theorem Th3:
  e is_a_unity_wrt o iff for a holds o.(e,a) = a & o.(a,e) = a
proof
  thus e is_a_unity_wrt o implies for a holds o.(e,a) = a & o.(a,e) = a
  proof
    assume e is_a_left_unity_wrt o & e is_a_right_unity_wrt o;
    hence thesis;
  end;
  assume for a holds o.(e,a) = a & o.(a,e) = a;
  hence (for a holds o.(e,a) = a) & for a holds o.(a,e) = a;
end;
