Article TRANSGEO, MML version 4.99.1005

:: TRANSGEO:funcnot 1 => TRANSGEO:func 1
definition
  let a1 be set;
  let a2, a3 be Function-like quasi_total bijective Relation of a1,a1;
  redefine func a3 * a2 -> Function-like quasi_total bijective Relation of a1,a1;
end;

:: TRANSGEO:th 2
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total bijective Relation of b1,b1 holds
   ex b4 being Element of b1 st
      b3 . b4 = b2;

:: TRANSGEO:th 4
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Function-like quasi_total bijective Relation of b1,b1 holds
      b4 . b2 = b3
   iff
      b4 " . b3 = b2;

:: TRANSGEO:funcnot 2 => TRANSGEO:func 2
definition
  let a1 be non empty set;
  let a2, a3 be Function-like quasi_total bijective Relation of a1,a1;
  func A2 \ A3 -> Function-like quasi_total bijective Relation of a1,a1 equals
    (a3 * a2) * (a3 ");
end;

:: TRANSGEO:def 1
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total bijective Relation of b1,b1 holds
b2 \ b3 = (b3 * b2) * (b3 ");

:: TRANSGEO:sch 1
scheme TRANSGEO:sch 1
{F1 -> non empty set}:
ex b1 being Function-like quasi_total bijective Relation of F1(),F1() st
   for b2, b3 being Element of F1() holds
      b1 . b2 = b3
   iff
      P1[b2, b3]
provided
   for b1 being Element of F1() holds
      ex b2 being Element of F1() st
         P1[b1, b2]
and
   for b1 being Element of F1() holds
      ex b2 being Element of F1() st
         P1[b2, b1]
and
   for b1, b2, b3 being Element of F1()
         st P1[b1, b2] & P1[b3, b2]
      holds b1 = b3
and
   for b1, b2, b3 being Element of F1()
         st P1[b1, b2] & P1[b1, b3]
      holds b2 = b3;


:: TRANSGEO:th 9
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total bijective Relation of b1,b1 holds
   b3 . (b3 " . b2) = b2 & b3 " . (b3 . b2) = b2;

:: TRANSGEO:th 11
theorem
for b1 being non empty set
for b2 being Function-like quasi_total bijective Relation of b1,b1 holds
   b2 * id b1 = (id b1) * b2;

:: TRANSGEO:th 13
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like quasi_total bijective Relation of b1,b1
      st (b2 * b3 = b4 * b3 or b3 * b2 = b3 * b4)
   holds b2 = b4;

:: TRANSGEO:th 16
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like quasi_total bijective Relation of b1,b1 holds
(b2 * b3) \ b4 = (b2 \ b4) * (b3 \ b4);

:: TRANSGEO:th 17
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total bijective Relation of b1,b1 holds
b2 " \ b3 = (b2 \ b3) ";

:: TRANSGEO:th 18
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like quasi_total bijective Relation of b1,b1 holds
b2 \ (b3 * b4) = (b2 \ b4) \ b3;

:: TRANSGEO:th 19
theorem
for b1 being non empty set
for b2 being Function-like quasi_total bijective Relation of b1,b1 holds
   (id b1) \ b2 = id b1;

:: TRANSGEO:th 20
theorem
for b1 being non empty set
for b2 being Function-like quasi_total bijective Relation of b1,b1 holds
   b2 \ id b1 = b2;

:: TRANSGEO:th 21
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total bijective Relation of b1,b1
      st b3 . b2 = b2
   holds (b3 \ b4) . (b4 . b2) = b4 . b2;

:: TRANSGEO:prednot 1 => TRANSGEO:pred 1
definition
  let a1 be non empty set;
  let a2 be Function-like quasi_total bijective Relation of a1,a1;
  let a3 be Relation of [:a1,a1:],[:a1,a1:];
  pred A2 is_FormalIz_of A3 means
    for b1, b2 being Element of a1 holds
    [[b1,b2],[a2 . b1,a2 . b2]] in a3;
end;

:: TRANSGEO:dfs 2
definiens
  let a1 be non empty set;
  let a2 be Function-like quasi_total bijective Relation of a1,a1;
  let a3 be Relation of [:a1,a1:],[:a1,a1:];
To prove
     a2 is_FormalIz_of a3
it is sufficient to prove
  thus for b1, b2 being Element of a1 holds
    [[b1,b2],[a2 . b1,a2 . b2]] in a3;

:: TRANSGEO:def 2
theorem
for b1 being non empty set
for b2 being Function-like quasi_total bijective Relation of b1,b1
for b3 being Relation of [:b1,b1:],[:b1,b1:] holds
      b2 is_FormalIz_of b3
   iff
      for b4, b5 being Element of b1 holds
      [[b4,b5],[b2 . b4,b2 . b5]] in b3;

:: TRANSGEO:th 23
theorem
for b1 being non empty set
for b2 being Relation of [:b1,b1:],[:b1,b1:]
      st b2 is_reflexive_in [:b1,b1:]
   holds id b1 is_FormalIz_of b2;

:: TRANSGEO:th 24
theorem
for b1 being non empty set
for b2 being Function-like quasi_total bijective Relation of b1,b1
for b3 being Relation of [:b1,b1:],[:b1,b1:]
      st b3 is_symmetric_in [:b1,b1:] & b2 is_FormalIz_of b3
   holds b2 " is_FormalIz_of b3;

:: TRANSGEO:th 25
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total bijective Relation of b1,b1
for b4 being Relation of [:b1,b1:],[:b1,b1:]
      st b4 is_transitive_in [:b1,b1:] & b2 is_FormalIz_of b4 & b3 is_FormalIz_of b4
   holds b2 * b3 is_FormalIz_of b4;

:: TRANSGEO:th 26
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total bijective Relation of b1,b1
for b4 being Relation of [:b1,b1:],[:b1,b1:]
      st (for b5, b6, b7, b8, b9, b10 being Element of b1
               st [[b7,b8],[b5,b6]] in b4 &
                  [[b5,b6],[b9,b10]] in b4 &
                  b5 <> b6
            holds [[b7,b8],[b9,b10]] in b4) &
         (for b5, b6, b7 being Element of b1 holds
         [[b5,b5],[b6,b7]] in b4) &
         b2 is_FormalIz_of b4 &
         b3 is_FormalIz_of b4
   holds b2 * b3 is_FormalIz_of b4;

:: TRANSGEO:prednot 2 => TRANSGEO:pred 2
definition
  let a1 be non empty set;
  let a2 be Function-like quasi_total bijective Relation of a1,a1;
  let a3 be Relation of [:a1,a1:],[:a1,a1:];
  pred A2 is_automorphism_of A3 means
    for b1, b2, b3, b4 being Element of a1 holds
       [[b1,b2],[b3,b4]] in a3
    iff
       [[a2 . b1,a2 . b2],[a2 . b3,a2 . b4]] in a3;
end;

:: TRANSGEO:dfs 3
definiens
  let a1 be non empty set;
  let a2 be Function-like quasi_total bijective Relation of a1,a1;
  let a3 be Relation of [:a1,a1:],[:a1,a1:];
To prove
     a2 is_automorphism_of a3
it is sufficient to prove
  thus for b1, b2, b3, b4 being Element of a1 holds
       [[b1,b2],[b3,b4]] in a3
    iff
       [[a2 . b1,a2 . b2],[a2 . b3,a2 . b4]] in a3;

:: TRANSGEO:def 3
theorem
for b1 being non empty set
for b2 being Function-like quasi_total bijective Relation of b1,b1
for b3 being Relation of [:b1,b1:],[:b1,b1:] holds
      b2 is_automorphism_of b3
   iff
      for b4, b5, b6, b7 being Element of b1 holds
         [[b4,b5],[b6,b7]] in b3
      iff
         [[b2 . b4,b2 . b5],[b2 . b6,b2 . b7]] in b3;

:: TRANSGEO:th 28
theorem
for b1 being non empty set
for b2 being Relation of [:b1,b1:],[:b1,b1:] holds
   id b1 is_automorphism_of b2;

:: TRANSGEO:th 29
theorem
for b1 being non empty set
for b2 being Function-like quasi_total bijective Relation of b1,b1
for b3 being Relation of [:b1,b1:],[:b1,b1:]
      st b2 is_automorphism_of b3
   holds b2 " is_automorphism_of b3;

:: TRANSGEO:th 30
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total bijective Relation of b1,b1
for b4 being Relation of [:b1,b1:],[:b1,b1:]
      st b2 is_automorphism_of b4 & b3 is_automorphism_of b4
   holds b3 * b2 is_automorphism_of b4;

:: TRANSGEO:th 31
theorem
for b1 being non empty set
for b2 being Function-like quasi_total bijective Relation of b1,b1
for b3 being Relation of [:b1,b1:],[:b1,b1:]
      st b3 is_symmetric_in [:b1,b1:] & b3 is_transitive_in [:b1,b1:] & b2 is_FormalIz_of b3
   holds b2 is_automorphism_of b3;

:: TRANSGEO:th 32
theorem
for b1 being non empty set
for b2 being Function-like quasi_total bijective Relation of b1,b1
for b3 being Relation of [:b1,b1:],[:b1,b1:]
      st (for b4, b5, b6, b7, b8, b9 being Element of b1
               st [[b6,b7],[b4,b5]] in b3 &
                  [[b4,b5],[b8,b9]] in b3 &
                  b4 <> b5
            holds [[b6,b7],[b8,b9]] in b3) &
         (for b4, b5, b6 being Element of b1 holds
         [[b4,b4],[b5,b6]] in b3) &
         b3 is_symmetric_in [:b1,b1:] &
         b2 is_FormalIz_of b3
   holds b2 is_automorphism_of b3;

:: TRANSGEO:th 33
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total bijective Relation of b1,b1
for b4 being Relation of [:b1,b1:],[:b1,b1:]
      st b2 is_FormalIz_of b4 & b3 is_automorphism_of b4
   holds b2 \ b3 is_FormalIz_of b4;

:: TRANSGEO:prednot 3 => TRANSGEO:pred 3
definition
  let a1 be non empty AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  pred A2 is_DIL_of A1 means
    a2 is_FormalIz_of the CONGR of a1;
end;

:: TRANSGEO:dfs 4
definiens
  let a1 be non empty AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is_DIL_of a1
it is sufficient to prove
  thus a2 is_FormalIz_of the CONGR of a1;

:: TRANSGEO:def 4
theorem
for b1 being non empty AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is_DIL_of b1
   iff
      b2 is_FormalIz_of the CONGR of b1;

:: TRANSGEO:th 35
theorem
for b1 being non empty AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is_DIL_of b1
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b3,b4 // b2 . b3,b2 . b4;

:: TRANSGEO:attrnot 1 => TRANSGEO:attr 1
definition
  let a1 be non empty AffinStruct;
  attr a1 is CongrSpace-like means
    (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b5,b6 & b5,b6 // b3,b4 & b5 <> b6
        holds b1,b2 // b3,b4) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     b1,b1 // b2,b3) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b1,b2 // b3,b4
        holds b3,b4 // b1,b2) &
     (for b1, b2 being Element of the carrier of a1 holds
     b1,b2 // b1,b2);
end;

:: TRANSGEO:dfs 5
definiens
  let a1 be non empty AffinStruct;
To prove
     a1 is CongrSpace-like
it is sufficient to prove
  thus (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b5,b6 & b5,b6 // b3,b4 & b5 <> b6
        holds b1,b2 // b3,b4) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     b1,b1 // b2,b3) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b1,b2 // b3,b4
        holds b3,b4 // b1,b2) &
     (for b1, b2 being Element of the carrier of a1 holds
     b1,b2 // b1,b2);

:: TRANSGEO:def 5
theorem
for b1 being non empty AffinStruct holds
      b1 is CongrSpace-like
   iff
      (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
             st b2,b3 // b6,b7 & b6,b7 // b4,b5 & b6 <> b7
          holds b2,b3 // b4,b5) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       b2,b2 // b3,b4) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b2,b3 // b4,b5
          holds b4,b5 // b2,b3) &
       (for b2, b3 being Element of the carrier of b1 holds
       b2,b3 // b2,b3);

:: TRANSGEO:exreg 1
registration
  cluster non empty strict CongrSpace-like AffinStruct;
end;

:: TRANSGEO:modenot 1
definition
  mode CongrSpace is non empty CongrSpace-like AffinStruct;
end;

:: TRANSGEO:th 37
theorem
for b1 being non empty CongrSpace-like AffinStruct holds
   id the carrier of b1 is_DIL_of b1;

:: TRANSGEO:th 38
theorem
for b1 being non empty CongrSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is_DIL_of b1
   holds b2 " is_DIL_of b1;

:: TRANSGEO:th 39
theorem
for b1 being non empty CongrSpace-like AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is_DIL_of b1 & b3 is_DIL_of b1
   holds b2 * b3 is_DIL_of b1;

:: TRANSGEO:th 40
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   b1 is CongrSpace-like;

:: TRANSGEO:attrnot 2 => TRANSGEO:attr 2
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  attr a2 is positive_dilatation means
    a2 is_DIL_of a1;
end;

:: TRANSGEO:dfs 6
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is positive_dilatation
it is sufficient to prove
  thus a2 is_DIL_of a1;

:: TRANSGEO:def 6
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is positive_dilatation(b1)
   iff
      b2 is_DIL_of b1;

:: TRANSGEO:prednot 4 => TRANSGEO:attr 2
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  synonym a2 is_CDil for positive_dilatation;
end;

:: TRANSGEO:th 42
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is positive_dilatation(b1)
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b3,b4 // b2 . b3,b2 . b4;

:: TRANSGEO:attrnot 3 => TRANSGEO:attr 3
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  attr a2 is negative_dilatation means
    for b1, b2 being Element of the carrier of a1 holds
    b1,b2 // a2 . b2,a2 . b1;
end;

:: TRANSGEO:dfs 7
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is negative_dilatation
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1 holds
    b1,b2 // a2 . b2,a2 . b1;

:: TRANSGEO:def 7
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is negative_dilatation(b1)
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b3,b4 // b2 . b4,b2 . b3;

:: TRANSGEO:prednot 5 => TRANSGEO:attr 3
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  synonym a2 is_SDil for negative_dilatation;
end;

:: TRANSGEO:th 44
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   id the carrier of b1 is positive_dilatation(b1);

:: TRANSGEO:th 45
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is positive_dilatation(b1)
   holds b2 " is positive_dilatation(b1);

:: TRANSGEO:th 46
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is positive_dilatation(b1) & b3 is positive_dilatation(b1)
   holds b2 * b3 is positive_dilatation(b1);

:: TRANSGEO:th 47
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is negative_dilatation(b1)
   holds b2 is not positive_dilatation(b1);

:: TRANSGEO:th 48
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is negative_dilatation(b1)
   holds b2 " is negative_dilatation(b1);

:: TRANSGEO:th 49
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is positive_dilatation(b1) & b3 is negative_dilatation(b1)
   holds b2 * b3 is negative_dilatation(b1) & b3 * b2 is negative_dilatation(b1);

:: TRANSGEO:attrnot 4 => TRANSGEO:attr 4
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  attr a2 is dilatation means
    a2 is_FormalIz_of lambda the CONGR of a1;
end;

:: TRANSGEO:dfs 8
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is dilatation
it is sufficient to prove
  thus a2 is_FormalIz_of lambda the CONGR of a1;

:: TRANSGEO:def 8
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is dilatation(b1)
   iff
      b2 is_FormalIz_of lambda the CONGR of b1;

:: TRANSGEO:prednot 6 => TRANSGEO:attr 4
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  synonym a2 is_Dil for dilatation;
end;

:: TRANSGEO:th 51
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is dilatation(b1)
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b3,b4 '||' b2 . b3,b2 . b4;

:: TRANSGEO:th 52
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st (b2 is positive_dilatation(b1) or b2 is negative_dilatation(b1))
   holds b2 is dilatation(b1);

:: TRANSGEO:th 53
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1)
   holds ex b3 being Function-like quasi_total bijective Relation of the carrier of Lambda b1,the carrier of Lambda b1 st
      b2 = b3 & b3 is_DIL_of Lambda b1;

:: TRANSGEO:th 54
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of Lambda b1,the carrier of Lambda b1
      st b2 is_DIL_of Lambda b1
   holds ex b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
      b2 = b3 & b3 is dilatation(b1);

:: TRANSGEO:th 55
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   id the carrier of b1 is dilatation(b1);

:: TRANSGEO:th 56
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1)
   holds b2 " is dilatation(b1);

:: TRANSGEO:th 57
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1) & b3 is dilatation(b1)
   holds b2 * b3 is dilatation(b1);

:: TRANSGEO:th 58
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
   st b2 is dilatation(b1)
for b3, b4, b5, b6 being Element of the carrier of b1 holds
   b3,b4 '||' b5,b6
iff
   b2 . b3,b2 . b4 '||' b2 . b5,b2 . b6;

:: TRANSGEO:th 59
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
   st b2 is dilatation(b1)
for b3, b4, b5 being Element of the carrier of b1 holds
   LIN b3,b4,b5
iff
   LIN b2 . b3,b2 . b4,b2 . b5;

:: TRANSGEO:th 60
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is dilatation(b1) & LIN b2,b4 . b2,b3
   holds LIN b2,b4 . b2,b4 . b3;

:: TRANSGEO:th 61
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 & not b2,b4 '||' b3,b5
   holds ex b6 being Element of the carrier of b1 st
      LIN b2,b4,b6 & LIN b3,b5,b6;

:: TRANSGEO:th 62
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1)
   holds    (b2 = id the carrier of b1 or for b3 being Element of the carrier of b1 holds
         b2 . b3 <> b3)
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b3,b2 . b3 '||' b4,b2 . b4;

:: TRANSGEO:th 63
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b5 is dilatation(b1) & b5 . b2 = b2 & b5 . b3 = b3 & not LIN b2,b3,b4
   holds b5 . b4 = b4;

:: TRANSGEO:th 64
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is dilatation(b1) & b4 . b2 = b2 & b4 . b3 = b3 & b2 <> b3
   holds b4 = id the carrier of b1;

:: TRANSGEO:th 65
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is dilatation(b1) & b5 is dilatation(b1) & b4 . b2 = b5 . b2 & b4 . b3 = b5 . b3 & b2 <> b3
   holds b4 = b5;

:: TRANSGEO:attrnot 5 => TRANSGEO:attr 5
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  attr a2 is translation means
    a2 is dilatation(a1) &
     (a2 = id the carrier of a1 or for b1 being Element of the carrier of a1 holds
        b1 <> a2 . b1);
end;

:: TRANSGEO:dfs 9
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is translation
it is sufficient to prove
  thus a2 is dilatation(a1) &
     (a2 = id the carrier of a1 or for b1 being Element of the carrier of a1 holds
        b1 <> a2 . b1);

:: TRANSGEO:def 9
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is translation(b1)
   iff
      b2 is dilatation(b1) &
       (b2 = id the carrier of b1 or for b3 being Element of the carrier of b1 holds
          b3 <> b2 . b3);

:: TRANSGEO:prednot 7 => TRANSGEO:attr 5
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  synonym a2 is_Tr for translation;
end;

:: TRANSGEO:th 67
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1)
   holds    b2 is translation(b1)
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b3,b2 . b3 '||' b4,b2 . b4;

:: TRANSGEO:th 69
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is translation(b1) & b5 is translation(b1) & b4 . b2 = b5 . b2 & not LIN b2,b4 . b2,b3
   holds b4 . b3 = b5 . b3;

:: TRANSGEO:th 70
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b3 is translation(b1) & b4 is translation(b1) & b3 . b2 = b4 . b2
   holds b3 = b4;

:: TRANSGEO:th 71
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is translation(b1)
   holds b2 " is translation(b1);

:: TRANSGEO:th 72
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is translation(b1) & b3 is translation(b1)
   holds b2 * b3 is translation(b1);

:: TRANSGEO:th 73
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is translation(b1)
   holds b2 is positive_dilatation(b1);

:: TRANSGEO:th 74
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b5 is dilatation(b1) & b5 . b2 = b2 & Mid b3,b2,b5 . b3 & not LIN b2,b3,b4
   holds Mid b4,b2,b5 . b4;

:: TRANSGEO:th 75
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b5 is dilatation(b1) & b5 . b2 = b2 & Mid b3,b2,b5 . b3 & b3 <> b2
   holds Mid b4,b2,b5 . b4;

:: TRANSGEO:th 76
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b6 is dilatation(b1) & b6 . b2 = b2 & b3 <> b2 & Mid b3,b2,b6 . b3 & not LIN b2,b4,b5
   holds b4,b5 // b6 . b5,b6 . b4;

:: TRANSGEO:th 77
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b6 is dilatation(b1) & b6 . b2 = b2 & b3 <> b2 & Mid b3,b2,b6 . b3 & LIN b2,b4,b5
   holds b4,b5 // b6 . b5,b6 . b4;

:: TRANSGEO:th 78
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is dilatation(b1) & b4 . b2 = b2 & b3 <> b2 & Mid b3,b2,b4 . b3
   holds b4 is negative_dilatation(b1);

:: TRANSGEO:th 79
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
   st b3 is dilatation(b1) &
      b3 . b2 = b2 &
      (for b4 being Element of the carrier of b1 holds
         b2,b4 // b2,b3 . b4)
for b4, b5 being Element of the carrier of b1 holds
b4,b5 // b3 . b4,b3 . b5;

:: TRANSGEO:th 80
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1) & b2 is not positive_dilatation(b1)
   holds b2 is negative_dilatation(b1);

:: TRANSGEO:th 82
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
   b1 is CongrSpace-like;

:: TRANSGEO:th 83
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   Lambda b1 is non empty CongrSpace-like AffinStruct;

:: TRANSGEO:attrnot 6 => TRANSGEO:attr 6
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  attr a2 is dilatation means
    a2 is_DIL_of a1;
end;

:: TRANSGEO:dfs 10
definiens
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is dilatation
it is sufficient to prove
  thus a2 is_DIL_of a1;

:: TRANSGEO:def 10
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is dilatation(b1)
   iff
      b2 is_DIL_of b1;

:: TRANSGEO:prednot 8 => TRANSGEO:attr 6
notation
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  synonym a2 is_Dil for dilatation;
end;

:: TRANSGEO:th 85
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is dilatation(b1)
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b3,b4 // b2 . b3,b2 . b4;

:: TRANSGEO:th 86
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
   id the carrier of b1 is dilatation(b1);

:: TRANSGEO:th 87
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1)
   holds b2 " is dilatation(b1);

:: TRANSGEO:th 88
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1) & b3 is dilatation(b1)
   holds b2 * b3 is dilatation(b1);

:: TRANSGEO:th 89
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
   st b2 is dilatation(b1)
for b3, b4, b5, b6 being Element of the carrier of b1 holds
   b3,b4 // b5,b6
iff
   b2 . b3,b2 . b4 // b2 . b5,b2 . b6;

:: TRANSGEO:th 90
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
   st b2 is dilatation(b1)
for b3, b4, b5 being Element of the carrier of b1 holds
   LIN b3,b4,b5
iff
   LIN b2 . b3,b2 . b4,b2 . b5;

:: TRANSGEO:th 91
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is dilatation(b1) & LIN b2,b4 . b2,b3
   holds LIN b2,b4 . b2,b4 . b3;

:: TRANSGEO:th 92
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5 & not b2,b4 // b3,b5
   holds ex b6 being Element of the carrier of b1 st
      LIN b2,b4,b6 & LIN b3,b5,b6;

:: TRANSGEO:th 93
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1)
   holds    (b2 = id the carrier of b1 or for b3 being Element of the carrier of b1 holds
         b2 . b3 <> b3)
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b3,b2 . b3 // b4,b2 . b4;

:: TRANSGEO:th 94
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b5 is dilatation(b1) & b5 . b2 = b2 & b5 . b3 = b3 & not LIN b2,b3,b4
   holds b5 . b4 = b4;

:: TRANSGEO:th 95
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is dilatation(b1) & b4 . b2 = b2 & b4 . b3 = b3 & b2 <> b3
   holds b4 = id the carrier of b1;

:: TRANSGEO:th 96
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is dilatation(b1) & b5 is dilatation(b1) & b4 . b2 = b5 . b2 & b4 . b3 = b5 . b3 & b2 <> b3
   holds b4 = b5;

:: TRANSGEO:th 97
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not LIN b2,b3,b4 & b2,b3 // b4,b5 & b2,b3 // b4,b6 & b2,b4 // b3,b5 & b2,b4 // b3,b6
   holds b5 = b6;

:: TRANSGEO:attrnot 7 => TRANSGEO:attr 7
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  attr a2 is translation means
    a2 is dilatation(a1) &
     (a2 = id the carrier of a1 or for b1 being Element of the carrier of a1 holds
        b1 <> a2 . b1);
end;

:: TRANSGEO:dfs 11
definiens
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is translation
it is sufficient to prove
  thus a2 is dilatation(a1) &
     (a2 = id the carrier of a1 or for b1 being Element of the carrier of a1 holds
        b1 <> a2 . b1);

:: TRANSGEO:def 11
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is translation(b1)
   iff
      b2 is dilatation(b1) &
       (b2 = id the carrier of b1 or for b3 being Element of the carrier of b1 holds
          b3 <> b2 . b3);

:: TRANSGEO:prednot 9 => TRANSGEO:attr 7
notation
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  synonym a2 is_Tr for translation;
end;

:: TRANSGEO:th 99
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
   id the carrier of b1 is translation(b1);

:: TRANSGEO:th 100
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is dilatation(b1)
   holds    b2 is translation(b1)
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b3,b2 . b3 // b4,b2 . b4;

:: TRANSGEO:th 102
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is translation(b1) & b5 is translation(b1) & b4 . b2 = b5 . b2 & not LIN b2,b4 . b2,b3
   holds b4 . b3 = b5 . b3;

:: TRANSGEO:th 103
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b3 is translation(b1) & b4 is translation(b1) & b3 . b2 = b4 . b2
   holds b3 = b4;

:: TRANSGEO:th 104
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is translation(b1)
   holds b2 " is translation(b1);

:: TRANSGEO:th 105
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is translation(b1) & b3 is translation(b1)
   holds b2 * b3 is translation(b1);

:: TRANSGEO:attrnot 8 => TRANSGEO:attr 8
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  attr a2 is collineation means
    a2 is_automorphism_of the CONGR of a1;
end;

:: TRANSGEO:dfs 12
definiens
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is collineation
it is sufficient to prove
  thus a2 is_automorphism_of the CONGR of a1;

:: TRANSGEO:def 12
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is collineation(b1)
   iff
      b2 is_automorphism_of the CONGR of b1;

:: TRANSGEO:prednot 10 => TRANSGEO:attr 8
notation
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  synonym a2 is_Col for collineation;
end;

:: TRANSGEO:th 107
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 holds
      b2 is collineation(b1)
   iff
      for b3, b4, b5, b6 being Element of the carrier of b1 holds
         b3,b4 // b5,b6
      iff
         b2 . b3,b2 . b4 // b2 . b5,b2 . b6;

:: TRANSGEO:th 108
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b5 is collineation(b1)
   holds    LIN b2,b3,b4
   iff
      LIN b5 . b2,b5 . b3,b5 . b4;

:: TRANSGEO:th 109
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b2 is collineation(b1) & b3 is collineation(b1)
   holds b2 " is collineation(b1) & b2 * b3 is collineation(b1) & id the carrier of b1 is collineation(b1);

:: TRANSGEO:th 110
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
for b4 being Element of bool the carrier of b1
      st b2 in b4
   holds b3 . b2 in b3 .: b4;

:: TRANSGEO:th 111
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
for b4 being Element of bool the carrier of b1 holds
      b2 in b3 .: b4
   iff
      ex b5 being Element of the carrier of b1 st
         b5 in b4 & b3 . b5 = b2;

:: TRANSGEO:th 112
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of bool the carrier of b1
      st b2 .: b3 = b2 .: b4
   holds b3 = b4;

:: TRANSGEO:th 113
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is collineation(b1)
   holds b4 .: Line(b2,b3) = Line(b4 . b2,b4 . b3);

:: TRANSGEO:th 114
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
      st b2 is collineation(b1) & b3 is being_line(b1)
   holds b2 .: b3 is being_line(b1);

:: TRANSGEO:th 115
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of bool the carrier of b1
      st b2 is collineation(b1) & b3 // b4
   holds b2 .: b3 // b2 .: b4;

:: TRANSGEO:th 116
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 is being_line(b1)
           holds b2 .: b3 is being_line(b1)
   holds b2 is collineation(b1);

:: TRANSGEO:th 117
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is collineation(b1) &
         b2 is being_line(b1) &
         (for b5 being Element of the carrier of b1
               st b5 in b2
            holds b4 . b5 = b5) &
         not b3 in b2 &
         b4 . b3 = b3
   holds b4 = id the carrier of b1;