Indexed Iterated Operation does not Change under Permutation

Theorem
Let $G$ be a commutative semigroup.

Let $a, b \in\Z$ be integers.

Let $\left[{a \,.\,.\, b}\right]$ be the integer interval between $a$ and $b$.

Let $f : \left[{a \,.\,.\, b}\right] \to G$ be a mapping.

Let $\sigma : \left[{a \,.\,.\, b}\right] \to \left[{a \,.\,.\, b}\right]$ be a permutation.

Nondegenerate case
Let $a \leq b$.

Then we have an equality of indexed iterated operations:


 * $\displaystyle \prod_{i \mathop = a}^b f(i) = \prod_{i \mathop = a}^b f(\sigma(i))$

General case
Let $G$ be a commutative monoid.

Then we have an equality of indexed iterated operations:


 * $\displaystyle \prod_{i \mathop = a}^b f(i) = \prod_{i \mathop = a}^b f(\sigma(i))$

Special cases

 * Indexed Summation does not Change under Permutation