Up-Complete Product/Lemma 2

Theorem
Let $X$ be a directed subset of $S \times T$.

Then
 * $\operatorname{pr}_1^\to\left({X}\right)$ and $\operatorname{pr}_2^\to\left({X}\right)$ are directed

where
 * $\operatorname{pr}_1$ denotes the first projection on $S \times T$
 * $\operatorname{pr}_2$ denotes the second projection on $S \times T$