Peano's axioms include: 0 is the first (that is, the lowest) natural number; the properties of equality. Also, the "successor" function S. S(n) means the natural number immediately following n, so n+1=S(n), implying S(n)+1=S(S(n)). This defines addition. n+0=n (0 is the additive identity) and S(0)=1, and there is no natural number x such that S(x)=0. The set of natural numbers ℕ is generated by recursive application of S; for example, 2=S(S(0)), 3=S(S(S(0))), etc. Natural number n would be n recursions of S.
Since n+1 is the same as n+S(0), then n+S(0)=S(n)=S(n+0), so, by induction, n+S(a)=S(n+a). n+a is defined as the sum of the two natural numbers n and a.
Multiplication is recursive addition. a×0=0, a×S(0)=a, which can be written a×1=a. Also, a×S(b)=a×(b+1)=a×b+a×1=a×b+a.
n=qm+r=q×m+r implies that the product q×m must be a natural number, since n and r are natural numbers, and the sum of two natural numbers is another natural number. Also S(q)×m=q×m+m and n=q×m+m+r. So q's successor S(q) (which we can call q') gives us n=q'×m+r. This implies that there are two natural numbers (one represented by q' and the other by r). By induction there could be a series of m's created by recursively applying the successor function S(S(S(...q), but ultimately we arrive at a natural number Q, such that n=Q×m+r, where 0≤r<m. That is, whenever r exceeds m, we simply apply S(q) until the inequality is satisfied. [For q×m to be a natural number we can eventually deduce that positive m=p/q or p=q×m, where p and q are natural numbers and m is confined to positive rational real numbers.]