theorem Th4:
  o is commutative implies (e is_a_unity_wrt o iff for a holds o.( e,a) = a)
proof
  assume
A1: o is commutative;
  now
    thus (for a holds o.(e,a) = a & o.(a,e) = a) implies for a holds o.(e,a) =
    a;
    assume
A2: for a holds o.(e,a) = a;
    let a;
    thus o.(e,a) = a by A2;
    thus o.(a,e) = o.(e,a) by A1
      .= a by A2;
  end;
  hence thesis by Th3;
end;
