Non-Zero Integers are Cancellable for Multiplication/Proof 3

Theorem
Every non-zero integer is cancellable for multiplication.

That is:
 * $\forall x, y, z \in \Z, x \ne 0: x y = x z \iff y = z$

Proof
Let $x y = x z$.

There are two cases to investigate: $x > 0$ and $x < 0$.

Let $x > 0$.

From Natural Numbers are Non-Negative Integers, $x \in \N_{> 0}$.

By the Extension Theorem for Distributive Operations and Ordering on Naturally Ordered Semigroup Product, $x$ is cancellable for multiplication. {{qed|lemma}

Let $x < 0$ and

We know that the Integers form Integral Domain and are thus a ring.

Then $-x > 0$ and so:

So whatever non-zero value $x$ takes, it is cancellable for multiplication.