Integer Addition is Cancellable

Theorem
The operation of addition on the set of integers $\Z$ is cancellable:


 * $\forall x, y, z \in \Z: x + z = y + z \implies x = y$

Proof
Let $x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$ for some $x, y, z\in \Z$.

Then: