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:


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

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