Permutation of Indices of Supremum

Theorem
Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$.

Let $R \left({i}\right)$ be a propositional functions of $i \in I$.

Let $\displaystyle \sup_{R \left({i}\right)} a_i$ be the indexed supremum on $\left \langle {a_i} \right \rangle$.

Then:
 * $\displaystyle \sum_{R \left({i}\right)} a_i = \sum_{R \left({\pi \left({i}\right)}\right)} a_{\pi \left({i}\right)}$

where $\pi$ is a permutation on the fiber of truth of $R$.