Argument of x to the n Equals n Times The Argument

Theorem
Let $z$ be a complex number.

Then:
 * $\forall n \in \N_{>0}: \map \arg {z^n} = n \map \arg z$

Proof
For $n = 1$


 * $\map \arg {z^1} = 1 \cdot \map \arg z$

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