Completely Multiplicative Function is Multiplicative

Theorem
Let $f: \Z \to \Z$ be a function on the integers $\Z$.

Let $f$ be completely multiplicative.

Then $f$ is multiplicative.

Proof
By definition of complete multiplicativity:


 * $\forall m, n \in \Z: \map f {m n} = \map f m \map f n$

Hence by True Statement is implied by Every Statement:


 * $\forall m, n \in \Z: m \perp n \implies \map f {m n} = \map f m \map f n$

So $f$ is multiplicative.