## Theorem

Each element $x$ of the set of integers $\Z$ has an inverse element $-x$ under the operation of integer addition:

$\forall x \in \Z: \exists -x \in \Z: x + \paren {-x} = 0 = \paren {-x} + x$

## Proof

Let us define $\eqclass {\tuple {a, b} } \boxtimes$ as in the formal definition of integers.

That is, $\eqclass {\tuple {a, b} } \boxtimes$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxtimes$.

$\boxtimes$ is the congruence relation defined on $\N \times \N$ by:

$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$

In order to streamline the notation, we will use $\eqclass {a, b} {}$ to mean $\eqclass {\tuple {a, b} } \boxtimes$, as suggested.

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

Thus:

 $\displaystyle \eqclass {a, a + x} {} + \eqclass {a + x, a} {}$ $=$ $\displaystyle \eqclass {a + a + x, a + x + a} {}$ $\displaystyle$ $=$ $\displaystyle \eqclass {a, a} {}$ Construction of Inverse Completion: Members of Equivalence Classes $\displaystyle$ $=$ $\displaystyle \eqclass {a + x + a , a + a + x} {}$ $\displaystyle$ $=$ $\displaystyle \eqclass {a + x, a} {} + \eqclass {a, a + x} {}$

So $\eqclass {a, a + x} {}$ has the inverse $\eqclass {a + x, a} {}$.

$\blacksquare$