Non-Zero Integers are Cancellable for Multiplication

Theorem
Every non-zero element of $\Z$ is cancellable for multiplication.

That is:


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

Proof
Let $x > 0$.

From Natural Numbers are Non-Negative Integers, $x \in \N^*$.

By the Extension Theorem for Distributive Operations and Ordering on Naturally Ordered Semigroup Product, $z$ is cancellable for multiplication.

Let $x < 0$ and let $x y = x z$.

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

Then $-x > 0$ and $\left({-x}\right) y = - \left({x y}\right) = - \left({x z}\right) = \left({- x}\right) z$ from Product with Ring Negative.

Thus from what we have proved, $y = z$.

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