# Translation of Index Variable of Product

## Theorem

$\displaystyle \prod_{R \left({j}\right)} a_j = \prod_{R \left({c \mathop + j}\right)} a_{c \mathop + j} = \prod_{R \left({c \mathop - j}\right)} a_{c \mathop - j}$

where:

$\displaystyle \prod_{R \left({j}\right)} a_j$ denotes the product over $a_j$ for all $j$ that satisfy the propositional function $R \left({j}\right)$
$c$ is an integer constant which is not dependent upon $j$.