# Non-Zero Integers are Cancellable for Multiplication/Proof 1

## 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, $x \in \N_{> 0}$.

$\Box$

Let $x < 0$.

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

Then $-x > 0$ and so:

 $\displaystyle \paren {-x} y$ $=$ $\displaystyle -\paren {x y}$ Product with Ring Negative $\displaystyle$ $=$ $\displaystyle -\paren {x z}$ $\struct {\Z, +}$ is a group: Group Axiom $\text G 3$: Existence of Inverse Element $\displaystyle$ $=$ $\displaystyle \paren {-x} z$ Product with Ring Negative $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle z$ from above: case where $x > 0$

$\Box$

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

$\blacksquare$