Inverse for Integer Addition

Theorem
Each element $x$ of the set of integers $\Z$ has an inverse element $-x$ under the operation of integer addition:
 * $\forall x \in \Z: \exists -x \in \Z: x + \left({-x}\right) = 0 = \left({-x}\right) + x$

Proof
Let us define $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ as in the formal definition of integers.

That is, $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.

$\boxminus$ is the congruence relation defined on $\N \times \N$ by $\left({x_1, y_1}\right) \boxminus \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$.

In order to streamline the notation, we will use $\left[\!\left[{a, b}\right]\!\right]$ to mean $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$, as suggested.

From the method of construction, the element $\left[\!\left[{a, a + x}\right]\!\right]$ has an inverse $\left[\!\left[{a + x, a}\right]\!\right]$ where $a$ and $x$ are elements of the natural numbers $\N$.

Thus:

So $\left[\!\left[{a, a + x}\right]\!\right]$ has the inverse $\left[\!\left[{a + x, a}\right]\!\right]$.