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Funcoid bases

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Refer to this draft for more details.

Conjecture Let S be a set of \mathbf{Rel}-morphisms. If \forall X, Y \in S: \operatorname{up} (X \sqcap^{\mathsf{FCD}} Y)
\subseteq S then \operatorname{up} (X_0 \sqcap^{\mathsf{FCD}} \ldots
\sqcap^{\mathsf{FCD}} X_n) \subseteq S for X_i \in S.

Trying to prove the above conjecture, first prove the following lemma:

Lemma For every funcoid f and filter Failed to parse (unknown function\mathscr): \mathcal{X} \in \mathscr{F} (\operatorname{Src} f)

\operatorname{up} \langle f \rangle \mathcal{X} = \bigcup_{F \in \operatorname{up} f}
   \operatorname{up} \langle F \rangle \mathcal{X} = \left\{ K \in \operatorname{up} \langle F
   \rangle \mathcal{X} \mid F \in \operatorname{up} f \right\}.

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