I am not really sure what you mean by "addition laws". It happens to be really easy to prove this by using the definition of the Riemann integral. For any subinterval (in the partition) there exists a rational and an irrational point. If for any partition we choose only the irrational points then the Riemann sum has value . If for any partition we choose only the rational points then the Riemann sum has value . The problem is that and cannot be made arbitrary close to one another with a fine enough parition (since it is independent of ).