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Funcoid bases/Basic results

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That $S$ is an upper set follows from every three supposedly equivalent conditions of the main conjecture. Thus we can assume that $S$ is an upper set while proving the conjecture.

We will denote $S' = S \cap \Gamma$ (where $\Gamma$ is the boolean lattice defined in "Funcoids are filters" chapter of the book).

Proposition (Under condition that $S$ is an upper set) we have $\bigwedge^{\mathsf{FCD}} S' = \bigwedge^{\mathsf{FCD}} S$.

Proof $\bigwedge^{\mathsf{FCD}} S' \sqsupseteq \bigwedge^{\mathsf{FCD}} S$ is obvious.

$\bigwedge^{\mathsf{FCD}} S = \bigwedge^{\mathsf{FCD}}_{K \in S} \bigwedge^{\mathsf{FCD}} \operatorname{up}^{\Gamma} K \sqsupseteq \bigwedge^{\mathsf{FCD}} S'$ because $\operatorname{up}^{\Gamma} K \subseteq S'$ and thus $\bigwedge^{\mathsf{FCD}} \operatorname{up}^{\Gamma} K \sqsupseteq \bigwedge^{\mathsf{FCD}} S'$ .