Quotient of Complex Conjugates

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

Let $\overline z$ be the complex conjugate of the complex number $z$.

Then:
 * $\overline {\paren {\dfrac {z_1} {z_2} } } = \dfrac {\paren {\overline {z_1} } } {\paren {\overline {z_2} } }$

for $z_2 \ne 0$.

Proof
Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.

Then: