Article PROJDES1, MML version 4.99.1005

:: PROJDES1:th 1
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3,b4 is_collinear
   holds b3,b4,b2 is_collinear & b4,b2,b3 is_collinear & b3,b2,b4 is_collinear & b2,b4,b3 is_collinear & b4,b3,b2 is_collinear;

:: PROJDES1:th 5
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2 being Element of the carrier of b1 holds
   ex b3, b4 being Element of the carrier of b1 st
      not b2,b3,b4 is_collinear;

:: PROJDES1:th 6
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2 <> b5
   holds not b2,b5,b4 is_collinear;

:: PROJDES1:th 7
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b4,b5 is_collinear
   holds b2 = b5;

:: PROJDES1:th 8
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b4,b5 is_collinear & b3,b4,b6 is_collinear & b4 <> b5 & b7,b5,b6 is_collinear & b2,b3,b7 is_collinear & b2 <> b7
   holds b6 <> b4;

:: PROJDES1:prednot 1 => PROJDES1:pred 1
definition
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3,A4,A5 are_coplanar means
    ex b1 being Element of the carrier of a1 st
       a2,a3,b1 is_collinear & a4,a5,b1 is_collinear;
end;

:: PROJDES1:dfs 1
definiens
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3,a4,a5 are_coplanar
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       a2,a3,b1 is_collinear & a4,a5,b1 is_collinear;

:: PROJDES1:def 1
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3,b4,b5 are_coplanar
iff
   ex b6 being Element of the carrier of b1 st
      b2,b3,b6 is_collinear & b4,b5,b6 is_collinear;

:: PROJDES1:th 10
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st (not b2,b3,b4 is_collinear & not b3,b4,b5 is_collinear & not b4,b5,b2 is_collinear implies b5,b2,b3 is_collinear)
   holds b2,b3,b4,b5 are_coplanar;

:: PROJDES1:th 11
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3,b4,b5 are_coplanar
   holds b3,b4,b5,b2 are_coplanar & b4,b5,b2,b3 are_coplanar & b5,b2,b3,b4 are_coplanar & b3,b2,b4,b5 are_coplanar & b4,b3,b5,b2 are_coplanar & b5,b4,b2,b3 are_coplanar & b2,b5,b3,b4 are_coplanar & b2,b4,b5,b3 are_coplanar & b3,b5,b2,b4 are_coplanar & b4,b2,b3,b5 are_coplanar & b5,b3,b4,b2 are_coplanar & b4,b2,b5,b3 are_coplanar & b5,b3,b2,b4 are_coplanar & b2,b4,b3,b5 are_coplanar & b3,b5,b4,b2 are_coplanar & b2,b3,b5,b4 are_coplanar & b2,b5,b4,b3 are_coplanar & b3,b4,b2,b5 are_coplanar & b3,b2,b5,b4 are_coplanar & b4,b3,b2,b5 are_coplanar & b4,b5,b3,b2 are_coplanar & b5,b2,b4,b3 are_coplanar & b5,b4,b3,b2 are_coplanar;

:: PROJDES1:th 12
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3,b4,b5 are_coplanar & b2,b3,b4,b6 are_coplanar & b2,b3,b4,b7 are_coplanar & b2,b3,b4,b8 are_coplanar
   holds b5,b6,b7,b8 are_coplanar;

:: PROJDES1:th 13
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b5,b6,b7,b2 are_coplanar & b5,b6,b7,b4 are_coplanar & b5,b6,b7,b3 are_coplanar & b2,b3,b4,b8 are_coplanar
   holds b5,b6,b7,b8 are_coplanar;

:: PROJDES1:th 14
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2 <> b3 & b2,b3,b4 is_collinear & b5,b6,b7,b2 are_coplanar & b5,b6,b7,b3 are_coplanar
   holds b5,b6,b7,b4 are_coplanar;

:: PROJDES1:th 15
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & b2,b3,b4,b5 are_coplanar & b2,b3,b4,b6 are_coplanar & b2,b3,b4,b7 are_coplanar & b2,b3,b4,b8 are_coplanar
   holds ex b9 being Element of the carrier of b1 st
      b5,b6,b9 is_collinear & b7,b8,b9 is_collinear;

:: PROJDES1:th 16
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr holds
   ex b2, b3, b4, b5 being Element of the carrier of b1 st
      not b2,b3,b4,b5 are_coplanar;

:: PROJDES1:th 17
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear
   holds ex b5 being Element of the carrier of b1 st
      not b2,b3,b4,b5 are_coplanar;

:: PROJDES1:th 18
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st (b2 <> b3 & b2 <> b4 & b3 <> b4 & b2 <> b5 & b3 <> b5 implies b5 = b4)
   holds b2,b3,b4,b5 are_coplanar;

:: PROJDES1:th 19
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not b2,b3,b4,b5 are_coplanar & b5,b2,b6 is_collinear & b2 <> b6
   holds not b2,b3,b4,b6 are_coplanar;

:: PROJDES1:th 20
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & not b5,b6,b7 is_collinear & b2,b3,b4,b8 are_coplanar & b2,b3,b4,b9 are_coplanar & b2,b3,b4,b10 are_coplanar & b5,b6,b7,b8 are_coplanar & b5,b6,b7,b9 are_coplanar & b5,b6,b7,b10 are_coplanar & not b2,b3,b4,b5 are_coplanar
   holds b8,b9,b10 is_collinear;

:: PROJDES1:th 21
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st b2 <> b3 & b4,b2,b3 is_collinear & not b2,b5,b6,b4 are_coplanar & not b3,b7,b8 is_collinear & b2,b5,b9 is_collinear & b3,b7,b9 is_collinear & b5,b6,b10 is_collinear & b7,b8,b10 is_collinear & b2,b6,b11 is_collinear & b3,b8,b11 is_collinear
   holds b9,b10,b11 is_collinear;

:: PROJDES1:th 22
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not b2,b3,b4,b5 are_coplanar & b2,b3,b4,b6 are_coplanar & not b2,b3,b6 is_collinear
   holds not b2,b3,b5,b6 are_coplanar;

:: PROJDES1:th 23
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st not b2,b3,b4,b5 are_coplanar & b5,b2,b6 is_collinear & b5,b3,b7 is_collinear & b5,b4,b8 is_collinear & b5 <> b6 & b5 <> b7 & b5 <> b8
   holds not b6,b7,b8 is_collinear & not b6,b7,b8,b5 are_coplanar;

:: PROJDES1:th 24
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13 being Element of the carrier of b1
      st b2,b3,b4,b5 are_coplanar & not b2,b3,b4,b6 are_coplanar & not b2,b3,b6,b5 are_coplanar & not b3,b4,b6,b5 are_coplanar & not b2,b4,b6,b5 are_coplanar & b5,b6,b7 is_collinear & b5,b2,b8 is_collinear & b5,b3,b9 is_collinear & b5,b4,b10 is_collinear & b2,b6,b11 is_collinear & b8,b7,b11 is_collinear & b3,b6,b12 is_collinear & b9,b7,b12 is_collinear & b4,b6,b13 is_collinear & b5 <> b8 & b5 <> b7 & b6 <> b7 & b5 <> b9
   holds not b11,b12,b13 is_collinear;

:: PROJDES1:prednot 2 => PROJDES1:pred 2
definition
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3,A4,A5 constitute_a_quadrangle means
    not a3,a4,a5 is_collinear & not a2,a3,a4 is_collinear & not a2,a4,a5 is_collinear & not a2,a5,a3 is_collinear;
end;

:: PROJDES1:dfs 2
definiens
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3,a4,a5 constitute_a_quadrangle
it is sufficient to prove
  thus not a3,a4,a5 is_collinear & not a2,a3,a4 is_collinear & not a2,a4,a5 is_collinear & not a2,a5,a3 is_collinear;

:: PROJDES1:def 2
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3,b4,b5 constitute_a_quadrangle
iff
   not b3,b4,b5 is_collinear & not b2,b3,b4 is_collinear & not b2,b4,b5 is_collinear & not b2,b5,b3 is_collinear;

:: PROJDES1:th 26
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st not b2,b3,b4 is_collinear & not b2,b4,b5 is_collinear & not b2,b3,b5 is_collinear & b2,b3,b6 is_collinear & b2,b4,b7 is_collinear & b2,b5,b8 is_collinear & b3,b4,b9 is_collinear & b6,b7,b9 is_collinear & b3 <> b6 & b4,b5,b10 is_collinear & b7,b8,b10 is_collinear & b3,b5,b11 is_collinear & b4 <> b7 & b6,b8,b11 is_collinear & b2 <> b6 & b2 <> b7 & b2 <> b8
   holds b10,b11,b9 is_collinear;

:: PROJDES1:th 27
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional CollStr holds
   b1 is Desarguesian;

:: PROJDES1:condreg 1
registration
  cluster non empty reflexive transitive proper Vebleian at_least_3rank up-3-dimensional -> Desarguesian (CollStr);
end;