Theorem 4.6 (The Trichotomy Law for Natural Numbers): If m and n are two natural numbers then either

or

Proof: Let m and n be two natural numbers. If

then m < n or m = n by Definition 4.6. So we can suppose that in the process of repeatedly applying the successor function to get to n, we haven't yet encountered m. So to get to m, we would have to keep going and in that case, n < m

.