Article ANALOAF, MML version 4.99.1005

:: ANALOAF:prednot 1 => ANALOAF:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3 // A4,A5 means
    (a2 <> a3 & a4 <> a5) implies ex b1, b2 being Element of REAL st
       0 < b1 &
        0 < b2 &
        b1 * (a3 - a2) = b2 * (a5 - a4);
end;

:: ANALOAF:dfs 1
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3 // a4,a5
it is sufficient to prove
  thus (a2 <> a3 & a4 <> a5) implies ex b1, b2 being Element of REAL st
       0 < b1 &
        0 < b2 &
        b1 * (a3 - a2) = b2 * (a5 - a4);

:: ANALOAF:def 1
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 holds
   b2,b3 // b4,b5
iff
   (b2 <> b3 & b4 <> b5 implies ex b6, b7 being Element of REAL st
      0 < b6 &
       0 < b7 &
       b6 * (b3 - b2) = b7 * (b5 - b4));

:: ANALOAF:th 4
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) + (b3 - b4) = b2 - b4;

:: ANALOAF:th 9
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 b2 + b3 = b4 + b5
   holds b2 - b5 = b4 - b3;

:: ANALOAF:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
   b4 * (b2 - b3) = - (b4 * (b3 - b2));

:: ANALOAF:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of REAL holds
(b4 - b5) * (b2 - b3) = (b5 - b4) * (b3 - b2);

:: ANALOAF:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b4 <> 0 & b4 * b2 = b3
   holds b2 = b4 " * b3;

:: ANALOAF:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of REAL holds
   (b4 <> 0 & b4 * b2 = b3 implies b2 = b4 " * b3) &
    (b4 <> 0 & b2 = b4 " * b3 implies b4 * b2 = b3);

:: ANALOAF:th 16
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 b2,b3 // b4,b5 & b2 <> b3 & b4 <> b5
   holds ex b6, b7 being Element of REAL st
      b6 * (b3 - b2) = b7 * (b5 - b4) &
       0 < b6 &
       0 < b7;

:: ANALOAF:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 // b2,b3;

:: ANALOAF:th 18
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,b4 & b2,b2 // b3,b4;

:: ANALOAF:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
      st b2,b3 // b3,b2
   holds b2 = b3;

:: ANALOAF:th 20
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
      st b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7
   holds b4,b5 // b6,b7;

:: ANALOAF:th 21
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 b2,b3 // b4,b5
   holds b3,b2 // b5,b4 & b4,b5 // b2,b3;

:: ANALOAF:th 22
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 // b3,b4
   holds b2,b3 // b2,b4;

:: ANALOAF:th 23
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 // b2,b4 & not b2,b3 // b3,b4
   holds b2,b4 // b4,b3;

:: ANALOAF:th 24
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 b2 - b3 = b4 - b5
   holds b3,b2 // b5,b4;

:: ANALOAF:th 25
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 b2 = (b3 + b4) - b5
   holds b5,b3 // b4,b2 & b5,b4 // b3,b2;

:: ANALOAF:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
   st ex b2, b3 being Element of the carrier of b1 st
        b2 <> b3
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 & b3 <> b5;

:: ANALOAF:th 27
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 b2 <> b3 & b3,b2 // b2,b4
   holds ex b6 being Element of the carrier of b1 st
      b5,b2 // b2,b6 & b5,b3 // b4,b6;

:: ANALOAF:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for 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 b2 <> b3 & b2 <> 0. b1 & b3 <> 0. b1;

:: ANALOAF:th 29
theorem
for b1 being non empty 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 ex b2, b3, b4, b5 being Element of the carrier of b1 st
      not b2,b3 // b4,b5 & not b2,b3 // b5,b4;

:: ANALOAF:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
   st ex b2, b3 being Element of the carrier of b1 st
        for b4 being Element of the carrier of b1 holds
           ex b5, b6 being Element of REAL st
              (b5 * b2) + (b6 * b3) = b4
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3 // b4,b5 & not b2,b3 // b5,b4
   holds ex b6 being Element of the carrier of b1 st
      (b2,b3 // b2,b6 or b2,b3 // b6,b2) & (b4,b5 // b4,b6 or b4,b5 // b6,b4);

:: ANALOAF:structnot 1 => ANALOAF:struct 1
definition
  struct(1-sorted) AffinStruct(#
    carrier -> set,
    CONGR -> Relation of [:the carrier of it,the carrier of it:],[:the carrier of it,the carrier of it:]
  #);
end;

:: ANALOAF:attrnot 1 => ANALOAF:attr 1
definition
  let a1 be AffinStruct;
  attr a1 is strict;
end;

:: ANALOAF:exreg 1
registration
  cluster strict AffinStruct;
end;

:: ANALOAF:aggrnot 1 => ANALOAF:aggr 1
definition
  let a1 be set;
  let a2 be Relation of [:a1,a1:],[:a1,a1:];
  aggr AffinStruct(#a1,a2#) -> strict AffinStruct;
end;

:: ANALOAF:selnot 1 => ANALOAF:sel 1
definition
  let a1 be AffinStruct;
  sel the CONGR of a1 -> Relation of [:the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1:];
end;

:: ANALOAF:exreg 2
registration
  cluster non empty strict AffinStruct;
end;

:: ANALOAF:prednot 2 => ANALOAF: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
    [[a2,a3],[a4,a5]] in the CONGR of a1;
end;

:: ANALOAF:dfs 2
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 [[a2,a3],[a4,a5]] in the CONGR of a1;

:: ANALOAF:def 2
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]] in the CONGR of b1;

:: ANALOAF:funcnot 1 => ANALOAF:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func DirPar A1 -> Relation of [:the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1:] means
    for b1, b2 being set holds
       [b1,b2] in it
    iff
       ex b3, b4, b5, b6 being Element of the carrier of a1 st
          b1 = [b3,b4] & b2 = [b5,b6] & b3,b4 // b5,b6;
end;

:: ANALOAF:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Relation of [:the carrier of b1,the carrier of b1:],[:the carrier of b1,the carrier of b1:] holds
      b2 = DirPar b1
   iff
      for b3, b4 being set holds
         [b3,b4] in b2
      iff
         ex b5, b6, b7, b8 being Element of the carrier of b1 st
            b3 = [b5,b6] & b4 = [b7,b8] & b5,b6 // b7,b8;

:: ANALOAF:th 33
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 holds
   [[b2,b3],[b4,b5]] in DirPar b1
iff
   b2,b3 // b4,b5;

:: ANALOAF:funcnot 2 => ANALOAF:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func OASpace A1 -> strict AffinStruct equals
    AffinStruct(#the carrier of a1,DirPar a1#);
end;

:: ANALOAF:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   OASpace b1 = AffinStruct(#the carrier of b1,DirPar b1#);

:: ANALOAF:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster OASpace a1 -> non empty strict;
end;

:: ANALOAF:th 35
theorem
for b1 being non empty 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 (ex b2, b3 being Element of the carrier of OASpace b1 st
       b2 <> b3) &
    (for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of OASpace b1 holds
    b2,b3 // b4,b4 &
     (b2,b3 // b3,b2 implies b2 = b3) &
     (b2 <> b3 & b2,b3 // b6,b7 & b2,b3 // b8,b9 implies b6,b7 // b8,b9) &
     (b2,b3 // b4,b5 implies b3,b2 // b5,b4) &
     (b2,b3 // b3,b4 implies b2,b3 // b2,b4) &
     (b2,b3 // b2,b4 & not b2,b3 // b3,b4 implies b2,b4 // b4,b3)) &
    (ex b2, b3, b4, b5 being Element of the carrier of OASpace b1 st
       not b2,b3 // b4,b5 & not b2,b3 // b5,b4) &
    (for b2, b3, b4 being Element of the carrier of OASpace b1 holds
    ex b5 being Element of the carrier of OASpace b1 st
       b2,b3 // b4,b5 & b2,b4 // b3,b5 & b3 <> b5) &
    (for b2, b3, b4, b5 being Element of the carrier of OASpace b1
          st b2 <> b4 & b4,b2 // b2,b5
       holds ex b6 being Element of the carrier of OASpace b1 st
          b3,b2 // b2,b6 & b3,b4 // b5,b6);

:: ANALOAF:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
   st ex b2, b3 being Element of the carrier of b1 st
        for b4 being Element of the carrier of b1 holds
           ex b5, b6 being Element of REAL st
              (b5 * b2) + (b6 * b3) = b4
for b2, b3, b4, b5 being Element of the carrier of OASpace b1
      st not b2,b3 // b4,b5 & not b2,b3 // b5,b4
   holds ex b6 being Element of the carrier of OASpace b1 st
      (b2,b3 // b2,b6 or b2,b3 // b6,b2) & (b4,b5 // b4,b6 or b4,b5 // b6,b4);

:: ANALOAF:attrnot 2 => ANALOAF:attr 2
definition
  let a1 be non empty AffinStruct;
  attr a1 is OAffinSpace-like means
    (for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
     b1,b2 // b3,b3 &
      (b1,b2 // b2,b1 implies b1 = b2) &
      (b1 <> b2 & b1,b2 // b5,b6 & b1,b2 // b7,b8 implies b5,b6 // b7,b8) &
      (b1,b2 // b3,b4 implies b2,b1 // b4,b3) &
      (b1,b2 // b2,b3 implies b1,b2 // b1,b3) &
      (b1,b2 // b1,b3 & not b1,b2 // b2,b3 implies b1,b3 // b3,b2)) &
     (ex b1, b2, b3, b4 being Element of the carrier of a1 st
        not b1,b2 // b3,b4 & not b1,b2 // b4,b3) &
     (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 & b2 <> b4) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b1 <> b3 & b3,b1 // b1,b4
        holds ex b5 being Element of the carrier of a1 st
           b2,b1 // b1,b5 & b2,b3 // b4,b5);
end;

:: ANALOAF:dfs 5
definiens
  let a1 be non empty AffinStruct;
To prove
     a1 is OAffinSpace-like
it is sufficient to prove
  thus (for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
     b1,b2 // b3,b3 &
      (b1,b2 // b2,b1 implies b1 = b2) &
      (b1 <> b2 & b1,b2 // b5,b6 & b1,b2 // b7,b8 implies b5,b6 // b7,b8) &
      (b1,b2 // b3,b4 implies b2,b1 // b4,b3) &
      (b1,b2 // b2,b3 implies b1,b2 // b1,b3) &
      (b1,b2 // b1,b3 & not b1,b2 // b2,b3 implies b1,b3 // b3,b2)) &
     (ex b1, b2, b3, b4 being Element of the carrier of a1 st
        not b1,b2 // b3,b4 & not b1,b2 // b4,b3) &
     (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 & b2 <> b4) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b1 <> b3 & b3,b1 // b1,b4
        holds ex b5 being Element of the carrier of a1 st
           b2,b1 // b1,b5 & b2,b3 // b4,b5);

:: ANALOAF:def 5
theorem
for b1 being non empty AffinStruct holds
      b1 is OAffinSpace-like
   iff
      (for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1 holds
       b2,b3 // b4,b4 &
        (b2,b3 // b3,b2 implies b2 = b3) &
        (b2 <> b3 & b2,b3 // b6,b7 & b2,b3 // b8,b9 implies b6,b7 // b8,b9) &
        (b2,b3 // b4,b5 implies b3,b2 // b5,b4) &
        (b2,b3 // b3,b4 implies b2,b3 // b2,b4) &
        (b2,b3 // b2,b4 & not b2,b3 // b3,b4 implies b2,b4 // b4,b3)) &
       (ex b2, b3, b4, b5 being Element of the carrier of b1 st
          not b2,b3 // b4,b5 & not b2,b3 // b5,b4) &
       (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 & b3 <> b5) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b2 <> b4 & b4,b2 // b2,b5
          holds ex b6 being Element of the carrier of b1 st
             b3,b2 // b2,b6 & b3,b4 // b5,b6);

:: ANALOAF:exreg 3
registration
  cluster non empty non trivial strict OAffinSpace-like AffinStruct;
end;

:: ANALOAF:modenot 1
definition
  mode OAffinSpace is non empty non trivial OAffinSpace-like AffinStruct;
end;

:: ANALOAF:th 37
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, b8, b9 being Element of the carrier of b1 holds
       b2,b3 // b4,b4 &
        (b2,b3 // b3,b2 implies b2 = b3) &
        (b2 <> b3 & b2,b3 // b6,b7 & b2,b3 // b8,b9 implies b6,b7 // b8,b9) &
        (b2,b3 // b4,b5 implies b3,b2 // b5,b4) &
        (b2,b3 // b3,b4 implies b2,b3 // b2,b4) &
        (b2,b3 // b2,b4 & not b2,b3 // b3,b4 implies b2,b4 // b4,b3)) &
       (ex b2, b3, b4, b5 being Element of the carrier of b1 st
          not b2,b3 // b4,b5 & not b2,b3 // b5,b4) &
       (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 & b3 <> b5) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b2 <> b4 & b4,b2 // b2,b5
          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 OAffinSpace-like AffinStruct;

:: ANALOAF:th 38
theorem
for b1 being non empty 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 OASpace b1 is non empty non trivial OAffinSpace-like AffinStruct;

:: ANALOAF:attrnot 3 => ANALOAF:attr 3
definition
  let a1 be non empty non trivial OAffinSpace-like 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 & not b1,b2 // b4,b3
       holds ex b5 being Element of the carrier of a1 st
          (b1,b2 // b1,b5 or b1,b2 // b5,b1) & (b3,b4 // b3,b5 or b3,b4 // b5,b3);
end;

:: ANALOAF:dfs 6
definiens
  let a1 be non empty non trivial OAffinSpace-like 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 & not b1,b2 // b4,b3
       holds ex b5 being Element of the carrier of a1 st
          (b1,b2 // b1,b5 or b1,b2 // b5,b1) & (b3,b4 // b3,b5 or b3,b4 // b5,b3);

:: ANALOAF:def 6
theorem
for b1 being non empty non trivial OAffinSpace-like 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 & not b2,b3 // b5,b4
         holds ex b6 being Element of the carrier of b1 st
            (b2,b3 // b2,b6 or b2,b3 // b6,b2) & (b4,b5 // b4,b6 or b4,b5 // b6,b4);

:: ANALOAF:exreg 4
registration
  cluster non empty non trivial strict OAffinSpace-like 2-dimensional AffinStruct;
end;

:: ANALOAF:modenot 2
definition
  mode OAffinPlane is non empty non trivial OAffinSpace-like 2-dimensional AffinStruct;
end;

:: ANALOAF:th 50
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, b8, b9 being Element of the carrier of b1 holds
       b2,b3 // b4,b4 &
        (b2,b3 // b3,b2 implies b2 = b3) &
        (b2 <> b3 & b2,b3 // b6,b7 & b2,b3 // b8,b9 implies b6,b7 // b8,b9) &
        (b2,b3 // b4,b5 implies b3,b2 // b5,b4) &
        (b2,b3 // b3,b4 implies b2,b3 // b2,b4) &
        (b2,b3 // b2,b4 & not b2,b3 // b3,b4 implies b2,b4 // b4,b3)) &
       (ex b2, b3, b4, b5 being Element of the carrier of b1 st
          not b2,b3 // b4,b5 & not b2,b3 // b5,b4) &
       (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 & b3 <> b5) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b2 <> b4 & b4,b2 // b2,b5
          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 & not b2,b3 // b5,b4
          holds ex b6 being Element of the carrier of b1 st
             (b2,b3 // b2,b6 or b2,b3 // b6,b2) & (b4,b5 // b4,b6 or b4,b5 // b6,b4))
   iff
      b1 is non empty non trivial OAffinSpace-like 2-dimensional AffinStruct;

:: ANALOAF:th 51
theorem
for b1 being non empty 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) &
            (for b4 being Element of the carrier of b1 holds
               ex b5, b6 being Element of REAL st
                  b4 = (b5 * b2) + (b6 * b3))
   holds OASpace b1 is non empty non trivial OAffinSpace-like 2-dimensional AffinStruct;