Article ANPROJ_2, MML version 4.99.1005

:: ANPROJ_2:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st for b5, b6, b7 being Element of REAL
              st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
           holds b5 = 0 & b6 = 0 & b7 = 0
   holds b2 is non-zero(b1) & b3 is non-zero(b1) & b4 is non-zero(b1) & not b2,b3,b4 are_LinDep & not are_Prop b2,b3;

:: ANPROJ_2:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st for b6, b7, b8, b9 being Element of REAL
              st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
           holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0
   holds b2 is non-zero(b1) & b3 is non-zero(b1) & not are_Prop b2,b3 & b4 is non-zero(b1) & b5 is non-zero(b1) & not are_Prop b4,b5 & not b2,b3,b4 are_LinDep & not b4,b5,b2 are_LinDep;

:: ANPROJ_2:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
   st (for b5 being Element of the carrier of b1 holds
         ex b6, b7, b8 being Element of REAL st
            b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4)) &
      (for b5, b6, b7 being Element of REAL
            st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
         holds b5 = 0 & b6 = 0 & b7 = 0)
for b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
   b2,b3,b7 are_LinDep & b5,b6,b7 are_LinDep & b7 is non-zero(b1);

:: ANPROJ_2:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
   st (for b6 being Element of the carrier of b1 holds
         ex b7, b8, b9, b10 being Element of REAL st
            b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5)) &
      (for b6, b7, b8, b9 being Element of REAL
            st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
         holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0)
for b6, b7 being Element of the carrier of b1
      st b6 is non-zero(b1) & b7 is non-zero(b1)
   holds ex b8, b9 being Element of the carrier of b1 st
      b6,b7,b9 are_LinDep & b3,b4,b8 are_LinDep & b2,b9,b8 are_LinDep & b8 is non-zero(b1) & b9 is non-zero(b1);

:: ANPROJ_2:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
   st for b6, b7, b8, b9 being Element of REAL
           st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
        holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0
for b6 being Element of the carrier of b1
      st b6 is non-zero(b1) & b2,b3,b6 are_LinDep
   holds not b4,b5,b6 are_LinDep;

:: ANPROJ_2:prednot 1 => ANPROJ_2:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  pred A2,A3,A4 are_Prop_Vect means
    a2 is non-zero(a1) & a3 is non-zero(a1) & a4 is non-zero(a1);
end;

:: ANPROJ_2:dfs 1
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4 be Element of the carrier of a1;
To prove
     a2,a3,a4 are_Prop_Vect
it is sufficient to prove
  thus a2 is non-zero(a1) & a3 is non-zero(a1) & a4 is non-zero(a1);

:: ANPROJ_2:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   b2,b3,b4 are_Prop_Vect
iff
   b2 is non-zero(b1) & b3 is non-zero(b1) & b4 is non-zero(b1);

:: ANPROJ_2:prednot 2 => ANPROJ_2:pred 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
  pred A2,A3,A4,A5,A6,A7 lie_on_a_triangle means
    a2,a3,a7 are_LinDep & a2,a4,a6 are_LinDep & a3,a4,a5 are_LinDep;
end;

:: ANPROJ_2:dfs 2
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
To prove
     a2,a3,a4,a5,a6,a7 lie_on_a_triangle
it is sufficient to prove
  thus a2,a3,a7 are_LinDep & a2,a4,a6 are_LinDep & a3,a4,a5 are_LinDep;

:: ANPROJ_2:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
   b2,b3,b4,b5,b6,b7 lie_on_a_triangle
iff
   b2,b3,b7 are_LinDep & b2,b4,b6 are_LinDep & b3,b4,b5 are_LinDep;

:: ANPROJ_2:prednot 3 => ANPROJ_2:pred 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
  pred A2,A3,A4,A5,A6,A7,A8 are_perspective means
    a2,a3,a6 are_LinDep & a2,a4,a7 are_LinDep & a2,a5,a8 are_LinDep;
end;

:: ANPROJ_2:dfs 3
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
To prove
     a2,a3,a4,a5,a6,a7,a8 are_perspective
it is sufficient to prove
  thus a2,a3,a6 are_LinDep & a2,a4,a7 are_LinDep & a2,a5,a8 are_LinDep;

:: ANPROJ_2:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1 holds
   b2,b3,b4,b5,b6,b7,b8 are_perspective
iff
   b2,b3,b6 are_LinDep & b2,b4,b7 are_LinDep & b2,b5,b8 are_LinDep;

:: ANPROJ_2:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3,b4 are_LinDep & not are_Prop b2,b3 & not are_Prop b2,b4 & not are_Prop b3,b4 & b2,b3,b4 are_Prop_Vect
   holds (ex b5, b6 being Element of REAL st
       b6 * b4 = b2 + (b5 * b3) & b5 <> 0 & b6 <> 0) &
    (ex b5, b6 being Element of REAL st
       b4 = (b6 * b2) + (b5 * b3) & b6 <> 0 & b5 <> 0);

:: ANPROJ_2:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3,b4 are_LinDep & not are_Prop b2,b3 & b2,b3,b4 are_Prop_Vect
   holds ex b5, b6 being Element of REAL st
      b4 = (b5 * b2) + (b6 * b3);

:: ANPROJ_2:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st b2 is non-zero(b1) & b3,b4,b5 are_Prop_Vect & b6,b7,b8 are_Prop_Vect & b9,b10,b11 are_Prop_Vect & b2,b3,b4,b5,b6,b7,b8 are_perspective & not are_Prop b2,b6 & not are_Prop b2,b7 & not are_Prop b2,b8 & not are_Prop b3,b6 & not are_Prop b4,b7 & not are_Prop b5,b8 & not b2,b3,b4 are_LinDep & not b2,b3,b5 are_LinDep & not b2,b4,b5 are_LinDep & b3,b4,b5,b9,b10,b11 lie_on_a_triangle & b6,b7,b8,b9,b10,b11 lie_on_a_triangle
   holds b9,b10,b11 are_LinDep;

:: ANPROJ_2:prednot 4 => ANPROJ_2:pred 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
  pred A2,A3,A4,A5,A6,A7,A8 lie_on_an_angle means
    not a2,a3,a6 are_LinDep & a2,a3,a4 are_LinDep & a2,a3,a5 are_LinDep & a2,a6,a7 are_LinDep & a2,a6,a8 are_LinDep;
end;

:: ANPROJ_2:dfs 4
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
To prove
     a2,a3,a4,a5,a6,a7,a8 lie_on_an_angle
it is sufficient to prove
  thus not a2,a3,a6 are_LinDep & a2,a3,a4 are_LinDep & a2,a3,a5 are_LinDep & a2,a6,a7 are_LinDep & a2,a6,a8 are_LinDep;

:: ANPROJ_2:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1 holds
   b2,b3,b4,b5,b6,b7,b8 lie_on_an_angle
iff
   not b2,b3,b6 are_LinDep & b2,b3,b4 are_LinDep & b2,b3,b5 are_LinDep & b2,b6,b7 are_LinDep & b2,b6,b8 are_LinDep;

:: ANPROJ_2:prednot 5 => ANPROJ_2:pred 5
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
  pred A2,A3,A4,A5,A6,A7,A8 are_half_mutually_not_Prop means
    not are_Prop a2,a4 & not are_Prop a2,a5 & not are_Prop a2,a7 & not are_Prop a2,a8 & not are_Prop a3,a4 & not are_Prop a3,a5 & not are_Prop a6,a7 & not are_Prop a6,a8 & not are_Prop a4,a5 & not are_Prop a7,a8;
end;

:: ANPROJ_2:dfs 5
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
To prove
     a2,a3,a4,a5,a6,a7,a8 are_half_mutually_not_Prop
it is sufficient to prove
  thus not are_Prop a2,a4 & not are_Prop a2,a5 & not are_Prop a2,a7 & not are_Prop a2,a8 & not are_Prop a3,a4 & not are_Prop a3,a5 & not are_Prop a6,a7 & not are_Prop a6,a8 & not are_Prop a4,a5 & not are_Prop a7,a8;

:: ANPROJ_2:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1 holds
   b2,b3,b4,b5,b6,b7,b8 are_half_mutually_not_Prop
iff
   not are_Prop b2,b4 & not are_Prop b2,b5 & not are_Prop b2,b7 & not are_Prop b2,b8 & not are_Prop b3,b4 & not are_Prop b3,b5 & not are_Prop b6,b7 & not are_Prop b6,b8 & not are_Prop b4,b5 & not are_Prop b7,b8;

:: ANPROJ_2:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st b2 is non-zero(b1) & b3,b4,b5 are_Prop_Vect & b6,b7,b8 are_Prop_Vect & b9,b10,b11 are_Prop_Vect & b2,b3,b4,b5,b6,b7,b8 lie_on_an_angle & b2,b3,b4,b5,b6,b7,b8 are_half_mutually_not_Prop & b3,b7,b11 are_LinDep & b6,b4,b11 are_LinDep & b3,b8,b10 are_LinDep & b5,b6,b10 are_LinDep & b4,b8,b9 are_LinDep & b5,b7,b9 are_LinDep
   holds b9,b10,b11 are_LinDep;

:: ANPROJ_2:th 10
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1 holds
ex b5, b6, b7 being Element of Funcs(b1,REAL) st
   (for b8 being set
          st b8 in b1
       holds (b8 = b2 implies b5 . b8 = 1) & (b8 = b2 or b5 . b8 = 0)) &
    (for b8 being set
          st b8 in b1
       holds (b8 = b3 implies b6 . b8 = 1) & (b8 = b3 or b6 . b8 = 0)) &
    (for b8 being set
          st b8 in b1
       holds (b8 = b4 implies b7 . b8 = 1) & (b8 = b4 or b7 . b8 = 0));

:: ANPROJ_2:th 11
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,REAL)
for b5, b6, b7 being Element of b1
   st b5 in b1 &
      b6 in b1 &
      b7 in b1 &
      b5 <> b6 &
      b5 <> b7 &
      b6 <> b7 &
      (for b8 being set
            st b8 in b1
         holds (b8 = b5 implies b2 . b8 = 1) & (b8 = b5 or b2 . b8 = 0)) &
      (for b8 being set
            st b8 in b1
         holds (b8 = b6 implies b3 . b8 = 1) & (b8 = b6 or b3 . b8 = 0)) &
      (for b8 being set
            st b8 in b1
         holds (b8 = b7 implies b4 . b8 = 1) & (b8 = b7 or b4 . b8 = 0))
for b8, b9, b10 being Element of REAL
      st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b8,b2],(RealFuncExtMult b1) . [b9,b3]),(RealFuncExtMult b1) . [b10,b4]) = RealFuncZero b1
   holds b8 = 0 & b9 = 0 & b10 = 0;

:: ANPROJ_2:th 12
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
      st b2 in b1 & b3 in b1 & b4 in b1 & b2 <> b3 & b2 <> b4 & b3 <> b4
   holds ex b5, b6, b7 being Element of Funcs(b1,REAL) st
      for b8, b9, b10 being Element of REAL
            st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b8,b5],(RealFuncExtMult b1) . [b9,b6]),(RealFuncExtMult b1) . [b10,b7]) = RealFuncZero b1
         holds b8 = 0 & b9 = 0 & b10 = 0;

:: ANPROJ_2:th 13
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,REAL)
for b5, b6, b7 being Element of b1
   st b1 = {b5,b6,b7} &
      b5 <> b6 &
      b5 <> b7 &
      b6 <> b7 &
      (for b8 being set
            st b8 in b1
         holds (b8 = b5 implies b2 . b8 = 1) & (b8 = b5 or b2 . b8 = 0)) &
      (for b8 being set
            st b8 in b1
         holds (b8 = b6 implies b3 . b8 = 1) & (b8 = b6 or b3 . b8 = 0)) &
      (for b8 being set
            st b8 in b1
         holds (b8 = b7 implies b4 . b8 = 1) & (b8 = b7 or b4 . b8 = 0))
for b8 being Element of Funcs(b1,REAL) holds
   ex b9, b10, b11 being Element of REAL st
      b8 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b9,b2],(RealFuncExtMult b1) . [b10,b3]),(RealFuncExtMult b1) . [b11,b4]);

:: ANPROJ_2:th 14
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
      st b1 = {b2,b3,b4} & b2 <> b3 & b2 <> b4 & b3 <> b4
   holds ex b5, b6, b7 being Element of Funcs(b1,REAL) st
      for b8 being Element of Funcs(b1,REAL) holds
         ex b9, b10, b11 being Element of REAL st
            b8 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b9,b5],(RealFuncExtMult b1) . [b10,b6]),(RealFuncExtMult b1) . [b11,b7]);

:: ANPROJ_2:th 15
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
      st b1 = {b2,b3,b4} & b2 <> b3 & b2 <> b4 & b3 <> b4
   holds ex b5, b6, b7 being Element of Funcs(b1,REAL) st
      (for b8, b9, b10 being Element of REAL
             st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b8,b5],(RealFuncExtMult b1) . [b9,b6]),(RealFuncExtMult b1) . [b10,b7]) = RealFuncZero b1
          holds b8 = 0 & b9 = 0 & b10 = 0) &
       (for b8 being Element of Funcs(b1,REAL) holds
          ex b9, b10, b11 being Element of REAL st
             b8 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b9,b5],(RealFuncExtMult b1) . [b10,b6]),(RealFuncExtMult b1) . [b11,b7]));

:: ANPROJ_2:th 16
theorem
ex b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct st
   ex b2, b3, b4 being Element of the carrier of b1 st
      (for b5, b6, b7 being Element of REAL
             st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
          holds b5 = 0 & b6 = 0 & b7 = 0) &
       (for b5 being Element of the carrier of b1 holds
          ex b6, b7, b8 being Element of REAL st
             b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4));

:: ANPROJ_2:th 17
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of b1 holds
ex b6, b7, b8, b9 being Element of Funcs(b1,REAL) st
   (for b10 being set
          st b10 in b1
       holds (b10 = b2 implies b6 . b10 = 1) & (b10 = b2 or b6 . b10 = 0)) &
    (for b10 being set
          st b10 in b1
       holds (b10 = b3 implies b7 . b10 = 1) & (b10 = b3 or b7 . b10 = 0)) &
    (for b10 being set
          st b10 in b1
       holds (b10 = b4 implies b8 . b10 = 1) & (b10 = b4 or b8 . b10 = 0)) &
    (for b10 being set
          st b10 in b1
       holds (b10 = b5 implies b9 . b10 = 1) & (b10 = b5 or b9 . b10 = 0));

:: ANPROJ_2:th 18
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,REAL)
for b6, b7, b8, b9 being Element of b1
   st b6 in b1 &
      b7 in b1 &
      b8 in b1 &
      b9 in b1 &
      b6 <> b7 &
      b6 <> b8 &
      b6 <> b9 &
      b7 <> b8 &
      b7 <> b9 &
      b8 <> b9 &
      (for b10 being set
            st b10 in b1
         holds (b10 = b6 implies b2 . b10 = 1) & (b10 = b6 or b2 . b10 = 0)) &
      (for b10 being set
            st b10 in b1
         holds (b10 = b7 implies b3 . b10 = 1) & (b10 = b7 or b3 . b10 = 0)) &
      (for b10 being set
            st b10 in b1
         holds (b10 = b8 implies b4 . b10 = 1) & (b10 = b8 or b4 . b10 = 0)) &
      (for b10 being set
            st b10 in b1
         holds (b10 = b9 implies b5 . b10 = 1) & (b10 = b9 or b5 . b10 = 0))
for b10, b11, b12, b13 being Element of REAL
      st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b10,b2],(RealFuncExtMult b1) . [b11,b3]),(RealFuncExtMult b1) . [b12,b4]),(RealFuncExtMult b1) . [b13,b5]) = RealFuncZero b1
   holds b10 = 0 & b11 = 0 & b12 = 0 & b13 = 0;

:: ANPROJ_2:th 19
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of b1
      st b2 in b1 & b3 in b1 & b4 in b1 & b5 in b1 & b2 <> b3 & b2 <> b4 & b2 <> b5 & b3 <> b4 & b3 <> b5 & b4 <> b5
   holds ex b6, b7, b8, b9 being Element of Funcs(b1,REAL) st
      for b10, b11, b12, b13 being Element of REAL
            st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b10,b6],(RealFuncExtMult b1) . [b11,b7]),(RealFuncExtMult b1) . [b12,b8]),(RealFuncExtMult b1) . [b13,b9]) = RealFuncZero b1
         holds b10 = 0 & b11 = 0 & b12 = 0 & b13 = 0;

:: ANPROJ_2:th 20
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,REAL)
for b6, b7, b8, b9 being Element of b1
   st b1 = {b6,b7,b8,b9} &
      b6 <> b7 &
      b6 <> b8 &
      b6 <> b9 &
      b7 <> b8 &
      b7 <> b9 &
      b8 <> b9 &
      (for b10 being set
            st b10 in b1
         holds (b10 = b6 implies b2 . b10 = 1) & (b10 = b6 or b2 . b10 = 0)) &
      (for b10 being set
            st b10 in b1
         holds (b10 = b7 implies b3 . b10 = 1) & (b10 = b7 or b3 . b10 = 0)) &
      (for b10 being set
            st b10 in b1
         holds (b10 = b8 implies b4 . b10 = 1) & (b10 = b8 or b4 . b10 = 0)) &
      (for b10 being set
            st b10 in b1
         holds (b10 = b9 implies b5 . b10 = 1) & (b10 = b9 or b5 . b10 = 0))
for b10 being Element of Funcs(b1,REAL) holds
   ex b11, b12, b13, b14 being Element of REAL st
      b10 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b11,b2],(RealFuncExtMult b1) . [b12,b3]),(RealFuncExtMult b1) . [b13,b4]),(RealFuncExtMult b1) . [b14,b5]);

:: ANPROJ_2:th 21
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of b1
      st b1 = {b2,b3,b4,b5} & b2 <> b3 & b2 <> b4 & b2 <> b5 & b3 <> b4 & b3 <> b5 & b4 <> b5
   holds ex b6, b7, b8, b9 being Element of Funcs(b1,REAL) st
      for b10 being Element of Funcs(b1,REAL) holds
         ex b11, b12, b13, b14 being Element of REAL st
            b10 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b11,b6],(RealFuncExtMult b1) . [b12,b7]),(RealFuncExtMult b1) . [b13,b8]),(RealFuncExtMult b1) . [b14,b9]);

:: ANPROJ_2:th 22
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of b1
      st b1 = {b2,b3,b4,b5} & b2 <> b3 & b2 <> b4 & b2 <> b5 & b3 <> b4 & b3 <> b5 & b4 <> b5
   holds ex b6, b7, b8, b9 being Element of Funcs(b1,REAL) st
      (for b10, b11, b12, b13 being Element of REAL
             st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b10,b6],(RealFuncExtMult b1) . [b11,b7]),(RealFuncExtMult b1) . [b12,b8]),(RealFuncExtMult b1) . [b13,b9]) = RealFuncZero b1
          holds b10 = 0 & b11 = 0 & b12 = 0 & b13 = 0) &
       (for b10 being Element of Funcs(b1,REAL) holds
          ex b11, b12, b13, b14 being Element of REAL st
             b10 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b11,b6],(RealFuncExtMult b1) . [b12,b7]),(RealFuncExtMult b1) . [b13,b8]),(RealFuncExtMult b1) . [b14,b9]));

:: ANPROJ_2:th 23
theorem
ex b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct st
   ex b2, b3, b4, b5 being Element of the carrier of b1 st
      (for b6, b7, b8, b9 being Element of REAL
             st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
          holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0) &
       (for b6 being Element of the carrier of b1 holds
          ex b7, b8, b9, b10 being Element of REAL st
             b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5));

:: ANPROJ_2:attrnot 1 => ANPROJ_2:attr 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  attr a1 is up-3-dimensional means
    ex b1, b2, b3 being Element of the carrier of a1 st
       for b4, b5, b6 being Element of REAL
             st ((b4 * b1) + (b5 * b2)) + (b6 * b3) = 0. a1
          holds b4 = 0 & b5 = 0 & b6 = 0;
end;

:: ANPROJ_2:dfs 6
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
To prove
     a1 is up-3-dimensional
it is sufficient to prove
  thus ex b1, b2, b3 being Element of the carrier of a1 st
       for b4, b5, b6 being Element of REAL
             st ((b4 * b1) + (b5 * b2)) + (b6 * b3) = 0. a1
          holds b4 = 0 & b5 = 0 & b6 = 0;

:: ANPROJ_2:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
      b1 is up-3-dimensional
   iff
      ex b2, b3, b4 being Element of the carrier of b1 st
         for b5, b6, b7 being Element of REAL
               st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
            holds b5 = 0 & b6 = 0 & b7 = 0;

:: ANPROJ_2:exreg 1
registration
  cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional RLSStruct;
end;

:: ANPROJ_2:condreg 1
registration
  cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional -> non trivial (RLSStruct);
end;

:: ANPROJ_2:attrnot 2 => COLLSP:attr 2
definition
  let a1 be non empty CollStr;
  attr a1 is reflexive means
    for b1, b2, b3 being Element of the carrier of a1 holds
    b1,b2,b1 is_collinear & b1,b1,b2 is_collinear & b1,b2,b2 is_collinear;
end;

:: ANPROJ_2:dfs 7
definiens
  let a1 be non empty CollStr;
To prove
     a1 is reflexive
it is sufficient to prove
  thus for b1, b2, b3 being Element of the carrier of a1 holds
    b1,b2,b1 is_collinear & b1,b1,b2 is_collinear & b1,b2,b2 is_collinear;

:: ANPROJ_2:def 7
theorem
for b1 being non empty CollStr holds
      b1 is reflexive
   iff
      for b2, b3, b4 being Element of the carrier of b1 holds
      b2,b3,b2 is_collinear & b2,b2,b3 is_collinear & b2,b3,b3 is_collinear;

:: ANPROJ_2:attrnot 3 => COLLSP:attr 3
definition
  let a1 be non empty CollStr;
  attr a1 is transitive means
    for b1, b2, b3, b4, b5 being Element of the carrier of a1
          st b1 <> b2 & b1,b2,b3 is_collinear & b1,b2,b4 is_collinear & b1,b2,b5 is_collinear
       holds b3,b4,b5 is_collinear;
end;

:: ANPROJ_2:dfs 8
definiens
  let a1 be non empty CollStr;
To prove
     a1 is transitive
it is sufficient to prove
  thus for b1, b2, b3, b4, b5 being Element of the carrier of a1
          st b1 <> b2 & b1,b2,b3 is_collinear & b1,b2,b4 is_collinear & b1,b2,b5 is_collinear
       holds b3,b4,b5 is_collinear;

:: ANPROJ_2:def 8
theorem
for b1 being non empty CollStr holds
      b1 is transitive
   iff
      for b2, b3, b4, b5, b6 being Element of the carrier of b1
            st b2 <> b3 & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b3,b6 is_collinear
         holds b4,b5,b6 is_collinear;

:: ANPROJ_2:attrnot 4 => ANPROJ_2:attr 2
definition
  let a1 be non empty CollStr;
  attr a1 is Vebleian means
    for b1, b2, b3, b4, b5 being Element of the carrier of a1
          st b1,b2,b4 is_collinear & b2,b3,b5 is_collinear
       holds ex b6 being Element of the carrier of a1 st
          b1,b3,b6 is_collinear & b4,b5,b6 is_collinear;
end;

:: ANPROJ_2:dfs 9
definiens
  let a1 be non empty CollStr;
To prove
     a1 is Vebleian
it is sufficient to prove
  thus for b1, b2, b3, b4, b5 being Element of the carrier of a1
          st b1,b2,b4 is_collinear & b2,b3,b5 is_collinear
       holds ex b6 being Element of the carrier of a1 st
          b1,b3,b6 is_collinear & b4,b5,b6 is_collinear;

:: ANPROJ_2:def 9
theorem
for b1 being non empty CollStr holds
      b1 is Vebleian
   iff
      for b2, b3, b4, b5, b6 being Element of the carrier of b1
            st b2,b3,b5 is_collinear & b3,b4,b6 is_collinear
         holds ex b7 being Element of the carrier of b1 st
            b2,b4,b7 is_collinear & b5,b6,b7 is_collinear;

:: ANPROJ_2:attrnot 5 => ANPROJ_2:attr 3
definition
  let a1 be non empty CollStr;
  attr a1 is at_least_3rank means
    for b1, b2 being Element of the carrier of a1 holds
    ex b3 being Element of the carrier of a1 st
       b1 <> b3 & b2 <> b3 & b1,b2,b3 is_collinear;
end;

:: ANPROJ_2:dfs 10
definiens
  let a1 be non empty CollStr;
To prove
     a1 is at_least_3rank
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1 holds
    ex b3 being Element of the carrier of a1 st
       b1 <> b3 & b2 <> b3 & b1,b2,b3 is_collinear;

:: ANPROJ_2:def 10
theorem
for b1 being non empty CollStr holds
      b1 is at_least_3rank
   iff
      for b2, b3 being Element of the carrier of b1 holds
      ex b4 being Element of the carrier of b1 st
         b2 <> b4 & b3 <> b4 & b2,b3,b4 is_collinear;

:: ANPROJ_2:th 24
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of ProjectiveSpace b1 holds
   b2,b3,b4 is_collinear
iff
   ex b5, b6, b7 being Element of the carrier of b1 st
      b2 = Dir b5 & b3 = Dir b6 & b4 = Dir b7 & b5 is non-zero(b1) & b6 is non-zero(b1) & b7 is non-zero(b1) & b5,b6,b7 are_LinDep;

:: ANPROJ_2:funcreg 1
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster ProjectiveSpace a1 -> strict reflexive transitive;
end;

:: ANPROJ_2:th 25
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of ProjectiveSpace b1
      st b2,b3,b4 is_collinear
   holds b2,b4,b3 is_collinear & b3,b2,b4 is_collinear & b4,b3,b2 is_collinear & b4,b2,b3 is_collinear & b3,b4,b2 is_collinear;

:: ANPROJ_2:funcreg 2
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster ProjectiveSpace a1 -> strict Vebleian;
end;

:: ANPROJ_2:funcreg 3
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional RLSStruct;
  cluster ProjectiveSpace a1 -> strict proper;
end;

:: ANPROJ_2:th 26
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st ex b2, b3 being Element of the carrier of b1 st
           for b4, b5 being Element of REAL
                 st (b4 * b2) + (b5 * b3) = 0. b1
              holds b4 = 0 & b5 = 0
   holds ProjectiveSpace b1 is at_least_3rank;

:: ANPROJ_2:funcreg 4
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional RLSStruct;
  cluster ProjectiveSpace a1 -> strict at_least_3rank;
end;

:: ANPROJ_2:exreg 2
registration
  cluster non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
end;

:: ANPROJ_2:modenot 1
definition
  mode CollProjectiveSpace is non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
end;

:: ANPROJ_2:attrnot 6 => ANPROJ_2:attr 4
definition
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
  attr a1 is Fanoian means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1,b2,b3 is_collinear & b4,b5,b3 is_collinear & b1,b4,b6 is_collinear & b2,b5,b6 is_collinear & b1,b5,b7 is_collinear & b2,b4,b7 is_collinear & b6,b3,b7 is_collinear & not b1,b2,b5 is_collinear & not b1,b2,b4 is_collinear & not b1,b4,b5 is_collinear
       holds b2,b4,b5 is_collinear;
end;

:: ANPROJ_2:dfs 11
definiens
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
To prove
     a1 is Fanoian
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1,b2,b3 is_collinear & b4,b5,b3 is_collinear & b1,b4,b6 is_collinear & b2,b5,b6 is_collinear & b1,b5,b7 is_collinear & b2,b4,b7 is_collinear & b6,b3,b7 is_collinear & not b1,b2,b5 is_collinear & not b1,b2,b4 is_collinear & not b1,b4,b5 is_collinear
       holds b2,b4,b5 is_collinear;

:: ANPROJ_2:def 11
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
      b1 is Fanoian
   iff
      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;

:: ANPROJ_2:attrnot 7 => ANPROJ_2:attr 5
definition
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
  attr a1 is Desarguesian means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
          st b1 <> b5 & b2 <> b5 & b1 <> b6 & b3 <> b6 & b1 <> b7 & b4 <> b7 & not b1,b2,b3 is_collinear & not b1,b2,b4 is_collinear & not b1,b3,b4 is_collinear & b2,b3,b10 is_collinear & b5,b6,b10 is_collinear & b3,b4,b8 is_collinear & b6,b7,b8 is_collinear & b2,b4,b9 is_collinear & b5,b7,b9 is_collinear & b1,b2,b5 is_collinear & b1,b3,b6 is_collinear & b1,b4,b7 is_collinear
       holds b8,b9,b10 is_collinear;
end;

:: ANPROJ_2:dfs 12
definiens
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
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 carrier of a1
          st b1 <> b5 & b2 <> b5 & b1 <> b6 & b3 <> b6 & b1 <> b7 & b4 <> b7 & not b1,b2,b3 is_collinear & not b1,b2,b4 is_collinear & not b1,b3,b4 is_collinear & b2,b3,b10 is_collinear & b5,b6,b10 is_collinear & b3,b4,b8 is_collinear & b6,b7,b8 is_collinear & b2,b4,b9 is_collinear & b5,b7,b9 is_collinear & b1,b2,b5 is_collinear & b1,b3,b6 is_collinear & b1,b4,b7 is_collinear
       holds b8,b9,b10 is_collinear;

:: ANPROJ_2:def 12
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
      b1 is Desarguesian
   iff
      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;

:: ANPROJ_2:attrnot 8 => ANPROJ_2:attr 6
definition
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
  attr a1 is Pappian means
    for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
          st b1 <> b3 & b1 <> b4 & b3 <> b4 & b2 <> b3 & b2 <> b4 & b1 <> b6 & b1 <> b7 & b6 <> b7 & b5 <> b6 & b5 <> b7 & not b1,b2,b5 is_collinear & b1,b2,b3 is_collinear & b1,b2,b4 is_collinear & b1,b5,b6 is_collinear & b1,b5,b7 is_collinear & b2,b6,b10 is_collinear & b5,b3,b10 is_collinear & b2,b7,b9 is_collinear & b4,b5,b9 is_collinear & b3,b7,b8 is_collinear & b4,b6,b8 is_collinear
       holds b8,b9,b10 is_collinear;
end;

:: ANPROJ_2:dfs 13
definiens
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
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 carrier of a1
          st b1 <> b3 & b1 <> b4 & b3 <> b4 & b2 <> b3 & b2 <> b4 & b1 <> b6 & b1 <> b7 & b6 <> b7 & b5 <> b6 & b5 <> b7 & not b1,b2,b5 is_collinear & b1,b2,b3 is_collinear & b1,b2,b4 is_collinear & b1,b5,b6 is_collinear & b1,b5,b7 is_collinear & b2,b6,b10 is_collinear & b5,b3,b10 is_collinear & b2,b7,b9 is_collinear & b4,b5,b9 is_collinear & b3,b7,b8 is_collinear & b4,b6,b8 is_collinear
       holds b8,b9,b10 is_collinear;

:: ANPROJ_2:def 13
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
      b1 is Pappian
   iff
      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;

:: ANPROJ_2:attrnot 9 => ANPROJ_2:attr 7
definition
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
  attr a1 is 2-dimensional means
    for b1, b2, b3, b4 being Element of the carrier of a1 holds
    ex b5 being Element of the carrier of a1 st
       b1,b2,b5 is_collinear & b3,b4,b5 is_collinear;
end;

:: ANPROJ_2:dfs 14
definiens
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
To prove
     a1 is 2-dimensional
it is sufficient to prove
  thus for b1, b2, b3, b4 being Element of the carrier of a1 holds
    ex b5 being Element of the carrier of a1 st
       b1,b2,b5 is_collinear & b3,b4,b5 is_collinear;

:: ANPROJ_2:def 14
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
      b1 is 2-dimensional
   iff
      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;

:: ANPROJ_2:attrnot 10 => ANPROJ_2:attr 7
notation
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
  antonym up-3-dimensional for 2-dimensional;
end;

:: ANPROJ_2:attrnot 11 => ANPROJ_2:attr 8
definition
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
  attr a1 is at_most-3-dimensional means
    for b1, b2, b3, b4, b5 being Element of the carrier of a1 holds
    ex b6, b7 being Element of the carrier of a1 st
       b1,b3,b6 is_collinear & b2,b4,b7 is_collinear & b5,b6,b7 is_collinear;
end;

:: ANPROJ_2:dfs 15
definiens
  let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
To prove
     a1 is at_most-3-dimensional
it is sufficient to prove
  thus for b1, b2, b3, b4, b5 being Element of the carrier of a1 holds
    ex b6, b7 being Element of the carrier of a1 st
       b1,b3,b6 is_collinear & b2,b4,b7 is_collinear & b5,b6,b7 is_collinear;

:: ANPROJ_2:def 15
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
      b1 is at_most-3-dimensional
   iff
      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;

:: ANPROJ_2:th 28
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of ProjectiveSpace 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;

:: ANPROJ_2:funcreg 5
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional RLSStruct;
  cluster ProjectiveSpace a1 -> strict Fanoian Desarguesian Pappian;
end;

:: ANPROJ_2:th 29
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st ex b2, b3, b4 being Element of the carrier of b1 st
           (for b5, b6, b7 being Element of REAL
                  st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
               holds b5 = 0 & b6 = 0 & b7 = 0) &
            (for b5 being Element of the carrier of b1 holds
               ex b6, b7, b8 being Element of REAL st
                  b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4))
   holds ex b2, b3 being Element of the carrier of ProjectiveSpace b1 st
      b2 <> b3 &
       (for b4, b5 being Element of the carrier of ProjectiveSpace b1 holds
       ex b6 being Element of the carrier of ProjectiveSpace b1 st
          b2,b3,b6 is_collinear & b4,b5,b6 is_collinear);

:: ANPROJ_2:th 30
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
   st ex b2, b3 being Element of the carrier of ProjectiveSpace b1 st
        b2 <> b3 &
         (for b4, b5 being Element of the carrier of ProjectiveSpace b1 holds
         ex b6 being Element of the carrier of ProjectiveSpace b1 st
            b2,b3,b6 is_collinear & b4,b5,b6 is_collinear)
for b2, b3, b4, b5 being Element of the carrier of ProjectiveSpace b1 holds
ex b6 being Element of the carrier of ProjectiveSpace b1 st
   b2,b3,b6 is_collinear & b4,b5,b6 is_collinear;

:: ANPROJ_2:th 31
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st ex b2, b3, b4 being Element of the carrier of b1 st
           (for b5, b6, b7 being Element of REAL
                  st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
               holds b5 = 0 & b6 = 0 & b7 = 0) &
            (for b5 being Element of the carrier of b1 holds
               ex b6, b7, b8 being Element of REAL st
                  b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4))
   holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
      b2 = ProjectiveSpace b1 & b2 is 2-dimensional;

:: ANPROJ_2:th 32
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st ex b2, b3, b4, b5 being Element of the carrier of b1 st
           (for b6 being Element of the carrier of b1 holds
               ex b7, b8, b9, b10 being Element of REAL st
                  b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5)) &
            (for b6, b7, b8, b9 being Element of REAL
                  st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
               holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0)
   holds ex b2, b3, b4 being Element of the carrier of ProjectiveSpace b1 st
      not b2,b3,b4 is_collinear &
       (for b5, b6 being Element of the carrier of ProjectiveSpace b1 holds
       ex b7, b8 being Element of the carrier of ProjectiveSpace b1 st
          b5,b6,b8 is_collinear & b3,b4,b7 is_collinear & b2,b8,b7 is_collinear);

:: ANPROJ_2:th 33
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st ProjectiveSpace b1 is proper &
         ProjectiveSpace b1 is at_least_3rank &
         (ex b2, b3, b4 being Element of the carrier of ProjectiveSpace b1 st
            not b2,b3,b4 is_collinear &
             (for b5, b6 being Element of the carrier of ProjectiveSpace b1 holds
             ex b7, b8 being Element of the carrier of ProjectiveSpace b1 st
                b5,b6,b8 is_collinear & b3,b4,b7 is_collinear & b2,b8,b7 is_collinear))
   holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
      b2 = ProjectiveSpace b1 & b2 is at_most-3-dimensional;

:: ANPROJ_2:th 34
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st ex b2, b3, b4, b5 being Element of the carrier of b1 st
           (for b6 being Element of the carrier of b1 holds
               ex b7, b8, b9, b10 being Element of REAL st
                  b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5)) &
            (for b6, b7, b8, b9 being Element of REAL
                  st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
               holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0)
   holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
      b2 = ProjectiveSpace b1 & b2 is at_most-3-dimensional;

:: ANPROJ_2:th 35
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st ex b2, b3, b4, b5 being Element of the carrier of b1 st
           for b6, b7, b8, b9 being Element of REAL
                 st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
              holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0
   holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
      b2 = ProjectiveSpace b1 & b2 is up-3-dimensional;

:: ANPROJ_2:th 36
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st ex b2, b3, b4, b5 being Element of the carrier of b1 st
           (for b6 being Element of the carrier of b1 holds
               ex b7, b8, b9, b10 being Element of REAL st
                  b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5)) &
            (for b6, b7, b8, b9 being Element of REAL
                  st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
               holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0)
   holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
      b2 = ProjectiveSpace b1 & b2 is up-3-dimensional & b2 is at_most-3-dimensional;

:: ANPROJ_2:exreg 3
registration
  cluster non empty strict reflexive transitive proper Vebleian at_least_3rank Fanoian Desarguesian Pappian 2-dimensional CollStr;
end;

:: ANPROJ_2:exreg 4
registration
  cluster non empty strict reflexive transitive proper Vebleian at_least_3rank Fanoian Desarguesian Pappian up-3-dimensional at_most-3-dimensional CollStr;
end;

:: ANPROJ_2:modenot 2
definition
  mode CollProjectivePlane is non empty reflexive transitive proper Vebleian at_least_3rank 2-dimensional CollStr;
end;

:: ANPROJ_2:th 37
theorem
for b1 being non empty CollStr holds
      b1 is non empty reflexive transitive proper Vebleian at_least_3rank 2-dimensional CollStr
   iff
      b1 is non empty reflexive transitive proper at_least_3rank CollStr &
       (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);