A new theorem about funcoids and generalizated filter bases

I proved the following theorem:

Theorem If S is a generalized filter base then \left\langle f \right\rangle \bigcap{\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} \left\langle\left\langle f \right\rangle \right\rangle S for every funcoid f.

The proof (presented in updated version of this online article) is short but not quite trivial.

It was originally formulated as a conjecture.

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