Finite Summation does not Change under Permutation

Theorem
Let $G$ be an abelian group.

Let $S$ be a finite set.

Let $f : S \to G$ be a mapping.

Let $\sigma : S\to S$ be a permutation.

Then we have the equality of summations over finite sets:


 * $\displaystyle \sum_{s \in S}f(s) = \sum_{s \in S} f(\sigma(s))$

Proof
This is a special case of Change of Variables in Summation over Finite Set.