Indexed Iterated Operation does not Change under Permutation
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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:
- $\ds \prod_{i \mathop = a}^b \map f i = \prod_{i \mathop = a}^b \map f {\map \sigma i}$
General case
Let $G$ be a commutative monoid.
Then we have an equality of indexed iterated operations:
- $\ds \prod_{i \mathop = a}^b \map f i = \prod_{i \mathop = a}^b \map f {\map \sigma i}$
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