### A distributive law for filter objects

I recently proved the following conjecture (now a theorem): Theorem $latex A\cap^{\mathfrak{F}}\bigcup{}^{\mathfrak{F}}S = \bigcup{}^{\mathfrak{F}} \{ A\cap^{\mathfrak{F}} X | X\in S \}$ for every set $latex A\in\mathscr{P}\mho$ and every $latex S\in\mathscr{P}\mathfrak{F}$ where $latex \mathfrak{F}$ is the set of filter objects on some set…