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

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