Article PROJPL_1, MML version 4.99.1005

:: PROJPL_1:prednot 1 => not INCSP_1:pred 1
notation
  let a1 be IncProjStr;
  let a2 be Element of the Points of a1;
  let a3 be Element of the Lines of a1;
  antonym a2 |' a3 for a2 on a3;
end;

:: PROJPL_1:prednot 2 => PROJPL_1:pred 1
definition
  let a1 be IncProjStr;
  let a2, a3 be Element of the Points of a1;
  let a4 be Element of the Lines of a1;
  pred A2,A3 |' A4 means
    not a2 on a4 & not a3 on a4;
end;

:: PROJPL_1:dfs 1
definiens
  let a1 be IncProjStr;
  let a2, a3 be Element of the Points of a1;
  let a4 be Element of the Lines of a1;
To prove
     a2,a3 |' a4
it is sufficient to prove
  thus not a2 on a4 & not a3 on a4;

:: PROJPL_1:def 1
theorem
for b1 being IncProjStr
for b2, b3 being Element of the Points of b1
for b4 being Element of the Lines of b1 holds
      b2,b3 |' b4
   iff
      not b2 on b4 & not b3 on b4;

:: PROJPL_1:prednot 3 => PROJPL_1:pred 2
definition
  let a1 be IncProjStr;
  let a2 be Element of the Points of a1;
  let a3, a4 be Element of the Lines of a1;
  pred A2 on A3,A4 means
    a2 on a3 & a2 on a4;
end;

:: PROJPL_1:dfs 2
definiens
  let a1 be IncProjStr;
  let a2 be Element of the Points of a1;
  let a3, a4 be Element of the Lines of a1;
To prove
     a2 on a3,a4
it is sufficient to prove
  thus a2 on a3 & a2 on a4;

:: PROJPL_1:def 2
theorem
for b1 being IncProjStr
for b2 being Element of the Points of b1
for b3, b4 being Element of the Lines of b1 holds
   b2 on b3,b4
iff
   b2 on b3 & b2 on b4;

:: PROJPL_1:prednot 4 => PROJPL_1:pred 3
definition
  let a1 be IncProjStr;
  let a2 be Element of the Points of a1;
  let a3, a4, a5 be Element of the Lines of a1;
  pred A2 on A3,A4,A5 means
    a2 on a3 & a2 on a4 & a2 on a5;
end;

:: PROJPL_1:dfs 3
definiens
  let a1 be IncProjStr;
  let a2 be Element of the Points of a1;
  let a3, a4, a5 be Element of the Lines of a1;
To prove
     a2 on a3,a4,a5
it is sufficient to prove
  thus a2 on a3 & a2 on a4 & a2 on a5;

:: PROJPL_1:def 3
theorem
for b1 being IncProjStr
for b2 being Element of the Points of b1
for b3, b4, b5 being Element of the Lines of b1 holds
   b2 on b3,b4,b5
iff
   b2 on b3 & b2 on b4 & b2 on b5;

:: PROJPL_1:th 1
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5, b6, b7 being Element of the Lines of b1 holds
({b2,b3} on b5 implies {b3,b2} on b5) &
 ({b2,b3,b4} on b5 implies {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) &
 (b2 on b5,b6 implies b2 on b6,b5) &
 (b2 on b5,b6,b7 implies b2 on b5,b7,b6 & b2 on b6,b5,b7 & b2 on b6,b7,b5 & b2 on b7,b5,b6 & b2 on b7,b6,b5);

:: PROJPL_1:attrnot 1 => PROJPL_1:attr 1
definition
  let a1 be IncProjStr;
  attr a1 is configuration 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;

:: PROJPL_1:dfs 4
definiens
  let a1 be IncProjStr;
To prove
     a1 is configuration
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;

:: PROJPL_1:def 4
theorem
for b1 being IncProjStr holds
      b1 is configuration
   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;

:: PROJPL_1:th 2
theorem
for b1 being IncProjStr holds
      b1 is configuration
   iff
      for b2, b3 being Element of the Points of b1
      for b4, b5 being Element of the Lines of b1
            st {b2,b3} on b4 & {b2,b3} on b5 & b2 <> b3
         holds b4 = b5;

:: PROJPL_1:th 3
theorem
for b1 being IncProjStr holds
      b1 is configuration
   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,b5 & b3 on b4,b5 & b2 <> b3
         holds b4 = b5;

:: PROJPL_1:th 4
theorem
for b1 being IncProjStr holds
      b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
   iff
      b1 is configuration &
       (for b2, b3 being Element of the Points of b1 holds
       ex b4 being Element of the Lines of b1 st
          {b2,b3} on b4) &
       (ex b2 being Element of the Points of b1 st
          ex b3 being Element of the Lines of b1 st
             not b2 on b3) &
       (for b2 being Element of the Lines of b1 holds
          ex b3, b4, b5 being Element of the Points of b1 st
             b3,b4,b5 are_mutually_different & {b3,b4,b5} on b2) &
       (for b2, b3, b4, b5, b6 being Element of the Points of b1
       for b7, b8, b9, b10 being Element of the Lines of b1
             st {b2,b3,b6} on b7 & {b4,b5,b6} on b8 & {b2,b4} on b9 & {b3,b5} on b10 & not b6 on b9 & not b6 on b10 & b7 <> b8
          holds ex b11 being Element of the Points of b1 st
             b11 on b9,b10);

:: PROJPL_1:modenot 1
definition
  mode IncProjectivePlane is linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr;
end;

:: PROJPL_1:prednot 5 => PROJPL_1:pred 4
definition
  let a1 be IncProjStr;
  let a2, a3, a4 be Element of the Points of a1;
  pred A2,A3,A4 is_collinear means
    ex b1 being Element of the Lines of a1 st
       {a2,a3,a4} on b1;
end;

:: PROJPL_1:dfs 5
definiens
  let a1 be IncProjStr;
  let a2, a3, a4 be Element of the Points of a1;
To prove
     a2,a3,a4 is_collinear
it is sufficient to prove
  thus ex b1 being Element of the Lines of a1 st
       {a2,a3,a4} on b1;

:: PROJPL_1:def 5
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1 holds
   b2,b3,b4 is_collinear
iff
   ex b5 being Element of the Lines of b1 st
      {b2,b3,b4} on b5;

:: PROJPL_1:prednot 6 => not PROJPL_1:pred 4
notation
  let a1 be IncProjStr;
  let a2, a3, a4 be Element of the Points of a1;
  antonym a2,a3,a4 is_a_triangle for a2,a3,a4 is_collinear;
end;

:: PROJPL_1:th 5
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1 holds
   b2,b3,b4 is_collinear
iff
   ex b5 being Element of the Lines of b1 st
      b2 on b5 & b3 on b5 & b4 on b5;

:: PROJPL_1:th 6
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1 holds
   b2,b3,b4 is_a_triangle
iff
   for b5 being Element of the Lines of b1
         st b2 on b5 & b3 on b5
      holds not b4 on b5;

:: PROJPL_1:prednot 7 => PROJPL_1:pred 5
definition
  let a1 be IncProjStr;
  let a2, a3, a4, a5 be Element of the Points of a1;
  pred A2,A3,A4,A5 is_a_quadrangle means
    a2,a3,a4 is_a_triangle & a3,a4,a5 is_a_triangle & a4,a5,a2 is_a_triangle & a5,a2,a3 is_a_triangle;
end;

:: PROJPL_1:dfs 6
definiens
  let a1 be IncProjStr;
  let a2, a3, a4, a5 be Element of the Points of a1;
To prove
     a2,a3,a4,a5 is_a_quadrangle
it is sufficient to prove
  thus a2,a3,a4 is_a_triangle & a3,a4,a5 is_a_triangle & a4,a5,a2 is_a_triangle & a5,a2,a3 is_a_triangle;

:: PROJPL_1:def 6
theorem
for b1 being IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1 holds
   b2,b3,b4,b5 is_a_quadrangle
iff
   b2,b3,b4 is_a_triangle & b3,b4,b5 is_a_triangle & b4,b5,b2 is_a_triangle & b5,b2,b3 is_a_triangle;

:: PROJPL_1:th 7
theorem
for b1 being IncProjStr
      st b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
   holds ex b2, b3 being Element of the Lines of b1 st
      b2 <> b3;

:: PROJPL_1:th 8
theorem
for b1 being IncProjStr
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1
      st b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr &
         b2 on b3
   holds ex b4, b5 being Element of the Points of b1 st
      {b4,b5} on b3 & b2,b4,b5 are_mutually_different;

:: PROJPL_1:th 9
theorem
for b1 being IncProjStr
for b2 being Element of the Points of b1
for b3, b4 being Element of the Lines of b1
      st b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr &
         b2 on b3 &
         b3 <> b4
   holds ex b5 being Element of the Points of b1 st
      b5 on b3 & not b5 on b4 & b2 <> b5;

:: PROJPL_1:th 10
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5 being Element of the Lines of b1
      st b1 is configuration & {b2,b3} on b5 & b2 <> b3 & not b4 on b5
   holds b2,b3,b4 is_a_triangle;

:: PROJPL_1:th 11
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
      st b2,b3,b4 is_collinear
   holds b2,b4,b3 is_collinear & b3,b2,b4 is_collinear & b3,b4,b2 is_collinear & b4,b2,b3 is_collinear & b4,b3,b2 is_collinear;

:: PROJPL_1:th 12
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
      st b2,b3,b4 is_a_triangle
   holds b2,b4,b3 is_a_triangle & b3,b2,b4 is_a_triangle & b3,b4,b2 is_a_triangle & b4,b2,b3 is_a_triangle & b4,b3,b2 is_a_triangle;

:: PROJPL_1:th 13
theorem
for b1 being IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
      st b2,b3,b4,b5 is_a_quadrangle
   holds b2,b4,b3,b5 is_a_quadrangle & b3,b2,b4,b5 is_a_quadrangle & b3,b4,b2,b5 is_a_quadrangle & b4,b2,b3,b5 is_a_quadrangle & b4,b3,b2,b5 is_a_quadrangle & b2,b3,b5,b4 is_a_quadrangle & b2,b4,b5,b3 is_a_quadrangle & b3,b2,b5,b4 is_a_quadrangle & b3,b4,b5,b2 is_a_quadrangle & b4,b2,b5,b3 is_a_quadrangle & b4,b3,b5,b2 is_a_quadrangle & b2,b5,b3,b4 is_a_quadrangle & b2,b5,b4,b3 is_a_quadrangle & b3,b5,b2,b4 is_a_quadrangle & b3,b5,b4,b2 is_a_quadrangle & b4,b5,b2,b3 is_a_quadrangle & b4,b5,b3,b2 is_a_quadrangle & b5,b2,b3,b4 is_a_quadrangle & b5,b2,b4,b3 is_a_quadrangle & b5,b3,b2,b4 is_a_quadrangle & b5,b3,b4,b2 is_a_quadrangle & b5,b4,b2,b3 is_a_quadrangle & b5,b4,b3,b2 is_a_quadrangle;

:: PROJPL_1:th 14
theorem
for b1 being IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
for b6, b7 being Element of the Lines of b1
      st b1 is configuration & {b2,b3} on b6 & {b4,b5} on b7 & b2,b3 |' b7 & b4,b5 |' b6 & b2 <> b3 & b4 <> b5
   holds b2,b3,b4,b5 is_a_quadrangle;

:: PROJPL_1:th 15
theorem
for b1 being IncProjStr
      st b1 is linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
   holds ex b2, b3, b4, b5 being Element of the Points of b1 st
      b2,b3,b4,b5 is_a_quadrangle;

:: PROJPL_1:modenot 2 => PROJPL_1:mode 1
definition
  let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
  mode Quadrangle of A1 -> Element of [:the Points of a1,the Points of a1,the Points of a1,the Points of a1:] means
    it `1,it `2,it `3,it `4 is_a_quadrangle;
end;

:: PROJPL_1:dfs 7
definiens
  let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
  let a2 be Element of [:the Points of a1,the Points of a1,the Points of a1,the Points of a1:];
To prove
     a2 is Quadrangle of a1
it is sufficient to prove
  thus a2 `1,a2 `2,a2 `3,a2 `4 is_a_quadrangle;

:: PROJPL_1:def 7
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian IncProjStr
for b2 being Element of [:the Points of b1,the Points of b1,the Points of b1,the Points of b1:] holds
      b2 is Quadrangle of b1
   iff
      b2 `1,b2 `2,b2 `3,b2 `4 is_a_quadrangle;

:: PROJPL_1:funcnot 1 => PROJPL_1:func 1
definition
  let a1 be linear partial up-2-dimensional up-3-rank Vebleian IncProjStr;
  let a2, a3 be Element of the Points of a1;
  assume a2 <> a3;
  func A2 * A3 -> Element of the Lines of a1 means
    {a2,a3} on it;
end;

:: PROJPL_1:def 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
   st b2 <> b3
for b4 being Element of the Lines of b1 holds
      b4 = b2 * b3
   iff
      {b2,b3} on b4;

:: PROJPL_1:th 16
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
      st b2 <> b3
   holds b2 on b2 * b3 &
    b3 on b2 * b3 &
    b2 * b3 = b3 * b2 &
    (b2 on b4 & b3 on b4 implies b4 = b2 * b3);

:: PROJPL_1:th 17
theorem
for b1 being IncProjStr
      st (ex b2, b3, b4 being Element of the Points of b1 st
            b2,b3,b4 is_a_triangle) &
         (for b2, b3 being Element of the Points of b1 holds
         ex b4 being Element of the Lines of b1 st
            {b2,b3} on b4)
   holds ex b2 being Element of the Points of b1 st
      ex b3 being Element of the Lines of b1 st
         not b2 on b3;

:: PROJPL_1:th 18
theorem
for b1 being IncProjStr
      st ex b2, b3, b4, b5 being Element of the Points of b1 st
           b2,b3,b4,b5 is_a_quadrangle
   holds ex b2, b3, b4 being Element of the Points of b1 st
      b2,b3,b4 is_a_triangle;

:: PROJPL_1:th 19
theorem
for b1 being IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5, b6 being Element of the Lines of b1
      st b2,b3,b4 is_a_triangle & {b2,b3} on b5 & {b2,b4} on b6
   holds b5 <> b6;

:: PROJPL_1:th 20
theorem
for b1 being IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
for b6, b7, b8 being Element of the Lines of b1
      st b2,b3,b4,b5 is_a_quadrangle & {b2,b3} on b6 & {b2,b4} on b7 & {b2,b5} on b8
   holds b6,b7,b8 are_mutually_different;

:: PROJPL_1:th 21
theorem
for b1 being 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 b1 is configuration & b2 on b6,b7,b8 & b6,b7,b8 are_mutually_different & not b2 on b9 & b3 on b9,b6 & b4 on b9,b7 & b5 on b9,b8
   holds b3,b4,b5 are_mutually_different;

:: PROJPL_1:th 22
theorem
for b1 being IncProjStr
   st b1 is configuration &
      (for b2, b3 being Element of the Points of b1 holds
      ex b4 being Element of the Lines of b1 st
         {b2,b3} on b4) &
      (for b2, b3 being Element of the Lines of b1 holds
      ex b4 being Element of the Points of b1 st
         b4 on b2,b3) &
      (ex b2, b3, b4, b5 being Element of the Points of b1 st
         b2,b3,b4,b5 is_a_quadrangle)
for b2 being Element of the Lines of b1 holds
   ex b3, b4, b5 being Element of the Points of b1 st
      b3,b4,b5 are_mutually_different & {b3,b4,b5} on b2;

:: PROJPL_1:th 23
theorem
for b1 being IncProjStr holds
      b1 is linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
   iff
      b1 is configuration &
       (for b2, b3 being Element of the Points of b1 holds
       ex b4 being Element of the Lines of b1 st
          {b2,b3} on b4) &
       (for b2, b3 being Element of the Lines of b1 holds
       ex b4 being Element of the Points of b1 st
          b4 on b2,b3) &
       (ex b2, b3, b4, b5 being Element of the Points of b1 st
          b2,b3,b4,b5 is_a_quadrangle);

:: PROJPL_1:funcnot 2 => PROJPL_1:func 2
definition
  let a1 be linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr;
  let a2, a3 be Element of the Lines of a1;
  assume a2 <> a3;
  func A2 * A3 -> Element of the Points of a1 means
    it on a2,a3;
end;

:: PROJPL_1:def 9
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3 being Element of the Lines of b1
   st b2 <> b3
for b4 being Element of the Points of b1 holds
      b4 = b2 * b3
   iff
      b4 on b2,b3;

:: PROJPL_1:th 24
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2 being Element of the Points of b1
for b3, b4 being Element of the Lines of b1
      st b3 <> b4
   holds b3 * b4 on b3 &
    b3 * b4 on b4 &
    b3 * b4 = b4 * b3 &
    (b2 on b3 & b2 on b4 implies b2 = b3 * b4);

:: PROJPL_1:th 25
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3 being Element of the Points of b1
for b4, b5 being Element of the Lines of b1
      st b4 <> b5 & b2 on b4 & not b3 on b4 & b2 <> b4 * b5
   holds (b3 * b2) * b5 on b5 & not (b3 * b2) * b5 on b4;

:: PROJPL_1:th 26
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
      st b2,b3,b4 is_a_triangle
   holds b2,b3,b4 are_mutually_different;

:: PROJPL_1:th 27
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
      st b2,b3,b4,b5 is_a_quadrangle
   holds b2,b3,b4,b5 are_mutually_different;

:: PROJPL_1:th 28
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
      st b2 * b4 = b3 * b5 & b2 <> b4 & b3 <> b5 & b4 <> b5
   holds b2 * b4 = b4 * b5;

:: PROJPL_1:th 29
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
      st b2 * b4 = b3 * b5 & b2 <> b4 & b3 <> b5 & b4 <> b5
   holds b2 on b4 * b5;

:: PROJPL_1:th 30
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4 being Element of the Points of b1
for b5 being Element of the Lines of b1
      st b3 on b5 & b4 on b5 & b3 <> b4 & not b2 on b5
   holds b3 * b2 <> b4 * b2 & b2 * b3 <> b2 * b4;

:: PROJPL_1:th 31
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2 being Element of the Points of b1
for b3, b4, b5 being Element of the Lines of b1
      st b2 on b4 & b2 on b5 & b4 <> b5 & not b2 on b3
   holds b3 * b4 <> b3 * b5 & b4 * b3 <> b5 * b3;

:: PROJPL_1:th 32
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
      st b4 on b2 * b3 & b4 on b2 * b5 & b4 <> b2 & b5 <> b2 & b2 <> b3
   holds b5 on b2 * b3;

:: PROJPL_1:th 33
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
      st b4 on b2 * b3 & b2 <> b4
   holds b3 on b2 * b4;

:: PROJPL_1:th 34
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4, b5 being Element of the Points of b1
for b6 being Element of the Lines of b1
      st b4 <> b5 & b4 on b6 & b5 on b6 & not b2 on b6 & not b3 on b6
   holds b2 * b4 <> b3 * b5;

:: PROJPL_1:th 35
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4 being Element of the Points of b1
      st b2 on b3 * b4 & b2 <> b4 & b3 <> b4
   holds b3 on b4 * b2;

:: PROJPL_1:th 36
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4 being Element of the Points of b1
      st b2 on b3 * b4 & b3 <> b2 & b3 <> b4
   holds b4 on b3 * b2;

:: PROJPL_1:th 37
theorem
for b1 being linear partial up-2-dimensional up-3-rank Vebleian 2-dimensional IncProjStr
for b2, b3, b4, b5, b6 being Element of the Points of b1
for b7, b8 being Element of the Lines of b1
      st b3 on b8 & b4 on b8 & b2 on b7 & not b2 on b8 & b3 <> b4 & b5 <> b2 & b6 <> b2 & b5 on b2 * b3 & b6 on b2 * b4
   holds ex b9 being Element of the Points of b1 st
      b9 on b5 * b6 & b9 on b7 & b9 <> b2;