Cardinality is Additive Function

Theorem
Let $$S$$ be a finite set.

Let $$\mathcal P \left({S}\right)$$ be the power set of $$S$$.

The function $$C: \mathcal P \left({S}\right) \to \R$$, where $$C$$ is defined as the cardinality of a set, is an additive function.

Proof
We have that $\mathcal P \left({S}\right)$ is an algebra of sets.