Compact Subset is Join Subsemilattice

Theorem
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

Let $K\left({L}\right)$ be a compact subset of $L$.

Then $K\left({L}\right)$ is join subsemilattice:
 * $\forall x, y \in K\left({L}\right): x \vee y \in K\left({L}\right)$

Proof
Let $x, y \in K\left({L}\right)$.

By definition of compact subset:
 * $x$ and $y$ are compact.

By definition of compact:
 * $x \ll x$ and $y \ll y$

By Way Below is Congruent for Join:
 * $x \vee y \ll x \vee y$

By definition of compact:
 * $x \vee y$ is compact.

Thus by definition compact subset:
 * $x \vee y \in K\left({L}\right)$