Unity Function is Completely Multiplicative

Theorem
Let $f_1: \Z_{> 0} \to \Z_{> 0}$ be the constant function:
 * $\forall n \in \Z_{> 0}: f_1 \left({n}\right) = 1$

Then $f_1$ is completely multiplicative.

Proof

 * $\forall m, n \in \Z_{> 0}: f_1 \left({m n}\right) = 1 = f_1 \left({m}\right) f_1 \left({n}\right)$