Non-Zero Integers are Cancellable for Multiplication

Theorem
Every non-zero element of $$\mathbb{Z}$$ is cancellable for multiplication. That is:

$$\forall x, y, z \in \mathbb{Z}, x \ne 0: x y = x z \iff y = z$$

Proof

 * Let $$x > 0$$.

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

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 Negative Product.

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

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