Indexed Iterated Operation does not Change under Permutation

Theorem
Let $G$ be a commutative semigroup.

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

Let $\closedint a b$ be the integer interval between $a$ and $b$.

Let $f: \closedint a b \to G$ be a mapping.

Let $\sigma: \closedint a b \to \closedint a b$ be a permutation.

Nondegenerate case
Let $a \le b$.

Then we have an equality of indexed iterated operations:


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

General case
Let $G$ be a commutative monoid.

Then we have an equality of indexed iterated operations:


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

Special cases

 * Indexed Summation does not Change under Permutation