Up-Complete Product/Lemma 1

Theorem

Let $\struct {S, \preceq_1}$, $\struct {T, \preceq_2}$ be non-empty ordered sets.

Let $\struct {S \times T, \preceq}$ be the simple order product of $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$.

Let $X$ be a directed subset of $S$.

Let $Y$ be a directed subset of $T$.

Then $X \times Y$ is also a directed subset of $S \times T$.

Let $\tuple {s_1, t_1}, \tuple {s_2, t_2} \in X \times Y$.

By definition of Cartesian product:

$s_1, s_2 \in X$ and $t_1, t_2 \in Y$

By definition of directed subset:

$\exists h_1 \in X: s_1 \preceq_1 h_1 \land s_2 \preceq_1 h_1$

and

$\exists h_2 \in X: t_1 \preceq_2 h_2 \land t_2 \preceq_2 h_2$

$\exists \tuple {h_1, h_2} \in X \times Y: \tuple {s_1, t_1} \preceq \tuple {h_1, h_2} \land \tuple {s_2, t_2} \preceq \tuple {h_1, h_2}$

Thus by definition:

$X \times Y$ is a directed subset of $S \times T$.

$\blacksquare$