Lemma 16.8.2. Let $R_ i \to \Lambda _ i$, $i = 1, 2$ be as in Situation 16.8.1. If PT holds for $R_ i \to \Lambda _ i$, $i = 1, 2$, then PT holds for $R_1 \times R_2 \to \Lambda _1 \times \Lambda _2$.

**Proof.**
Omitted. Hint: A product of filtered colimits is a filtered colimit.
$\square$

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