Real Part of Integer Power of Complex Number is Harmonic

Theorem
Let $z \in \C$ be a complex number.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $z^n$ denote $z$ raised to the $n$th power.

Then the real part $\map \Re {z^n}$ of $z^n$ is a harmonic polynomial.

Proof
Let $z = x + i y$.

Then: