Summation over Finite Set Equals Summation over Support

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

Let $S$ be a finite set.

Let $f: S \to \mathbb A$ be a mapping.

Let $\operatorname{Supp} \left({f}\right)$ be its support.

Then we have an equality of summations over finite sets:


 * $\displaystyle \sum_{s \mathop \in S} f \left({s}\right) = \sum_{s \mathop \in \operatorname{Supp} \left({f}\right)} f \left({s}\right)$

Proof
Note that by Subset of Finite Set is Finite, $\operatorname{Supp} \left({f}\right)$ is indeed finite.

The result now follows from:
 * Sum over Complement of Finite Set
 * Sum of Zero over Finite Set
 * Identity Element of Addition on Numbers

Also see

 * Definition:Summation over Set with Finite Support