MY ATTEMPT
Given two integers $a,b$, there is just one possibility among three: $a - b$ is zero; $a - b$ is a positive natural number $n$; or $a - b$ is the negation of a positive natural number $n$. In the first case, one has $a = b$; in the second case, one has that $a - b = n > 0$, that is to say, $a > b$; in the third case, $a - b = -n$, that is to say, $b = a + n$, thus $b > a$.
Could someone verify if I am reasoning rightly?
EDIT
The main result on which I am basing my answer is:
Trichotomy of integers
Let $x$ be an integer. Then exactly one of the following three statements is true: (a) $x$ is zero; (b) $x$ is equal to a positive natural number $n$; or (c) $x$ is the negation $-n$ of a positive natural number $n$.