A new proposition proved

I’ve proved the following lemma:

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

I spent much time (probably a few hours) to prove it, but the found proof is really simple, almost trivial.

The proof is currently located in this PDF file.