Arithmetic iff Compact Subset form Lattice in Algebraic Lattice

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an algebraic lattice.

Then $L$ is arithmetic $\left({K\left({L}\right), \precsim}\right)$ is a lattice,

where $K\left({L}\right)$ denotes the compact subset of $L$,
 * $\mathord \precsim = \mathord \preceq \cap \left({K\left({L}\right) \times K\left({L}\right)}\right)$