reserve AFV for WeakAffVect;
reserve a,b,c,d,e,f,a9,b9,c9,d9,f9,p,q,r,o,x99 for Element of AFV;
reserve a,b,c for Element of GroupVect(AFV,o);
reserve a,b for Element of GroupVect(AFV,o);
reserve AFV for AffVect,
  o for Element of AFV;
reserve ADG for Proper_Uniquely_Two_Divisible_Group;
reserve f for Function of the carrier of ADG,the carrier of ADG;

theorem Th54:
  for o9 being Element of ADG, o being Element of AV(ADG) st (for
  b being Element of ADG holds f.b = o9+b) holds rng(f) = the carrier of
  GroupVect(AV(ADG),o)
proof
  set X = the carrier of ADG;
A1: X = dom(f) by FUNCT_2:def 1;
  let o9 be Element of ADG, o be Element of AV(ADG) such that
A2: for b being Element of ADG holds f.b = o9+b;
  now
    let y be object;
    assume y in X;
    then reconsider y9=y as Element of X;
    set x9=y9-o9;
    f.x9 = o9+((-o9)+y9) by A2
      .= (o9+(-o9))+y9 by RLVECT_1:def 3
      .= y9+(0.ADG) by RLVECT_1:5
      .= y by RLVECT_1:4;
    hence y in rng(f) by A1,FUNCT_1:def 3;
  end;
  then
A3: X c= rng(f) by TARSKI:def 3;
  rng(f) c= X & X = the carrier of GroupVect(AV(ADG),o) by RELAT_1:def 19
,TDGROUP:4;
  hence thesis by A3,XBOOLE_0:def 10;
end;
