theorem
  G is Abelian addGroup iff for A,g st A <> {} holds g * A = {g}
proof
  thus G is Abelian addGroup implies for A,g st A <> {} holds
  g * A = {g} by
Th55;
  assume
A1: for A,g st A <> {} holds g * A = {g};
  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;
