reserve A for set;
reserve X,Y,Z for set,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve u for UnOp of A,
  o,o9 for BinOp of A,
  a,b,c,e,e1,e2 for Element of A;

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;
