
theorem Th4:
  for x,y being Element of REAL+ holds x - y in REAL
proof
  let x,y be Element of REAL+;
  per cases;
  suppose
    y <=' x;
    then x - y = x -' y by ARYTM_1:def 2;
    then x - y in REAL+;
    hence thesis by Th1;
  end;
  suppose
A1: not y <=' x;
    then x - y = [{},y -' x] by ARYTM_1:def 2;
    hence thesis by A1,Th2,ARYTM_1:9;
  end;
end;
