Currently, I am trying to understand Hrushovski's proof of the Manin-Mumford Conjecture from the article 'Théorie des modèles et conjecture de Manin-Mumford' by Bouscaren. My question is about a step in the proof of crl. 2.13.
Let $(K,\sigma)$ be a difference field and let $A$ be a semi-abelian variety defined over the fixed field of $\sigma$ and let $F(T)\in \mathbb{Z}[T]$. The automorphism $\sigma\colon K\rightarrow K$ induces an endomorphism $\sigma\colon A\rightarrow A$ so $F(\sigma)\colon A\rightarrow A$ is an endomorphism as well. In the proof is used that $\ker F(\sigma)$ is a stable set in $K^n$ for some $n$, where stable is defined as follows:
A definable $D\subseteq K^n$ is stable if for all $E=acl_\sigma(E)\subseteq K$, for all finite tuples $\bar{a},\bar{b}\subset D$ with $\operatorname{tp}(\bar{a}/E)=\operatorname{tp}(\bar{b}/E)$ and all extensions $F=acl_\sigma(F) \subseteq K$ of $E$, if $\bar{a}$ and $F$ are independent over $E$, and $\bar{b}$ and $F$ are independent over $E$, then $\operatorname{tp}(\bar{a}/F)=\operatorname{tp}(\bar{b}/F)$.
How do I prove that $\ker (F\sigma)$ is indeed stable in this sense?