Supremum of Suprema over Overlapping Domains

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

Let $\displaystyle \sup_{R \left({i}\right)} a_i$ and $\displaystyle \sup_{S \left({i}\right)} a_i$ be the indexed suprema on $\left \langle {a_i} \right \rangle$ over $R \left({i}\right)$ and $S \left({i}\right)$ respectively.

Then:
 * $\displaystyle \sup \left({\sup_{R \left({i}\right)} a_i, \sup_{S \left({i}\right)} a_i}\right) = \sup_{R \left({i}\right) \mathop \lor S \left({i}\right)} a_i$