Compact Subset is Bounded Below Join Semilattice

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

Let $P = \left({K\left({L}\right), \precsim}\right)$ be an ordered subset of $L$,

where $K\left({L}\right)$ denotes the compact subset of $L$.

Then $P$ is a bounded below join semilattice.

Proof
By Bottom is Compact:
 * $\bot_L$ is a compact element,

where $\bot_L$ denotes the smallest element in $L$.

By definition of compact subset:
 * $\bot_L \in K \left({L} \right)$

By definition of the smallest element:
 * $\forall x \in K\left({L}\right): \bot_L \preceq x$

By definition of ordered subset:
 * $\forall x \in K\left({L}\right): \bot_L \precsim x$

Thus by definition:
 * $P$ is bounded below.

Thus by Compact Subset is Join Subsemilattice:
 * $P$ is join semilattice.