theorem
  o is commutative implies (e is_a_left_unity_wrt o iff e
  is_a_right_unity_wrt o)
proof
  assume
A1: o is commutative;
  then e is_a_unity_wrt o iff e is_a_left_unity_wrt o by Th6;
  hence thesis by A1,Th7;
end;
