Article PROJRED1, MML version 4.99.1005

:: PROJRED1:th 1
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of the Lines of b1 holds
   ex b3 being Element of the Points of b1 st
      not b3 on b2;

:: PROJRED1:th 2
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of the Points of b1 holds
   ex b3 being Element of the Lines of b1 st
      not b2 on b3;

:: PROJRED1:th 3
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Lines of b1
      st b2 <> b3
   holds ex b4, b5 being Element of the Points of b1 st
      b4 on b2 & not b4 on b3 & b5 on b3 & not b5 on b2;

:: PROJRED1:th 4
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Points of b1
      st b2 <> b3
   holds ex b4, b5 being Element of the Lines of b1 st
      b2 on b4 & not b2 on b5 & b3 on b5 & not b3 on b4;

:: PROJRED1:th 5
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of the Points of b1 holds
   ex b3, b4, b5 being Element of the Lines of b1 st
      b2 on b3 & b2 on b4 & b2 on b5 & b3 <> b4 & b4 <> b5 & b5 <> b3;

:: PROJRED1:th 6
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Lines of b1 holds
ex b4 being Element of the Points of b1 st
   not b4 on b2 & not b4 on b3;

:: PROJRED1:th 7
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of the Lines of b1 holds
   ex b3 being Element of the Points of b1 st
      b3 on b2;

:: PROJRED1:th 8
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Points of b1
for b4 being Element of the Lines of b1 holds
   ex b5 being Element of the Points of b1 st
      b5 on b4 & b5 <> b2 & b5 <> b3;

:: PROJRED1:th 9
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3 being Element of the Points of b1 holds
ex b4 being Element of the Lines of b1 st
   not b2 on b4 & not b3 on b4;

:: PROJRED1:th 12
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
for b6, b7, b8, b9 being Element of the Lines of b1
      st b2 on b6 & b2 on b7 & b6 <> b7 & b3 on b6 & b2 <> b3 & b4 on b7 & b5 on b7 & b4 <> b5 & b3 on b8 & b4 on b8 & b3 on b9 & b5 on b9
   holds b8 <> b9;

:: PROJRED1:th 13
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5 being Element of the Lines of b1
      st {b2,b3,b4} on b5
   holds {b2,b4,b3} on b5 & {b3,b2,b4} on b5 & {b3,b4,b2} on b5 & {b4,b2,b3} on b5 & {b4,b3,b2} on b5;

:: PROJRED1:th 14
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Desarguesian IncProjStr
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 <> b8 & b2 <> b3 & b2 <> b5 & b6 <> b5
   holds ex b21 being Element of the Lines of b1 st
      {b9,b10,b11} on b21;

:: PROJRED1:th 15
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
   st ex b2 being Element of the Lines of b1 st
        ex b3, b4, b5, b6 being Element of the Points of b1 st
           b3 on b2 & b4 on b2 & b5 on b2 & b6 on b2 & b3,b4,b5,b6 are_mutually_different
for b2 being Element of the Lines of b1 holds
   ex b3, b4, b5, b6 being Element of the Points of b1 st
      b3 on b2 & b4 on b2 & b5 on b2 & b6 on b2 & b3,b4,b5,b6 are_mutually_different;

:: PROJRED1:th 16
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Fanoian IncProjStr holds
   ex b2, b3, b4, b5, b6, b7, b8 being Element of the Points of b1 st
      ex b9, b10, b11, b12, b13, b14, b15, b16 being Element of the Lines of b1 st
         not b3 on b13 & not b4 on b13 & not b2 on b12 & not b5 on b12 & not b2 on b14 & not b4 on b14 & not b3 on b15 & not b5 on b15 & {b6,b2,b5} on b13 & {b6,b3,b4} on b12 & {b7,b3,b5} on b14 & {b7,b2,b4} on b15 & {b8,b2,b3} on b9 & {b8,b4,b5} on b10 & {b6,b7} on b11 & not b8 on b11;

:: PROJRED1:th 17
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Fanoian IncProjStr holds
   ex b2 being Element of the Points of b1 st
      ex b3, b4, b5, b6 being Element of the Lines of b1 st
         b2 on b3 & b2 on b4 & b2 on b5 & b2 on b6 & b3,b4,b5,b6 are_mutually_different;

:: PROJRED1:th 18
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Fanoian IncProjStr holds
   ex b2, b3, b4, b5 being Element of the Points of b1 st
      ex b6 being Element of the Lines of b1 st
         b2 on b6 & b3 on b6 & b4 on b6 & b5 on b6 & b2,b3,b4,b5 are_mutually_different;

:: PROJRED1:th 19
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian Fanoian IncProjStr
for b2 being Element of the Lines of b1 holds
   ex b3, b4, b5, b6 being Element of the Points of b1 st
      b3 on b2 & b4 on b2 & b5 on b2 & b6 on b2 & b3,b4,b5,b6 are_mutually_different;

:: PROJRED1:funcnot 1 => PROJRED1:func 1
definition
  let a1 be linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr;
  let a2, a3 be Element of the Lines of a1;
  let a4 be Element of the Points of a1;
  assume not a4 on a2 & not a4 on a3;
  func IncProj(A2,A4,A3) -> Function-like Relation of the Points of a1,the Points of a1 means
    dom it c= the Points of a1 &
     (for b1 being Element of the Points of a1 holds
           b1 in dom it
        iff
           b1 on a2) &
     (for b1, b2 being Element of the Points of a1
           st b1 on a2 & b2 on a3
        holds    it . b1 = b2
        iff
           ex b3 being Element of the Lines of a1 st
              a4 on b3 & b1 on b3 & b2 on b3);
end;

:: PROJRED1:def 1
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Lines of b1
for b4 being Element of the Points of b1
   st not b4 on b2 & not b4 on b3
for b5 being Function-like Relation of the Points of b1,the Points of b1 holds
      b5 = IncProj(b2,b4,b3)
   iff
      dom b5 c= the Points of b1 &
       (for b6 being Element of the Points of b1 holds
             b6 in dom b5
          iff
             b6 on b2) &
       (for b6, b7 being Element of the Points of b1
             st b6 on b2 & b7 on b3
          holds    b5 . b6 = b7
          iff
             ex b8 being Element of the Lines of b1 st
                b4 on b8 & b6 on b8 & b7 on b8);

:: PROJRED1:th 21
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1
   st not b2 on b3
for b4 being Element of the Points of b1
      st b4 on b3
   holds (IncProj(b3,b2,b3)) . b4 = b4;

:: PROJRED1:th 22
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
      st not b2 on b4 & not b2 on b5 & b3 on b4
   holds (IncProj(b4,b2,b5)) . b3 is Element of the Points of b1;

:: PROJRED1:th 23
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5, b6 being Element of the Lines of b1
      st not b2 on b5 & not b2 on b6 & b3 on b5 & b4 = (IncProj(b5,b2,b6)) . b3
   holds b4 on b6;

:: PROJRED1:th 24
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
      st not b2 on b4 & not b2 on b5 & b3 in proj2 IncProj(b4,b2,b5)
   holds b3 on b5;

:: PROJRED1:th 25
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5, b6 being Element of the Lines of b1
      st not b2 on b4 & not b2 on b5 & not b3 on b5 & not b3 on b6
   holds proj1 ((IncProj(b4,b2,b5)) * IncProj(b5,b3,b6)) = dom IncProj(b4,b2,b5) &
    proj2 ((IncProj(b4,b2,b5)) * IncProj(b5,b3,b6)) = proj2 IncProj(b5,b3,b6);

:: PROJRED1:th 26
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2 being Element of the Points of b1
for b3, b4 being Element of the Lines of b1
for b5, b6, b7, b8 being Element of the Points of b1
      st not b2 on b3 & not b2 on b4 & b5 on b3 & b6 on b3 & (IncProj(b3,b2,b4)) . b5 = b7 & (IncProj(b3,b2,b4)) . b6 = b8 & b7 = b8
   holds b5 = b6;

:: PROJRED1:th 27
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
      st not b2 on b4 & not b2 on b5 & b3 on b4 & b3 on b5
   holds (IncProj(b4,b2,b5)) . b3 = b3;

:: PROJRED1:th 28
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional Desarguesian IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5, b6, b7 being Element of the Lines of b1
      st not b2 on b5 & not b2 on b6 & not b3 on b6 & not b3 on b7 & b4 on b5 & b4 on b6 & b4 on b7 & b5 <> b7
   holds ex b8 being Element of the Points of b1 st
      not b8 on b5 &
       not b8 on b7 &
       (IncProj(b5,b2,b6)) * IncProj(b6,b3,b7) = IncProj(b5,b8,b7);