Up-Complete Product

Theorem
Let $\left({S, \preceq_1}\right)$, $\left({T, \preceq_2}\right)$ be ordered sets.

Let $\left({S \times T, \preceq}\right)$ be product of $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$.

Then
 * $\left({S \times T, \preceq}\right)$ is up-complete both $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ are also up-complete.