Compact Subset is Join Subsemilattice

Theorem
Let $L = \struct {S, \vee, \preceq}$ be a bounded below join semilattice.

Let $\map K L$ be a compact subset of $L$.

Then $\map K L$ is join subsemilattice:
 * $\forall x, y \in \map K L: x \vee y \in \map K L$

Proof
Let $x, y \in \map K L$.

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:
 * $x \vee y$ is compact.

Thus by definition compact subset:
 * $x \vee y \in \map K L$