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;