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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 $ \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\}. $