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.