theorem
  G is Abelian addGroup iff for A,g holds A * g = A
proof
  thus G is Abelian addGroup implies for A,g holds A * g = A by Th55;
  assume
A1: for A,g holds A * g = A;
  now
    let a,b;
    {a} = {a} * b by A1
      .= {a * b} by ThB37;
    hence a = a * b by ZFMISC_1:3;
  end;
  hence thesis by ThB30;
end;
