Summation of Zero/Set

Theorem
Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $S$ be a set.

Let $0: S \to \mathbb A$ be the zero mapping.

Then the summation with finite support of $0$ over $S$ equals zero:
 * $\displaystyle \sum_{s \mathop \in S} \map 0 s = 0$

Proof
By Support of Zero Mapping, the support of $0$ is empty.

By Empty Set is Finite, the support of $0$ is indeed finite.

By Summation over Empty Set, $\displaystyle \sum_{s \mathop \in S} \map 0 s = \sum_{s \mathop \in \O} \map 0 s = 0$