Every Pseudoprime Element is Prime implies Lattice is Arithmetic

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below Definition:Algebraic Ordered Set}algebraic distributive lattice.

Assume that
 * for every element $p$ of $S$ if $p$ is pseudoprime element, then $p$ is prime element.

Then $L$ is arithmetic.

Proof
By If Every Element Pseudoprime is Prime then Way Below Relation is Multiplicative:
 * $\ll$ is multiplicative relation.

Thus by Arithmetic iff Way Below Relation is Multiplicative in Algebraic Lattice:
 * the result holds.