Inverse for Integer Addition

Theorem
Each element $$x$$ of the set of integers 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 $$\left[\!\left[{a + x, a}\right]\!\right]$$.