Non-Zero Complex Numbers are Closed under Multiplication/Proof 3

Theorem
The set of non-zero complex numbers is closed under multiplication.

Proof
Equivalently this is to say:
 * $z_1 z_2 = 0 \implies z_1 = 0 \lor z_2 = 0$

Let $z_1 z_2 = 0$.

WLOG, let $\left({x_2, y_2}\right) \ne \left({0, 0}\right)$.

Suppose also that $\left({x_1, y_1}\right) \ne \left({0, 0}\right)$.

Then:

From this contradiction it follows that:
 * $\left({x_1, y_1}\right) = \left({0, 0}\right)$