Power Function is Completely Multiplicative/Integers

Theorem
Let $c \in \Z$ be an integer.

Let $f_c: \Z \to \Z$ be the mapping defined as:
 * $\forall n \in \Z: \map {f_c} n = n^c$

Then $f_c$ is completely multiplicative.

Proof
Let $r, s \in \Z$.

Then: