 If \(\mm\) is a structure and \(A\subseteq M\), then \(\mm^{A}\) is obtained by restricting the relations \(R^\mm\) and functions \(F^\mm\) to A. This does not always result in a structure but if it does, it is called the relativization of \(\mm\) to A. Suppose \(\phi=\phi(x_1,\ldots,x_n,X_1,\ldots,X_m,F_1,\ldots,F_k)\) is a second-order formula and U is a unary relation variable. There is a second-order formula \(\phi^{U}\), the relativization of \(\phi\) to U, such that for all \(\mm\) for which \(\mm^{U^\mm}\) is a structure: 