Absolute Value Function is Completely Multiplicative/Proof 1

Proof
Let either $x = 0$ or $y = 0$, or both.

We have that $\size 0 = 0$ by definition of absolute value.

Hence:
 * $\size x \size y = 0 = x y = \size {x y}$

Let $x > 0$ and $y > 0$.

Then:

and:

Let $x < 0$ and $y < 0$.

Then:

and:

The final case is where one of $x$ and $y$ is positive, and one is negative.

, let $x < 0$ and $y > 0$.

Then:

and:

The case where $x > 0$ and $y < 0$ is the same.