Exchange of Order of Supremum Operators

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

Let $\family {b_j}_{j \mathop \in J}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $J$.

Let $\map R i$ and $\map S j$ be propositional functions of $i \in I$, $j \in J$.

Let $\ds \sup_{\map R i} a_i$ and $\ds \sup_{\map S j} b_j$ be the indexed suprema on $\family {a_i}$ and $\family {b_j}$ respectively.

Then:
 * $\ds \sup_{\map R i} \paren {\sup_{\map S j} a_{i j} } = \sup_{\map S j} \paren {\sup_{\map R i} a_{i j} }$