Finite Summation does not Change under Permutation

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

Then we have the equality of summations over finite sets:


 * $\ds \sum_{s \mathop \in S} \map f s = \sum_{s \mathop \in S} \map f {\map \sigma s}$

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