Argument of x to the n Equals n Times The Argument

Theorem
Let $z$ be a complex number.

For all $n \in \N_{>0}$, $\arg(z^n) = n\arg(z)$.

Proof
For $n = 1$,
 * $\displaystyle \arg(z^1) = 1\cdot\arg(z)$

Assuming the result is true for $n=k$, we have