 Theorem 25. Downward Löwenheim-Skolem Theorem. Let \(M = \langle d,I\rangle\) be an interpretation of the language \(\LKe\). Let \(d_1\) be any subset of \(d\), and let \(\kappa\) be the maximum of the size of \(K\), the size of \(d_1\), and denumerably infinite. Then there is a submodel \(M' = \langle d',I'\rangle\) of \(M\) such that (1) \(d'\) is not larger than \(\kappa\), and (2) \(M\) and \(M'\) are equivalent. In particular, if the set \(K\) of non-logical terminology is either finite or denumerably infinite, then any interpretation has an equivalent submodel whose domain is either finite or denumerably infinite.