Article DIRAF, MML version 4.99.1005

:: DIRAF:funcnot 1 => DIRAF:func 1
definition
  let a1 be non empty set;
  let a2 be Relation of [:a1,a1:],[:a1,a1:];
  func lambda A2 -> Relation of [:a1,a1:],[:a1,a1:] means
    for b1, b2, b3, b4 being Element of a1 holds
       [[b1,b2],[b3,b4]] in it
    iff
       ([[b1,b2],[b3,b4]] in a2 or [[b1,b2],[b4,b3]] in a2);
end;

:: DIRAF:def 1
theorem
for b1 being non empty set
for b2, b3 being Relation of [:b1,b1:],[:b1,b1:] holds
   b3 = lambda b2
iff
   for b4, b5, b6, b7 being Element of b1 holds
      [[b4,b5],[b6,b7]] in b3
   iff
      ([[b4,b5],[b6,b7]] in b2 or [[b4,b5],[b7,b6]] in b2);

:: DIRAF:funcnot 2 => DIRAF:func 2
definition
  let a1 be non empty AffinStruct;
  func Lambda A1 -> strict AffinStruct equals
    AffinStruct(#the carrier of a1,lambda the CONGR of a1#);
end;

:: DIRAF:def 2
theorem
for b1 being non empty AffinStruct holds
   Lambda b1 = AffinStruct(#the carrier of b1,lambda the CONGR of b1#);

:: DIRAF:funcreg 1
registration
  let a1 be non empty AffinStruct;
  cluster Lambda a1 -> non empty strict;
end;

:: DIRAF:th 4
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 // b2,b3;

:: DIRAF:th 5
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b3,b2 // b5,b4 & b4,b5 // b2,b3 & b5,b4 // b3,b2;

:: DIRAF:th 6
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2 <> b3 & b4,b5 // b2,b3 & b2,b3 // b6,b7
   holds b4,b5 // b6,b7;

:: DIRAF:th 7
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b2 // b3,b4 & b3,b4 // b2,b2;

:: DIRAF:th 8
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5 & b2,b3 // b5,b4 & b2 <> b3
   holds b4 = b5;

:: DIRAF:th 9
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   b2,b3 // b2,b4
iff
   (b2,b3 // b3,b4 or b2,b4 // b4,b3);

:: DIRAF:prednot 1 => DIRAF:pred 1
definition
  let a1 be non empty AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  pred Mid A2,A3,A4 means
    a2,a3 // a3,a4;
end;

:: DIRAF:dfs 3
definiens
  let a1 be non empty AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
To prove
     Mid a2,a3,a4
it is sufficient to prove
  thus a2,a3 // a3,a4;

:: DIRAF:def 3
theorem
for b1 being non empty AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   Mid b2,b3,b4
iff
   b2,b3 // b3,b4;

:: DIRAF:th 11
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   b2,b3 // b2,b4
iff
   (Mid b2,b3,b4 or Mid b2,b4,b3);

:: DIRAF:th 12
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st Mid b2,b3,b2
   holds b2 = b3;

:: DIRAF:th 13
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st Mid b2,b3,b4
   holds Mid b4,b3,b2;

:: DIRAF:th 14
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
Mid b2,b2,b3 & Mid b2,b3,b3;

:: DIRAF:th 15
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st Mid b2,b3,b4 & Mid b2,b4,b5
   holds Mid b3,b4,b5;

:: DIRAF:th 16
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <> b3 & Mid b4,b2,b3 & Mid b2,b3,b5
   holds Mid b4,b3,b5;

:: DIRAF:th 17
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
   Mid b2,b3,b4 & b3 <> b4;

:: DIRAF:th 18
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st Mid b2,b3,b4 & Mid b3,b2,b4
   holds b2 = b3;

:: DIRAF:th 19
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <> b3 & Mid b2,b3,b4 & Mid b2,b3,b5 & not Mid b3,b4,b5
   holds Mid b3,b5,b4;

:: DIRAF:th 20
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <> b3 & Mid b2,b3,b4 & Mid b2,b3,b5 & not Mid b2,b4,b5
   holds Mid b2,b5,b4;

:: DIRAF:th 21
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st Mid b2,b3,b4 & Mid b2,b5,b4 & not Mid b2,b3,b5
   holds Mid b2,b5,b3;

:: DIRAF:prednot 2 => DIRAF:pred 2
definition
  let a1 be non empty AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3 '||' A4,A5 means
    (not a2,a3 // a4,a5) implies a2,a3 // a5,a4;
end;

:: DIRAF:dfs 4
definiens
  let a1 be non empty AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3 '||' a4,a5
it is sufficient to prove
  thus (not a2,a3 // a4,a5) implies a2,a3 // a5,a4;

:: DIRAF:def 4
theorem
for b1 being non empty AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3 '||' b4,b5
iff
   (b2,b3 // b4,b5 or b2,b3 // b5,b4);

:: DIRAF:th 23
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3 '||' b4,b5
iff
   [[b2,b3],[b4,b5]] in lambda the CONGR of b1;

:: DIRAF:th 24
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 '||' b3,b2 & b2,b3 '||' b2,b3;

:: DIRAF:th 25
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 '||' b4,b4 & b4,b4 '||' b2,b3;

:: DIRAF:th 26
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3 '||' b2,b4
   holds b3,b2 '||' b3,b4;

:: DIRAF:th 27
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 '||' b4,b5
   holds b2,b3 '||' b5,b4 & b3,b2 '||' b4,b5 & b3,b2 '||' b5,b4 & b4,b5 '||' b2,b3 & b4,b5 '||' b3,b2 & b5,b4 '||' b2,b3 & b5,b4 '||' b3,b2;

:: DIRAF:th 28
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2 <> b3 &
         ((b2,b3 '||' b4,b5 implies not b2,b3 '||' b6,b7) & (b2,b3 '||' b4,b5 implies not b6,b7 '||' b2,b3) & (b4,b5 '||' b2,b3 implies not b6,b7 '||' b2,b3) implies b4,b5 '||' b2,b3 & b2,b3 '||' b6,b7)
   holds b4,b5 '||' b6,b7;

:: DIRAF:th 29
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   ex b2, b3, b4 being Element of the carrier of b1 st
      not b2,b3 '||' b2,b4;

:: DIRAF:th 30
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
   b2,b3 '||' b4,b5 & b4 <> b5;

:: DIRAF:th 31
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
   b2,b3 '||' b4,b5 & b2,b4 '||' b3,b5;

:: DIRAF:th 32
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 '||' b3,b4 & b3 <> b2
   holds ex b6 being Element of the carrier of b1 st
      b5,b3 '||' b3,b6 & b5,b2 '||' b4,b6;

:: DIRAF:prednot 3 => DIRAF:pred 3
definition
  let a1 be non empty AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  pred LIN A2,A3,A4 means
    a2,a3 '||' a2,a4;
end;

:: DIRAF:dfs 5
definiens
  let a1 be non empty AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
To prove
     LIN a2,a3,a4
it is sufficient to prove
  thus a2,a3 '||' a2,a4;

:: DIRAF:def 5
theorem
for b1 being non empty AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   LIN b2,b3,b4
iff
   b2,b3 '||' b2,b4;

:: DIRAF:prednot 4 => DIRAF:pred 3
notation
  let a1 be non empty AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  synonym a2,a3,a4 is_collinear for LIN a2,a3,a4;
end;

:: DIRAF:th 34
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st Mid b2,b3,b4
   holds LIN b2,b3,b4;

:: DIRAF:th 35
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st LIN b2,b3,b4 & not Mid b2,b3,b4 & not Mid b3,b2,b4
   holds Mid b2,b4,b3;

:: DIRAF:th 36
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st LIN b2,b3,b4
   holds LIN b2,b4,b3 & LIN b3,b2,b4 & LIN b3,b4,b2 & LIN b4,b2,b3 & LIN b4,b3,b2;

:: DIRAF:th 37
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
LIN b2,b2,b3 & LIN b2,b3,b3 & LIN b2,b3,b2;

:: DIRAF:th 38
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2 <> b3 & LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b3,b6
   holds LIN b4,b5,b6;

:: DIRAF:th 39
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <> b3 & LIN b2,b3,b4 & b2,b3 '||' b4,b5
   holds LIN b2,b3,b5;

:: DIRAF:th 40
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st LIN b2,b3,b4 & LIN b2,b3,b5
   holds b2,b3 '||' b4,b5;

:: DIRAF:th 41
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2 <> b3 & LIN b4,b5,b2 & LIN b4,b5,b3 & LIN b2,b3,b6
   holds LIN b4,b5,b6;

:: DIRAF:th 42
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   ex b2, b3, b4 being Element of the carrier of b1 st
      not LIN b2,b3,b4;

:: DIRAF:th 43
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
      st b2 <> b3
   holds ex b4 being Element of the carrier of b1 st
      not LIN b2,b3,b4;

:: DIRAF:th 45
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being non empty AffinStruct
   st b2 = Lambda b1
for b3, b4, b5, b6 being Element of the carrier of b1
for b7, b8, b9, b10 being Element of the carrier of b2
      st b3 = b7 & b4 = b8 & b5 = b9 & b6 = b10
   holds    b7,b8 // b9,b10
   iff
      b3,b4 '||' b5,b6;

:: DIRAF:th 46
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being non empty AffinStruct
      st b2 = Lambda b1
   holds (ex b3, b4 being Element of the carrier of b2 st
       b3 <> b4) &
    (for b3, b4, b5, b6, b7, b8 being Element of the carrier of b2 holds
    b3,b4 // b4,b3 &
     b3,b4 // b5,b5 &
     (b3 <> b4 & b3,b4 // b5,b6 & b3,b4 // b7,b8 implies b5,b6 // b7,b8) &
     (b3,b4 // b3,b5 implies b4,b3 // b4,b5)) &
    (ex b3, b4, b5 being Element of the carrier of b2 st
       not b3,b4 // b3,b5) &
    (for b3, b4, b5 being Element of the carrier of b2 holds
    ex b6 being Element of the carrier of b2 st
       b3,b5 // b4,b6 & b4 <> b6) &
    (for b3, b4, b5 being Element of the carrier of b2 holds
    ex b6 being Element of the carrier of b2 st
       b3,b4 // b5,b6 & b3,b5 // b4,b6) &
    (for b3, b4, b5, b6 being Element of the carrier of b2
          st b5,b3 // b3,b6 & b3 <> b5
       holds ex b7 being Element of the carrier of b2 st
          b4,b3 // b3,b7 & b4,b5 // b6,b7);

:: DIRAF:attrnot 1 => DIRAF:attr 1
definition
  let a1 be non empty AffinStruct;
  attr a1 is AffinSpace-like means
    (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1 holds
     b1,b2 // b2,b1 &
      b1,b2 // b3,b3 &
      (b1 <> b2 & b1,b2 // b3,b4 & b1,b2 // b5,b6 implies b3,b4 // b5,b6) &
      (b1,b2 // b1,b3 implies b2,b1 // b2,b3)) &
     (ex b1, b2, b3 being Element of the carrier of a1 st
        not b1,b2 // b1,b3) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b3 // b2,b4 & b2 <> b4) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b2 // b3,b4 & b1,b3 // b2,b4) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b3,b1 // b1,b4 & b1 <> b3
        holds ex b5 being Element of the carrier of a1 st
           b2,b1 // b1,b5 & b2,b3 // b4,b5);
end;

:: DIRAF:dfs 6
definiens
  let a1 be non empty AffinStruct;
To prove
     a1 is AffinSpace-like
it is sufficient to prove
  thus (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1 holds
     b1,b2 // b2,b1 &
      b1,b2 // b3,b3 &
      (b1 <> b2 & b1,b2 // b3,b4 & b1,b2 // b5,b6 implies b3,b4 // b5,b6) &
      (b1,b2 // b1,b3 implies b2,b1 // b2,b3)) &
     (ex b1, b2, b3 being Element of the carrier of a1 st
        not b1,b2 // b1,b3) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b3 // b2,b4 & b2 <> b4) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b2 // b3,b4 & b1,b3 // b2,b4) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b3,b1 // b1,b4 & b1 <> b3
        holds ex b5 being Element of the carrier of a1 st
           b2,b1 // b1,b5 & b2,b3 // b4,b5);

:: DIRAF:def 7
theorem
for b1 being non empty AffinStruct holds
      b1 is AffinSpace-like
   iff
      (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
       b2,b3 // b3,b2 &
        b2,b3 // b4,b4 &
        (b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7 implies b4,b5 // b6,b7) &
        (b2,b3 // b2,b4 implies b3,b2 // b3,b4)) &
       (ex b2, b3, b4 being Element of the carrier of b1 st
          not b2,b3 // b2,b4) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b2,b4 // b3,b5 & b3 <> b5) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b2,b3 // b4,b5 & b2,b4 // b3,b5) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b4,b2 // b2,b5 & b2 <> b4
          holds ex b6 being Element of the carrier of b1 st
             b3,b2 // b2,b6 & b3,b4 // b5,b6);

:: DIRAF:exreg 1
registration
  cluster non empty non trivial strict AffinSpace-like AffinStruct;
end;

:: DIRAF:modenot 1
definition
  mode AffinSpace is non empty non trivial AffinSpace-like AffinStruct;
end;

:: DIRAF:th 47
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
   (ex b2, b3 being Element of the carrier of b1 st
       b2 <> b3) &
    (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
    b2,b3 // b3,b2 &
     b2,b3 // b4,b4 &
     (b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7 implies b4,b5 // b6,b7) &
     (b2,b3 // b2,b4 implies b3,b2 // b3,b4)) &
    (ex b2, b3, b4 being Element of the carrier of b1 st
       not b2,b3 // b2,b4) &
    (for b2, b3, b4 being Element of the carrier of b1 holds
    ex b5 being Element of the carrier of b1 st
       b2,b4 // b3,b5 & b3 <> b5) &
    (for b2, b3, b4 being Element of the carrier of b1 holds
    ex b5 being Element of the carrier of b1 st
       b2,b3 // b4,b5 & b2,b4 // b3,b5) &
    (for b2, b3, b4, b5 being Element of the carrier of b1
          st b4,b2 // b2,b5 & b2 <> b4
       holds ex b6 being Element of the carrier of b1 st
          b3,b2 // b2,b6 & b3,b4 // b5,b6);

:: DIRAF:th 48
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   Lambda b1 is non empty non trivial AffinSpace-like AffinStruct;

:: DIRAF:th 49
theorem
for b1 being non empty AffinStruct holds
      (ex b2, b3 being Element of the carrier of b1 st
          b2 <> b3) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
       b2,b3 // b3,b2 &
        b2,b3 // b4,b4 &
        (b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7 implies b4,b5 // b6,b7) &
        (b2,b3 // b2,b4 implies b3,b2 // b3,b4)) &
       (ex b2, b3, b4 being Element of the carrier of b1 st
          not b2,b3 // b2,b4) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b2,b4 // b3,b5 & b3 <> b5) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b2,b3 // b4,b5 & b2,b4 // b3,b5) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b4,b2 // b2,b5 & b2 <> b4
          holds ex b6 being Element of the carrier of b1 st
             b3,b2 // b2,b6 & b3,b4 // b5,b6)
   iff
      b1 is non empty non trivial AffinSpace-like AffinStruct;

:: DIRAF:th 50
theorem
for b1 being non empty non trivial OAffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3 '||' b4,b5
   holds ex b6 being Element of the carrier of b1 st
      b2,b3 '||' b2,b6 & b4,b5 '||' b4,b6;

:: DIRAF:th 51
theorem
for b1 being non empty AffinStruct
for b2 being non empty non trivial OAffinSpace-like 2-dimensional AffinStruct
   st b1 = Lambda b2
for b3, b4, b5, b6 being Element of the carrier of b1
      st not b3,b4 // b5,b6
   holds ex b7 being Element of the carrier of b1 st
      b3,b4 // b3,b7 & b5,b6 // b5,b7;

:: DIRAF:attrnot 2 => DIRAF:attr 2
definition
  let a1 be non empty AffinStruct;
  attr a1 is 2-dimensional means
    for b1, b2, b3, b4 being Element of the carrier of a1
          st not b1,b2 // b3,b4
       holds ex b5 being Element of the carrier of a1 st
          b1,b2 // b1,b5 & b3,b4 // b3,b5;
end;

:: DIRAF:dfs 7
definiens
  let a1 be non empty AffinStruct;
To prove
     a1 is 2-dimensional
it is sufficient to prove
  thus for b1, b2, b3, b4 being Element of the carrier of a1
          st not b1,b2 // b3,b4
       holds ex b5 being Element of the carrier of a1 st
          b1,b2 // b1,b5 & b3,b4 // b3,b5;

:: DIRAF:def 8
theorem
for b1 being non empty AffinStruct holds
      b1 is 2-dimensional
   iff
      for b2, b3, b4, b5 being Element of the carrier of b1
            st not b2,b3 // b4,b5
         holds ex b6 being Element of the carrier of b1 st
            b2,b3 // b2,b6 & b4,b5 // b4,b6;

:: DIRAF:exreg 2
registration
  cluster non empty non trivial strict AffinSpace-like 2-dimensional AffinStruct;
end;

:: DIRAF:modenot 2
definition
  mode AffinPlane is non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
end;

:: DIRAF:th 53
theorem
for b1 being non empty non trivial OAffinSpace-like 2-dimensional AffinStruct holds
   Lambda b1 is non empty non trivial AffinSpace-like 2-dimensional AffinStruct;

:: DIRAF:th 54
theorem
for b1 being non empty AffinStruct holds
      b1 is non empty non trivial AffinSpace-like 2-dimensional AffinStruct
   iff
      (ex b2, b3 being Element of the carrier of b1 st
          b2 <> b3) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
       b2,b3 // b3,b2 &
        b2,b3 // b4,b4 &
        (b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7 implies b4,b5 // b6,b7) &
        (b2,b3 // b2,b4 implies b3,b2 // b3,b4)) &
       (ex b2, b3, b4 being Element of the carrier of b1 st
          not b2,b3 // b2,b4) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b2,b4 // b3,b5 & b3 <> b5) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b2,b3 // b4,b5 & b2,b4 // b3,b5) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b4,b2 // b2,b5 & b2 <> b4
          holds ex b6 being Element of the carrier of b1 st
             b3,b2 // b2,b6 & b3,b4 // b5,b6) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st not b2,b3 // b4,b5
          holds ex b6 being Element of the carrier of b1 st
             b2,b3 // b2,b6 & b4,b5 // b4,b6);