# Non-Zero Integers are Cancellable for Multiplication/Proof 2

## 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 $y, z \in \Z: y \ne z$.

 $\displaystyle y$ $\ne$ $\displaystyle z$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle y - z$ $\ne$ $\displaystyle 0$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x \paren {y - z}$ $\ne$ $\displaystyle 0$ Ring of Integers has no Zero Divisors‎ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x y - x z$ $\ne$ $\displaystyle 0$ Integer Multiplication Distributes over Subtraction

The result follows by transposition.

$\blacksquare$