theorem
  for ADG being Proper_Uniquely_Two_Divisible_Group, o9 being Element of
ADG, o being Element of AV(ADG) st o=o9 holds ADG,GroupVect(AV(ADG),o) are_Iso
proof
  let ADG be Proper_Uniquely_Two_Divisible_Group, o9 be Element of ADG, o be
  Element of AV(ADG) such that
A1: o=o9;
  set AS = AV(ADG);
  set X = the carrier of ADG,Z=GroupVect(AS,o);
  set T = the carrier of GroupVect(AS,o);
  deffunc F(Element of X) = o9+$1;
  consider g being UnOp of X such that
A2: for a being Element of X holds g.a = F(a) from FUNCT_2:sch 4;
  X = T by TDGROUP:4;
  then reconsider f = g as Function of X,T;
A3: now
    let a,b be Element of ADG;
    reconsider fa = f.a as Element of AV(ADG);
    thus f.(a+b) = (f.a)+(f.b) by A1,A2,Th52;
    thus f.(0.ADG) = 0.Z by A1,A2,Th52;
    thus f.(-a) = (Pcom(o)).fa by A1,A2,Th52
      .= -(f.a) by Th44;
  end;
  f is one-to-one & rng(f) = T by A2,Th53,Th54;
  then f is_Iso_of ADG,Z by A3;
  hence thesis;
end;
