A new theorem about funcoids and generalizated filter bases

I proved the following theorem:

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

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

It was originally formulated as a conjecture.