Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 144 / 2 / c / Solution

Choose $b\in N^m$ and an $L$-formula $\phi (\bar{x};y)$ such thatLet $p(y)=\mathrm{tp}(b/M)$. Stability and the Fundamental theorem of stability make this a definable type. Apply its definition to the formula $\psi (y;\bar{x})=\phi (\bar{x};y)$. There is an $M$-formula $d_p\psi (\bar{x})$ such that, for every $\bar{a}\in M^n$,Thus $d_p\psi$ defines $X\cap M^n$ in $M$, which is the stable trace of a definable set property.