A new proposition proved

I’ve proved the following lemma:

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.

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.

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