Article AFF_3, MML version 4.99.1005

:: AFF_3:attrnot 1 => AFF_3:attr 1
definition
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  attr a1 is satisfying_DES1 means
    for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b2 <> b1 & b2 <> b3 & b1 <> b3 & b4 in b1 & b5 in b1 & b6 in b1 & b4 in b2 & b7 in b2 & b8 in b2 & b4 in b3 & b9 in b3 & b10 in b3 & b4 <> b5 & b4 <> b7 & b4 <> b9 & b11 <> b12 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b5 <> b6 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b6,b10
       holds b5,b9 // b11,b12;
end;

:: AFF_3:dfs 1
definiens
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
     a1 is satisfying_DES1
it is sufficient to prove
  thus for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b2 <> b1 & b2 <> b3 & b1 <> b3 & b4 in b1 & b5 in b1 & b6 in b1 & b4 in b2 & b7 in b2 & b8 in b2 & b4 in b3 & b9 in b3 & b10 in b3 & b4 <> b5 & b4 <> b7 & b4 <> b9 & b11 <> b12 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b5 <> b6 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b6,b10
       holds b5,b9 // b11,b12;

:: AFF_3:def 1
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES1
   iff
      for b2, b3, b4 being Element of bool the carrier of b1
      for b5, b6, b7, b8, b9, b10, b11, b12, b13 being Element of the carrier of b1
            st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b3 <> b2 & b3 <> b4 & b2 <> b4 & b5 in b2 & b6 in b2 & b7 in b2 & b5 in b3 & b8 in b3 & b9 in b3 & b5 in b4 & b10 in b4 & b11 in b4 & b5 <> b6 & b5 <> b8 & b5 <> b10 & b12 <> b13 & not LIN b8,b6,b10 & not LIN b9,b7,b11 & b6 <> b7 & LIN b8,b6,b12 & LIN b9,b7,b12 & LIN b8,b10,b13 & LIN b9,b11,b13 & b6,b10 // b7,b11
         holds b6,b10 // b12,b13;

:: AFF_3:prednot 1 => AFF_3:attr 1
notation
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  synonym a1 satisfies_DES1 for satisfying_DES1;
end;

:: AFF_3:attrnot 2 => AFF_3:attr 2
definition
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  attr a1 is satisfying_DES1_1 means
    for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b2 <> b1 & b2 <> b3 & b1 <> b3 & b4 in b1 & b5 in b1 & b6 in b1 & b4 in b2 & b7 in b2 & b8 in b2 & b4 in b3 & b9 in b3 & b10 in b3 & b4 <> b5 & b4 <> b7 & b4 <> b9 & b11 <> b12 & b9 <> b12 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b11,b12
       holds b5,b9 // b6,b10;
end;

:: AFF_3:dfs 2
definiens
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
     a1 is satisfying_DES1_1
it is sufficient to prove
  thus for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b2 <> b1 & b2 <> b3 & b1 <> b3 & b4 in b1 & b5 in b1 & b6 in b1 & b4 in b2 & b7 in b2 & b8 in b2 & b4 in b3 & b9 in b3 & b10 in b3 & b4 <> b5 & b4 <> b7 & b4 <> b9 & b11 <> b12 & b9 <> b12 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b11,b12
       holds b5,b9 // b6,b10;

:: AFF_3:def 2
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES1_1
   iff
      for b2, b3, b4 being Element of bool the carrier of b1
      for b5, b6, b7, b8, b9, b10, b11, b12, b13 being Element of the carrier of b1
            st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b3 <> b2 & b3 <> b4 & b2 <> b4 & b5 in b2 & b6 in b2 & b7 in b2 & b5 in b3 & b8 in b3 & b9 in b3 & b5 in b4 & b10 in b4 & b11 in b4 & b5 <> b6 & b5 <> b8 & b5 <> b10 & b12 <> b13 & b10 <> b13 & not LIN b8,b6,b10 & not LIN b9,b7,b11 & LIN b8,b6,b12 & LIN b9,b7,b12 & LIN b8,b10,b13 & LIN b9,b11,b13 & b6,b10 // b12,b13
         holds b6,b10 // b7,b11;

:: AFF_3:prednot 2 => AFF_3:attr 2
notation
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  synonym a1 satisfies_DES1_1 for satisfying_DES1_1;
end;

:: AFF_3:attrnot 3 => AFF_3:attr 3
definition
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  attr a1 is satisfying_DES1_2 means
    for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b2 <> b1 & b2 <> b3 & b1 <> b3 & b4 in b1 & b5 in b1 & b6 in b1 & b4 in b2 & b7 in b2 & b8 in b2 & b9 in b3 & b10 in b3 & b4 <> b5 & b4 <> b7 & b4 <> b9 & b11 <> b12 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b9 <> b10 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b6,b10 & b5,b9 // b11,b12
       holds b4 in b3;
end;

:: AFF_3:dfs 3
definiens
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
     a1 is satisfying_DES1_2
it is sufficient to prove
  thus for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b2 <> b1 & b2 <> b3 & b1 <> b3 & b4 in b1 & b5 in b1 & b6 in b1 & b4 in b2 & b7 in b2 & b8 in b2 & b9 in b3 & b10 in b3 & b4 <> b5 & b4 <> b7 & b4 <> b9 & b11 <> b12 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b9 <> b10 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b6,b10 & b5,b9 // b11,b12
       holds b4 in b3;

:: AFF_3:def 3
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES1_2
   iff
      for b2, b3, b4 being Element of bool the carrier of b1
      for b5, b6, b7, b8, b9, b10, b11, b12, b13 being Element of the carrier of b1
            st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b3 <> b2 & b3 <> b4 & b2 <> b4 & b5 in b2 & b6 in b2 & b7 in b2 & b5 in b3 & b8 in b3 & b9 in b3 & b10 in b4 & b11 in b4 & b5 <> b6 & b5 <> b8 & b5 <> b10 & b12 <> b13 & not LIN b8,b6,b10 & not LIN b9,b7,b11 & b10 <> b11 & LIN b8,b6,b12 & LIN b9,b7,b12 & LIN b8,b10,b13 & LIN b9,b11,b13 & b6,b10 // b7,b11 & b6,b10 // b12,b13
         holds b5 in b4;

:: AFF_3:prednot 3 => AFF_3:attr 3
notation
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  synonym a1 satisfies_DES1_2 for satisfying_DES1_2;
end;

:: AFF_3:attrnot 4 => AFF_3:attr 4
definition
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  attr a1 is satisfying_DES1_3 means
    for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b2 <> b1 & b2 <> b3 & b1 <> b3 & b4 in b1 & b5 in b1 & b6 in b1 & b7 in b2 & b8 in b2 & b4 in b3 & b9 in b3 & b10 in b3 & b4 <> b5 & b4 <> b7 & b4 <> b9 & b11 <> b12 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b7 <> b8 & b5 <> b6 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b6,b10 & b5,b9 // b11,b12
       holds b4 in b2;
end;

:: AFF_3:dfs 4
definiens
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
     a1 is satisfying_DES1_3
it is sufficient to prove
  thus for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b2 <> b1 & b2 <> b3 & b1 <> b3 & b4 in b1 & b5 in b1 & b6 in b1 & b7 in b2 & b8 in b2 & b4 in b3 & b9 in b3 & b10 in b3 & b4 <> b5 & b4 <> b7 & b4 <> b9 & b11 <> b12 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b7 <> b8 & b5 <> b6 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b6,b10 & b5,b9 // b11,b12
       holds b4 in b2;

:: AFF_3:def 4
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES1_3
   iff
      for b2, b3, b4 being Element of bool the carrier of b1
      for b5, b6, b7, b8, b9, b10, b11, b12, b13 being Element of the carrier of b1
            st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b3 <> b2 & b3 <> b4 & b2 <> b4 & b5 in b2 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b5 in b4 & b10 in b4 & b11 in b4 & b5 <> b6 & b5 <> b8 & b5 <> b10 & b12 <> b13 & not LIN b8,b6,b10 & not LIN b9,b7,b11 & b8 <> b9 & b6 <> b7 & LIN b8,b6,b12 & LIN b9,b7,b12 & LIN b8,b10,b13 & LIN b9,b11,b13 & b6,b10 // b7,b11 & b6,b10 // b12,b13
         holds b5 in b3;

:: AFF_3:prednot 4 => AFF_3:attr 4
notation
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  synonym a1 satisfies_DES1_3 for satisfying_DES1_3;
end;

:: AFF_3:attrnot 5 => AFF_3:attr 5
definition
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  attr a1 is satisfying_DES2 means
    for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b2 <> b3 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b1 // b2 & b1 // b3 & not LIN b6,b4,b8 & not LIN b7,b5,b9 & b10 <> b11 & b4 <> b5 & LIN b6,b4,b10 & LIN b7,b5,b10 & LIN b6,b8,b11 & LIN b7,b9,b11 & b4,b8 // b5,b9
       holds b4,b8 // b10,b11;
end;

:: AFF_3:dfs 5
definiens
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
     a1 is satisfying_DES2
it is sufficient to prove
  thus for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b2 <> b3 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b1 // b2 & b1 // b3 & not LIN b6,b4,b8 & not LIN b7,b5,b9 & b10 <> b11 & b4 <> b5 & LIN b6,b4,b10 & LIN b7,b5,b10 & LIN b6,b8,b11 & LIN b7,b9,b11 & b4,b8 // b5,b9
       holds b4,b8 // b10,b11;

:: AFF_3:def 5
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES2
   iff
      for b2, b3, b4 being Element of bool the carrier of b1
      for b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of b1
            st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b2 <> b3 & b2 <> b4 & b3 <> b4 & b5 in b2 & b6 in b2 & b7 in b3 & b8 in b3 & b9 in b4 & b10 in b4 & b2 // b3 & b2 // b4 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b11 <> b12 & b5 <> b6 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b6,b10
         holds b5,b9 // b11,b12;

:: AFF_3:prednot 5 => AFF_3:attr 5
notation
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  synonym a1 satisfies_DES2 for satisfying_DES2;
end;

:: AFF_3:attrnot 6 => AFF_3:attr 6
definition
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  attr a1 is satisfying_DES2_1 means
    for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b2 <> b3 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b1 // b2 & b1 // b3 & not LIN b6,b4,b8 & not LIN b7,b5,b9 & b10 <> b11 & LIN b6,b4,b10 & LIN b7,b5,b10 & LIN b6,b8,b11 & LIN b7,b9,b11 & b4,b8 // b10,b11
       holds b4,b8 // b5,b9;
end;

:: AFF_3:dfs 6
definiens
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
     a1 is satisfying_DES2_1
it is sufficient to prove
  thus for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b2 <> b3 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b1 // b2 & b1 // b3 & not LIN b6,b4,b8 & not LIN b7,b5,b9 & b10 <> b11 & LIN b6,b4,b10 & LIN b7,b5,b10 & LIN b6,b8,b11 & LIN b7,b9,b11 & b4,b8 // b10,b11
       holds b4,b8 // b5,b9;

:: AFF_3:def 6
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES2_1
   iff
      for b2, b3, b4 being Element of bool the carrier of b1
      for b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of b1
            st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b2 <> b3 & b2 <> b4 & b3 <> b4 & b5 in b2 & b6 in b2 & b7 in b3 & b8 in b3 & b9 in b4 & b10 in b4 & b2 // b3 & b2 // b4 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b11 <> b12 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b11,b12
         holds b5,b9 // b6,b10;

:: AFF_3:prednot 6 => AFF_3:attr 6
notation
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  synonym a1 satisfies_DES2_1 for satisfying_DES2_1;
end;

:: AFF_3:attrnot 7 => AFF_3:attr 7
definition
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  attr a1 is satisfying_DES2_2 means
    for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b2 <> b3 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b1 // b3 & not LIN b6,b4,b8 & not LIN b7,b5,b9 & b10 <> b11 & b4 <> b5 & LIN b6,b4,b10 & LIN b7,b5,b10 & LIN b6,b8,b11 & LIN b7,b9,b11 & b4,b8 // b5,b9 & b4,b8 // b10,b11
       holds b1 // b2;
end;

:: AFF_3:dfs 7
definiens
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
     a1 is satisfying_DES2_2
it is sufficient to prove
  thus for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b2 <> b3 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b1 // b3 & not LIN b6,b4,b8 & not LIN b7,b5,b9 & b10 <> b11 & b4 <> b5 & LIN b6,b4,b10 & LIN b7,b5,b10 & LIN b6,b8,b11 & LIN b7,b9,b11 & b4,b8 // b5,b9 & b4,b8 // b10,b11
       holds b1 // b2;

:: AFF_3:def 7
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES2_2
   iff
      for b2, b3, b4 being Element of bool the carrier of b1
      for b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of b1
            st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b2 <> b3 & b2 <> b4 & b3 <> b4 & b5 in b2 & b6 in b2 & b7 in b3 & b8 in b3 & b9 in b4 & b10 in b4 & b2 // b4 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b11 <> b12 & b5 <> b6 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b6,b10 & b5,b9 // b11,b12
         holds b2 // b3;

:: AFF_3:prednot 7 => AFF_3:attr 7
notation
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  synonym a1 satisfies_DES2_2 for satisfying_DES2_2;
end;

:: AFF_3:attrnot 8 => AFF_3:attr 8
definition
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  attr a1 is satisfying_DES2_3 means
    for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b2 <> b3 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b1 // b2 & not LIN b6,b4,b8 & not LIN b7,b5,b9 & b10 <> b11 & b8 <> b9 & LIN b6,b4,b10 & LIN b7,b5,b10 & LIN b6,b8,b11 & LIN b7,b9,b11 & b4,b8 // b5,b9 & b4,b8 // b10,b11
       holds b1 // b3;
end;

:: AFF_3:dfs 8
definiens
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
To prove
     a1 is satisfying_DES2_3
it is sufficient to prove
  thus for b1, b2, b3 being Element of bool the carrier of a1
    for b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of a1
          st b1 is being_line(a1) & b2 is being_line(a1) & b3 is being_line(a1) & b1 <> b2 & b1 <> b3 & b2 <> b3 & b4 in b1 & b5 in b1 & b6 in b2 & b7 in b2 & b8 in b3 & b9 in b3 & b1 // b2 & not LIN b6,b4,b8 & not LIN b7,b5,b9 & b10 <> b11 & b8 <> b9 & LIN b6,b4,b10 & LIN b7,b5,b10 & LIN b6,b8,b11 & LIN b7,b9,b11 & b4,b8 // b5,b9 & b4,b8 // b10,b11
       holds b1 // b3;

:: AFF_3:def 8
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES2_3
   iff
      for b2, b3, b4 being Element of bool the carrier of b1
      for b5, b6, b7, b8, b9, b10, b11, b12 being Element of the carrier of b1
            st b2 is being_line(b1) & b3 is being_line(b1) & b4 is being_line(b1) & b2 <> b3 & b2 <> b4 & b3 <> b4 & b5 in b2 & b6 in b2 & b7 in b3 & b8 in b3 & b9 in b4 & b10 in b4 & b2 // b3 & not LIN b7,b5,b9 & not LIN b8,b6,b10 & b11 <> b12 & b9 <> b10 & LIN b7,b5,b11 & LIN b8,b6,b11 & LIN b7,b9,b12 & LIN b8,b10,b12 & b5,b9 // b6,b10 & b5,b9 // b11,b12
         holds b2 // b4;

:: AFF_3:prednot 8 => AFF_3:attr 8
notation
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  synonym a1 satisfies_DES2_3 for satisfying_DES2_3;
end;

:: AFF_3:th 9
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st b1 is satisfying_DES1
   holds b1 is satisfying_DES1_1;

:: AFF_3:th 10
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st b1 is satisfying_DES1_1
   holds b1 is satisfying_DES1;

:: AFF_3:th 11
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st b1 is Desarguesian
   holds b1 is satisfying_DES1;

:: AFF_3:th 12
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st b1 is Desarguesian
   holds b1 is satisfying_DES1_2;

:: AFF_3:th 13
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st b1 is satisfying_DES1_2
   holds b1 is satisfying_DES1_3;

:: AFF_3:th 14
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st b1 is satisfying_DES1_2
   holds b1 is Desarguesian;

:: AFF_3:th 15
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st b1 is satisfying_DES2_1
   holds b1 is satisfying_DES2;

:: AFF_3:th 16
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES2_1
   iff
      b1 is satisfying_DES2_3;

:: AFF_3:th 17
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is satisfying_DES2
   iff
      b1 is satisfying_DES2_2;

:: AFF_3:th 18
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st b1 is satisfying_DES1_3
   holds b1 is satisfying_DES2_1;