
theorem for a,b be Integer holds
  parity (a+b) = (parity a) + (parity b) - 2*(parity a)*(parity b) &
  parity a - parity b = parity (a+b) - 2*(parity (a+b))*(parity b) &
  parity a - parity b = 2*(parity a)*parity (a+b) - parity (a+b)
  proof
    let a,b be Integer;
    per cases;
    suppose a is even & b is even; then
      parity a = 0 & parity b = 0 & parity (a+b) = 0;
      hence thesis;
    end;
    suppose
      a is odd & b is odd; then
      parity a = 1 & parity b = 1 & parity (a+b) = 0;
      hence thesis;
    end;
    suppose
      A1: a is even & b is odd; then
      a + b is odd;
      hence thesis by A1;
    end;
    suppose
      A1: a is odd & b is even; then
      a + b is odd;
      hence thesis by A1;
    end;
  end;
