Non-Zero Complex Numbers are Closed under Multiplication/Proof 2

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

Proof
Let $z_1, z_2 \in \C_{\ne 0}$.

Then:
 * $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$

for some $x_1, y_1, x_2, y_2 \in \R$ such that:
 * $x_1 \ne 0$ or $y_1 \ne 0$
 * $x_2 \ne 0$ or $y_2 \ne 0$

Then:
 * $z = r_1 e^{i \theta_1}$ and $w = r_2 e^{i \theta_2}$

for some $r_1, r_2, \theta_1, \theta_2 \in \R$.

and:
 * $z_1 \times z_2 = \left({r_1 \times r_2}\right) e^{i \left({\theta_1 + \theta_2}\right)}$

By definition:
 * $r_1 = \sqrt {x_1^2 + y_1^2}$
 * $r_2 = \sqrt {x_2^2 + y_2^2}$

Thus $r_1 > 0$ and $r_2 > 0$.

Hence $r_1 \times r_2 > 0$ and so $z_1 \times z_2 \ne 0$.