Lemma Let for every $X, Y \in S$ and $Z \in \mathrm{up} (X \sqcap^{\mathsf{FCD}} Y)$ there is a $T \in S$ such that $T \sqsubseteq Z$.
Then for every $X_0, \ldots, X_n \in S$ and $Z \in \mathrm{up} (X_0 \sqcap^{\mathsf{FCD}} \ldots \sqcap^{\mathsf{FCD}} X_n)$ there is a $T \in S$ such that $T \sqsubseteq Z$.