Supremum of Suprema over Overlapping Domains

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 $\map R i$ and $\map S i$ be propositional functions of $i \in I$.

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

Then:
 * $\ds \map \sup {\sup_{\map R i} a_i, \sup_{\map S i} a_i} = \sup_{\map R i \mathop \lor \map S i} a_i$