Summation of Zero/Finite Set

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

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

Then the summation of $0$ over $S$ equals zero:
 * $\displaystyle \sum_{s \mathop \in S} 0 \left({s}\right) = 0$

Proof
At least three proofs are possible:
 * Using the definition of summation and Indexed Summation of Zero
 * using Indexed Summation of Sum of Mappings
 * using Summation of Multiple of Mapping on Finite Set