attr a1 is Fanoian means
for b1, b2, b3, b4, b5, b6, b7 being Element of the Points of a1
for b8, b9, b10, b11, b12, b13, b14 being Element of the Lines of a1
st not b2 on b8 & not b3 on b8 & not b1 on b9 & not b4 on b9 & not b1 on b10 & not b3 on b10 & not b2 on b11 & not b4 on b11 & {b5,b1,b4} on b8 & {b5,b2,b3} on b9 & {b6,b2,b4} on b10 & {b6,b1,b3} on b11 & {b7,b1,b2} on b12 & {b7,b3,b4} on b13 & {b5,b6} on b14
holds not b7 on b14;
end;
:: INCPROJ:dfs 12
definiens
let a1 be IncProjStr;
To prove
a1 is Fanoian
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7 being Element of the Points of a1
for b8, b9, b10, b11, b12, b13, b14 being Element of the Lines of a1
st not b2 on b8 & not b3 on b8 & not b1 on b9 & not b4 on b9 & not b1 on b10 & not b3 on b10 & not b2 on b11 & not b4 on b11 & {b5,b1,b4} on b8 & {b5,b2,b3} on b9 & {b6,b2,b4} on b10 & {b6,b1,b3} on b11 & {b7,b1,b2} on b12 & {b7,b3,b4} on b13 & {b5,b6} on b14
holds not b7 on b14;
:: INCPROJ:def 18
theorem
for b1 being IncProjStr holds
b1 is Fanoian
iff
for b2, b3, b4, b5, b6, b7, b8 being Element of the Points of b1
for b9, b10, b11, b12, b13, b14, b15 being Element of the Lines of b1
st not b3 on b9 & not b4 on b9 & not b2 on b10 & not b5 on b10 & not b2 on b11 & not b4 on b11 & not b3 on b12 & not b5 on b12 & {b6,b2,b5} on b9 & {b6,b3,b4} on b10 & {b7,b3,b5} on b11 & {b7,b2,b4} on b12 & {b8,b2,b3} on b13 & {b8,b4,b5} on b14 & {b6,b7} on b15
holds not b8 on b15;
:: INCPROJ:attrnot 11 => INCPROJ:attr 9
definition
let a1 be IncProjStr;
attr a1 is Desarguesian means
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the Points of a1
for b11, b12, b13, b14, b15, b16, b17, b18, b19 being Element of the Lines of a1
st {b1,b2,b3} on b11 & {b1,b5,b4} on b12 & {b1,b7,b6} on b13 & {b7,b5,b10} on b14 & {b7,b8,b3} on b15 & {b5,b9,b3} on b16 & {b10,b4,b6} on b17 & {b2,b8,b6} on b18 & {b2,b9,b4} on b19 & b11,b12,b13 are_mutually_different & b1 <> b3 & b1 <> b5 & b1 <> b7 & b1 <> b2 & b1 <> b4 & b1 <> b6 & b3 <> b2 & b5 <> b4 & b7 <> b6
holds ex b20 being Element of the Lines of a1 st
{b8,b9,b10} on b20;
end;
:: INCPROJ:dfs 13
definiens
let a1 be IncProjStr;
To prove
a1 is Desarguesian
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the Points of a1
for b11, b12, b13, b14, b15, b16, b17, b18, b19 being Element of the Lines of a1
st {b1,b2,b3} on b11 & {b1,b5,b4} on b12 & {b1,b7,b6} on b13 & {b7,b5,b10} on b14 & {b7,b8,b3} on b15 & {b5,b9,b3} on b16 & {b10,b4,b6} on b17 & {b2,b8,b6} on b18 & {b2,b9,b4} on b19 & b11,b12,b13 are_mutually_different & b1 <> b3 & b1 <> b5 & b1 <> b7 & b1 <> b2 & b1 <> b4 & b1 <> b6 & b3 <> b2 & b5 <> b4 & b7 <> b6
holds ex b20 being Element of the Lines of a1 st
{b8,b9,b10} on b20;
:: INCPROJ:def 19
theorem
for b1 being IncProjStr holds
b1 is Desarguesian
iff
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the Points of b1
for b12, b13, b14, b15, b16, b17, b18, b19, b20 being Element of the Lines of b1
st {b2,b3,b4} on b12 & {b2,b6,b5} on b13 & {b2,b8,b7} on b14 & {b8,b6,b11} on b15 & {b8,b9,b4} on b16 & {b6,b10,b4} on b17 & {b11,b5,b7} on b18 & {b3,b9,b7} on b19 & {b3,b10,b5} on b20 & b12,b13,b14 are_mutually_different & b2 <> b4 & b2 <> b6 & b2 <> b8 & b2 <> b3 & b2 <> b5 & b2 <> b7 & b4 <> b3 & b6 <> b5 & b8 <> b7
holds ex b21 being Element of the Lines of b1 st
{b9,b10,b11} on b21;
:: INCPROJ:attrnot 12 => INCPROJ:attr 10
definition
let a1 be IncProjStr;
attr a1 is Pappian means
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the Points of a1
for b11, b12, b13, b14, b15, b16, b17, b18, b19 being Element of the Lines of a1
st b1,b2,b3,b4 are_mutually_different & b1,b5,b6,b7 are_mutually_different & b13 <> b16 & b1 on b13 & b1 on b16 & {b3,b7,b8} on b11 & {b4,b5,b9} on b14 & {b2,b6,b10} on b17 & {b2,b7,b9} on b12 & {b4,b6,b8} on b15 & {b3,b5,b10} on b18 & {b5,b6,b7} on b13 & {b2,b3,b4} on b16 & {b8,b9} on b19
holds b10 on b19;
end;
:: INCPROJ:dfs 14
definiens
let a1 be IncProjStr;
To prove
a1 is Pappian
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the Points of a1
for b11, b12, b13, b14, b15, b16, b17, b18, b19 being Element of the Lines of a1
st b1,b2,b3,b4 are_mutually_different & b1,b5,b6,b7 are_mutually_different & b13 <> b16 & b1 on b13 & b1 on b16 & {b3,b7,b8} on b11 & {b4,b5,b9} on b14 & {b2,b6,b10} on b17 & {b2,b7,b9} on b12 & {b4,b6,b8} on b15 & {b3,b5,b10} on b18 & {b5,b6,b7} on b13 & {b2,b3,b4} on b16 & {b8,b9} on b19
holds b10 on b19;
:: INCPROJ:def 20
theorem
for b1 being IncProjStr holds
b1 is Pappian
iff
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the Points of b1
for b12, b13, b14, b15, b16, b17, b18, b19, b20 being Element of the Lines of b1
st b2,b3,b4,b5 are_mutually_different & b2,b6,b7,b8 are_mutually_different & b14 <> b17 & b2 on b14 & b2 on b17 & {b4,b8,b9} on b12 & {b5,b6,b10} on b15 & {b3,b7,b11} on b18 & {b3,b8,b10} on b13 & {b5,b7,b9} on b16 & {b4,b6,b11} on b19 & {b6,b7,b8} on b14 & {b3,b4,b5} on b17 & {b9,b10} on b20
holds b11 on b20;
:: INCPROJ:exreg 2
registration
cluster linear partial up-2-dimensional up-3-rank Vebleian up-3-dimensional at_most-3-dimensional Fanoian Desarguesian IncProjStr;
end;
:: INCPROJ:exreg 3
registration
cluster linear partial up-2-dimensional up-3-rank Vebleian up-3-dimensional at_most-3-dimensional Fanoian Pappian IncProjStr;
end;
:: INCPROJ:exreg 4
registration
cluster linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Fanoian Desarguesian IncProjStr;
end;
:: INCPROJ:exreg 5
registration
cluster linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Fanoian Pappian IncProjStr;
end;