I mentioned Peano axioms as the foundation on which these operations are defined and their properties established. Let's see how it can be done. This says that the set N is not empty. There is at least one natural number. For every x in N there exists a number x' known as the successor of x.
Different numbers have different successors.
In other words, M contains all natural numbers. It's not my purpose to pursue this subject in every detail. A definition of subtraction can be found on-line in An Interactive Real Analysis book. Theorem 2 x' x. According to Axioms 1 and 3, 1' 1.