 There is a stronger version of Corollary 23. Let \(M_1 =\langle d_1,I_1\rangle\) and \(M_2 =\langle d_2,I_2\rangle\) be interpretations of the language \(\LKe\). Define \(M_1\) to be a submodel of \(M_2\) if \(d_1 \subseteq d_2, I_1 (c) = I_2 (c)\) for each constant \(c\), and \(I_1\) is the restriction of \(I_2\) to \(d_1\). For example, if \(R\) is a binary relation letter in \(K\), then for all \(a,b\) in \(d_1\), the pair \(\langle a,b\rangle\) is in \(I_1 (R)\) if and only if \(\langle a,b\rangle\) is in \(I_2 (R)\). If we had included function letters among the non-logical terminology, we would also require that \(d_1\) be closed under their interpretations in \(M_2\). Notice that if \(M_1\) is a submodel of \(M_2\), then any variable-assignment on \(M_1\) is also a variable-assignment on \(M_2\).