Translation of Index Variable of Summation/Infinite Series

Theorem
Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers.

Let $\ds \sum_{\map R j} a_j$ denote a summation over $R$.

Let the fiber of truth of $R$ be infinite.

Then:
 * $\ds \sum_{\map R j} a_j = \sum_{\map R {c \mathop + j} } a_{c \mathop + j} = \sum_{\map R {c \mathop - j} } a_{c \mathop - j}$

where $c$ is an integer constant which is not dependent upon $j$.