A simple theorem not noticed before

Today I have noticed a new simple to prove theorem which is missing in my article:

Theorem If ( \mathfrak{A}; \mathfrak{Z}) is a join-closed filtrator and both \mathfrak{A} and \mathfrak{Z} are complete lattices, then for every S \in \mathscr{P} \mathfrak{A}

\mathrm{Cor}'  \bigsqcap^{\mathfrak{A}} S = \bigsqcap^{\mathfrak{Z}} \langle \mathrm{Cor}' \rangle S.

Particularly if S \in \mathscr{P} \mathfrak{Z}, this theorem reduced to the formula:

\mathrm{Cor}'  \bigsqcap^{\mathfrak{A}} S = \bigsqcap^{\mathfrak{Z}} S.

Particularly the above formulas hold for filters on complete lattices.

I’ve added these theorems and their simple consequences to my book.

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