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:


 * $\forall m, n \in \Z: f \left({m n}\right) = f \left({m}\right) f \left({n}\right)$

Hence by True Statement is implied by Every Statement:


 * $\forall m, n \in \Z: m \perp n \implies f \left({m n}\right) = f \left({m}\right) f \left({n}\right)$

So $f$ is multiplicative.