Complex Modulus of Product of Complex Numbers/Proof 2

Theorem
Let $z_1, z_2 \in \C$ be complex numbers.

Let $\left\vert{z}\right\vert$ be the modulus of $z$.

Then:
 * $\left\vert{z_1 z_2}\right\vert = \left\vert{z_1}\right\vert \cdot \left\vert{z_2}\right\vert$

Proof
Let $\overline z$ denote the complex conjugate of $z$.

Then: