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:
 * $\displaystyle \sum_{s \mathop \in S} 0(s) = 0$

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, $\displaystyle \sum_{s \mathop \in S} 0(s) = \sum_{s \mathop \in \varnothing} 0(s) = 0$