Product of Multiplicative Functions is Multiplicative

Theorem
Let $f: \Z \to \Z$ and $g: \Z \to \Z$ be multiplicative functions.

Then their pointwise product:
 * $f \times g: \Z \to \Z: \forall s \in S: \left({f \times g}\right) \left({s}\right) := f \left({s}\right) \times g \left({s}\right)$

is also multiplicative.

Proof
Let $f$ and $g$ be multiplicative.

Let $m \perp n$.

Then:

Hence the result by definition of multiplicative function.