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

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

Let $z_1 z_2 = 0$.

, let $\tuple {x_2, y_2} \ne \tuple {0, 0}$.

Suppose also that $\tuple {x_1, y_1} \ne \tuple {0, 0}$.

Then:

From this contradiction it follows that:
 * $\tuple {x_1, y_1} = \tuple {0, 0}$