Article AFVECT0, MML version 4.99.1005

:: AFVECT0:attrnot 1 => AFVECT0:attr 1
definition
  let a1 be non empty AffinStruct;
  attr a1 is WeakAffVect-like means
    (for b1, b2, b3 being Element of the carrier of a1
           st b1,b2 // b3,b3
        holds b1 = b2) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b5,b6 & b3,b4 // b5,b6
        holds b1,b2 // b3,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) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b4,b5 & b1,b3 // b4,b6
        holds b2,b3 // b5,b6) &
     (for b1, b2 being Element of the carrier of a1 holds
     ex b3 being Element of the carrier of a1 st
        b1,b3 // b3,b2) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b1,b2 // b3,b4
        holds b1,b3 // b2,b4);
end;

:: AFVECT0:dfs 1
definiens
  let a1 be non empty AffinStruct;
To prove
     a1 is WeakAffVect-like
it is sufficient to prove
  thus (for b1, b2, b3 being Element of the carrier of a1
           st b1,b2 // b3,b3
        holds b1 = b2) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b5,b6 & b3,b4 // b5,b6
        holds b1,b2 // b3,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) &
     (for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
           st b1,b2 // b4,b5 & b1,b3 // b4,b6
        holds b2,b3 // b5,b6) &
     (for b1, b2 being Element of the carrier of a1 holds
     ex b3 being Element of the carrier of a1 st
        b1,b3 // b3,b2) &
     (for b1, b2, b3, b4 being Element of the carrier of a1
           st b1,b2 // b3,b4
        holds b1,b3 // b2,b4);

:: AFVECT0:def 1
theorem
for b1 being non empty AffinStruct holds
      b1 is WeakAffVect-like
   iff
      (for b2, b3, b4 being Element of the carrier of b1
             st b2,b3 // b4,b4
          holds b2 = b3) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
             st b2,b3 // b6,b7 & b4,b5 // b6,b7
          holds b2,b3 // b4,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) &
       (for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
             st b2,b3 // b5,b6 & b2,b4 // b5,b7
          holds b3,b4 // b6,b7) &
       (for b2, b3 being Element of the carrier of b1 holds
       ex b4 being Element of the carrier of b1 st
          b2,b4 // b4,b3) &
       (for b2, b3, b4, b5 being Element of the carrier of b1
             st b2,b3 // b4,b5
          holds b2,b4 // b3,b5);

:: AFVECT0:exreg 1
registration
  cluster non empty non trivial strict WeakAffVect-like AffinStruct;
end;

:: AFVECT0:modenot 1
definition
  mode WeakAffVect is non empty non trivial WeakAffVect-like AffinStruct;
end;

:: AFVECT0:condreg 1
registration
  cluster non empty AffVect-like -> WeakAffVect-like (AffinStruct);
end;

:: AFVECT0:th 2
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 // b2,b3;

:: AFVECT0:th 3
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2 being Element of the carrier of b1 holds
   b2,b2 // b2,b2;

:: AFVECT0:th 4
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b4,b5 // b2,b3;

:: AFVECT0:th 5
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3 // b2,b4
   holds b3 = b4;

:: AFVECT0:th 6
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2,b3 // b4,b5 & b2,b3 // b4,b6
   holds b5 = b6;

:: AFVECT0:th 7
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b2 // b3,b3;

:: AFVECT0:th 8
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b3,b2 // b5,b4;

:: AFVECT0:th 9
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st b2,b3 // b4,b5 & b2,b4 // b6,b5
   holds b3 = b6;

:: AFVECT0:th 10
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st b2,b3 // b4,b5 & b6,b7 // b2,b3 & b6,b8 // b4,b5
   holds b7 = b8;

:: AFVECT0:th 11
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st b2,b3 // b4,b5 & b6,b7 // b3,b2 & b6,b8 // b5,b4
   holds b7 = b8;

:: AFVECT0:th 12
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st b2,b3 // b4,b5 & b6,b7 // b8,b9 & b3,b10 // b6,b7 & b5,b11 // b8,b9
   holds b2,b10 // b4,b11;

:: AFVECT0:th 13
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2,b3 // b4,b5 & b2,b6 // b7,b5
   holds b3,b6 // b7,b4;

:: AFVECT0:prednot 1 => AFVECT0:pred 1
definition
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  pred MDist A2,A3 means
    a2,a3 // a3,a2 & a2 <> a3;
  symmetry;
::  for a1 being non empty non trivial WeakAffVect-like AffinStruct
::  for a2, a3 being Element of the carrier of a1
::        st MDist a2,a3
::     holds MDist a3,a2;
  irreflexivity;
::  for a1 being non empty non trivial WeakAffVect-like AffinStruct
::  for a2 being Element of the carrier of a1 holds
::     not MDist a2,a2;
end;

:: AFVECT0:dfs 2
definiens
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
To prove
     MDist a2,a3
it is sufficient to prove
  thus a2,a3 // a3,a2 & a2 <> a3;

:: AFVECT0:def 2
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
   MDist b2,b3
iff
   b2,b3 // b3,b2 & b2 <> b3;

:: AFVECT0:th 16
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct holds
   ex b2, b3 being Element of the carrier of b1 st
      b2 <> b3 & not MDist b2,b3;

:: AFVECT0:th 18
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st MDist b2,b3 & MDist b2,b4 & b3 <> b4
   holds MDist b3,b4;

:: AFVECT0:th 19
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st MDist b2,b3 & b2,b3 // b4,b5
   holds MDist b4,b5;

:: AFVECT0:prednot 2 => AFVECT0:pred 2
definition
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  pred Mid A2,A3,A4 means
    a2,a3 // a3,a4;
end;

:: AFVECT0:dfs 3
definiens
  let a1 be non empty non trivial WeakAffVect-like 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;

:: AFVECT0:def 3
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   Mid b2,b3,b4
iff
   b2,b3 // b3,b4;

:: AFVECT0:th 21
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st Mid b2,b3,b4
   holds Mid b4,b3,b2;

:: AFVECT0:th 22
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
   Mid b2,b3,b3
iff
   b2 = b3;

:: AFVECT0:th 23
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
   Mid b2,b3,b2
iff
   (b2 = b3 or MDist b2,b3);

:: AFVECT0:th 24
theorem
for b1 being non empty non trivial WeakAffVect-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,b4,b3;

:: AFVECT0:th 25
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st Mid b2,b3,b4 & Mid b2,b5,b4 & b3 <> b5
   holds MDist b3,b5;

:: AFVECT0:th 26
theorem
for b1 being non empty non trivial WeakAffVect-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;

:: AFVECT0:th 27
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st Mid b2,b3,b4 & Mid b2,b3,b5
   holds b4 = b5;

:: AFVECT0:th 28
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st Mid b2,b3,b4 & MDist b3,b5
   holds Mid b2,b5,b4;

:: AFVECT0:th 29
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st Mid b2,b3,b4 & Mid b2,b5,b6 & MDist b3,b5
   holds b4 = b6;

:: AFVECT0:th 30
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st Mid b2,b3,b4 & Mid b5,b3,b6
   holds b2,b5 // b6,b4;

:: AFVECT0:th 31
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st Mid b2,b3,b4 & Mid b5,b6,b7 & MDist b3,b6
   holds b2,b5 // b7,b4;

:: AFVECT0:funcnot 1 => AFVECT0:func 1
definition
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  func PSym(A2,A3) -> Element of the carrier of a1 means
    Mid a3,a2,it;
end;

:: AFVECT0:def 4
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   b4 = PSym(b2,b3)
iff
   Mid b3,b2,b4;

:: AFVECT0:th 33
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   PSym(b2,b3) = b4
iff
   b3,b2 // b2,b4;

:: AFVECT0:th 35
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
   PSym(b2,b3) = b3
iff
   (b3 = b2 or MDist b3,b2);

:: AFVECT0:th 36
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
PSym(b2,PSym(b2,b3)) = b3;

:: AFVECT0:th 37
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
      st PSym(b2,b3) = PSym(b2,b4)
   holds b3 = b4;

:: AFVECT0:th 38
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
   PSym(b2,b4) = b3;

:: AFVECT0:th 39
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 // PSym(b4,b3),PSym(b4,b2);

:: AFVECT0:th 40
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
   b2,b3 // b4,b5
iff
   PSym(b6,b2),PSym(b6,b3) // PSym(b6,b4),PSym(b6,b5);

:: AFVECT0:th 41
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   MDist b2,b3
iff
   MDist PSym(b4,b2),PSym(b4,b3);

:: AFVECT0:th 42
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   Mid b2,b3,b4
iff
   Mid PSym(b5,b2),PSym(b5,b3),PSym(b5,b4);

:: AFVECT0:th 43
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   PSym(b2,b3) = PSym(b4,b3)
iff
   (b2 = b4 or MDist b2,b4);

:: AFVECT0:th 44
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
PSym(b2,PSym(b3,PSym(b2,b4))) = PSym(PSym(b2,b3),b4);

:: AFVECT0:th 45
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
   PSym(b2,PSym(b3,b4)) = PSym(b3,PSym(b2,b4))
iff
   (b2 <> b3 & not MDist b2,b3 implies MDist b3,PSym(b2,b3));

:: AFVECT0:th 46
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
PSym(b2,PSym(b3,PSym(b4,b5))) = PSym(b4,PSym(b3,PSym(b2,b5)));

:: AFVECT0:th 47
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
ex b6 being Element of the carrier of b1 st
   PSym(b2,PSym(b3,PSym(b4,b5))) = PSym(b6,b5);

:: AFVECT0:th 48
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
   PSym(b2,PSym(b5,b3)) = PSym(b5,PSym(b4,b3));

:: AFVECT0:funcnot 2 => AFVECT0:func 2
definition
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  func Padd(A2,A3,A4) -> Element of the carrier of a1 means
    a2,a3 // a4,it;
end;

:: AFVECT0:def 5
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b5 = Padd(b2,b3,b4)
iff
   b2,b3 // b4,b5;

:: AFVECT0:funcnot 3 => AFVECT0:func 1
notation
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2, a3 be Element of the carrier of a1;
  synonym Pcom(a2,a3) for PSym(a2,a3);
end;

:: AFVECT0:funcnot 4 => AFVECT0:func 3
definition
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2 be Element of the carrier of a1;
  func Padd A2 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of a1 means
    for b1, b2 being Element of the carrier of a1 holds
    it .(b1,b2) = Padd(a2,b1,b2);
end;

:: AFVECT0:def 7
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of b1 holds
      b3 = Padd b2
   iff
      for b4, b5 being Element of the carrier of b1 holds
      b3 .(b4,b5) = Padd(b2,b4,b5);

:: AFVECT0:funcnot 5 => AFVECT0:func 4
definition
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2 be Element of the carrier of a1;
  func Pcom A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = PSym(a2,b1);
end;

:: AFVECT0:def 8
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b3 = Pcom b2
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = PSym(b2,b4);

:: AFVECT0:funcnot 6 => AFVECT0:func 5
definition
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2 be Element of the carrier of a1;
  func GroupVect(A1,A2) -> strict addLoopStr equals
    addLoopStr(#the carrier of a1,Padd a2,a2#);
end;

:: AFVECT0:def 9
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2 being Element of the carrier of b1 holds
   GroupVect(b1,b2) = addLoopStr(#the carrier of b1,Padd b2,b2#);

:: AFVECT0:funcreg 1
registration
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2 be Element of the carrier of a1;
  cluster GroupVect(a1,a2) -> non empty strict;
end;

:: AFVECT0:th 55
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2 being Element of the carrier of b1 holds
   the carrier of GroupVect(b1,b2) = the carrier of b1 & the addF of GroupVect(b1,b2) = Padd b2 & 0. GroupVect(b1,b2) = b2;

:: AFVECT0:th 57
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of the carrier of GroupVect(b1,b2)
for b5, b6 being Element of the carrier of b1
      st b3 = b5 & b4 = b6
   holds b3 + b4 = (Padd b2) .(b5,b6);

:: AFVECT0:funcreg 2
registration
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2 be Element of the carrier of a1;
  cluster GroupVect(a1,a2) -> strict right_complementable Abelian add-associative right_zeroed;
end;

:: AFVECT0:th 58
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of the carrier of GroupVect(b1,b2)
for b4 being Element of the carrier of b1
      st b3 = b4
   holds - b3 = (Pcom b2) . b4;

:: AFVECT0:th 59
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2 being Element of the carrier of b1 holds
   0. GroupVect(b1,b2) = b2;

:: AFVECT0:th 66
theorem
for b1 being non empty non trivial WeakAffVect-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of the carrier of GroupVect(b1,b2) holds
   ex b4 being Element of the carrier of GroupVect(b1,b2) st
      b4 + b4 = b3;

:: AFVECT0:funcreg 3
registration
  let a1 be non empty non trivial WeakAffVect-like AffinStruct;
  let a2 be Element of the carrier of a1;
  cluster GroupVect(a1,a2) -> strict Two_Divisible;
end;

:: AFVECT0:th 67
theorem
for b1 being non empty non trivial AffVect-like AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of the carrier of GroupVect(b1,b2)
      st b3 + b3 = 0. GroupVect(b1,b2)
   holds b3 = 0. GroupVect(b1,b2);

:: AFVECT0:funcreg 4
registration
  let a1 be non empty non trivial AffVect-like AffinStruct;
  let a2 be Element of the carrier of a1;
  cluster GroupVect(a1,a2) -> strict Fanoian;
end;

:: AFVECT0:exreg 2
registration
  cluster non empty non trivial strict right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr;
end;

:: AFVECT0:modenot 2
definition
  mode Proper_Uniquely_Two_Divisible_Group is non empty non trivial right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr;
end;

:: AFVECT0:th 69
theorem
for b1 being non empty non trivial AffVect-like AffinStruct
for b2 being Element of the carrier of b1 holds
   GroupVect(b1,b2) is non empty non trivial right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr;

:: AFVECT0:funcreg 5
registration
  let a1 be non empty non trivial AffVect-like AffinStruct;
  let a2 be Element of the carrier of a1;
  cluster GroupVect(a1,a2) -> non trivial strict;
end;

:: AFVECT0:th 70
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr holds
   AV b1 is non empty non trivial AffVect-like AffinStruct;

:: AFVECT0:funcreg 6
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr;
  cluster AV a1 -> non trivial strict AffVect-like;
end;

:: AFVECT0:th 71
theorem
for b1 being non empty non trivial strict AffVect-like AffinStruct
for b2 being Element of the carrier of b1 holds
   b1 = AV GroupVect(b1,b2);

:: AFVECT0:th 72
theorem
for b1 being strict AffinStruct holds
      b1 is non empty non trivial AffVect-like AffinStruct
   iff
      ex b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr st
         b1 = AV b2;

:: AFVECT0:prednot 3 => AFVECT0:pred 3
definition
  let a1, a2 be non empty addLoopStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  pred A3 is_Iso_of A1,A2 means
    a3 is one-to-one &
     proj2 a3 = the carrier of a2 &
     (for b1, b2 being Element of the carrier of a1 holds
     a3 . (b1 + b2) = (a3 . b1) + (a3 . b2) &
      a3 . 0. a1 = 0. a2 &
      a3 . - b1 = - (a3 . b1));
end;

:: AFVECT0:dfs 9
definiens
  let a1, a2 be non empty addLoopStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is_Iso_of a1,a2
it is sufficient to prove
  thus a3 is one-to-one &
     proj2 a3 = the carrier of a2 &
     (for b1, b2 being Element of the carrier of a1 holds
     a3 . (b1 + b2) = (a3 . b1) + (a3 . b2) &
      a3 . 0. a1 = 0. a2 &
      a3 . - b1 = - (a3 . b1));

:: AFVECT0:def 10
theorem
for b1, b2 being non empty addLoopStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is_Iso_of b1,b2
   iff
      b3 is one-to-one &
       proj2 b3 = the carrier of b2 &
       (for b4, b5 being Element of the carrier of b1 holds
       b3 . (b4 + b5) = (b3 . b4) + (b3 . b5) &
        b3 . 0. b1 = 0. b2 &
        b3 . - b4 = - (b3 . b4));

:: AFVECT0:prednot 4 => AFVECT0:pred 4
definition
  let a1, a2 be non empty addLoopStr;
  pred A1,A2 are_Iso means
    ex b1 being Function-like quasi_total Relation of the carrier of a1,the carrier of a2 st
       b1 is_Iso_of a1,a2;
end;

:: AFVECT0:dfs 10
definiens
  let a1, a2 be non empty addLoopStr;
To prove
     a1,a2 are_Iso
it is sufficient to prove
  thus ex b1 being Function-like quasi_total Relation of the carrier of a1,the carrier of a2 st
       b1 is_Iso_of a1,a2;

:: AFVECT0:def 11
theorem
for b1, b2 being non empty addLoopStr holds
   b1,b2 are_Iso
iff
   ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 st
      b3 is_Iso_of b1,b2;

:: AFVECT0:th 75
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of AV b1
   st (for b5 being Element of the carrier of b1 holds
         b2 . b5 = b3 + b5) &
      b4 = b3
for b5, b6 being Element of the carrier of b1 holds
b2 . (b5 + b6) = (Padd b4) .(b2 . b5,b2 . b6) &
 b2 . 0. b1 = 0. GroupVect(AV b1,b4) &
 b2 . - b5 = (Pcom b4) . (b2 . b5);

:: AFVECT0:th 76
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
      st for b4 being Element of the carrier of b1 holds
           b2 . b4 = b3 + b4
   holds b2 is one-to-one;

:: AFVECT0:th 77
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of AV b1
      st for b5 being Element of the carrier of b1 holds
           b2 . b5 = b3 + b5
   holds proj2 b2 = the carrier of GroupVect(AV b1,b4);

:: AFVECT0:th 78
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed Fanoian Two_Divisible addLoopStr
for b2 being Element of the carrier of b1
for b3 being Element of the carrier of AV b1
      st b3 = b2
   holds b1,GroupVect(AV b1,b3) are_Iso;