Let $M$ be a monster model of a stable theory $T$ in a countable language, and assume that $A$ is an infinite algebraically closed set and that $A\subseteq{N}$ is an $|A|^+$ saturated model. Let $A\subseteq{A_0}\subseteq{N}$ be s.t. $|A_0|=|A|+\aleph_0$. I want to show that there exists a realization $A_0'$ of $\text{stp}(A_0/A)$ inside of $N$ with $A_0'\overset{\vert}{\smile}_{A}A_0$.
I'm not sure how to obtain this. It should reduce to a compactness argument that adds constant symbols for elements of $A_0'$ and then states it realizes the strong type of $A_0'$ over $A$ and doesn't $A_0'$ fork with $A_0$ over $A$. I'm not sure how to code in the latter part of the argument: i.e. $A_0'$ and doesn't fork with $A_0$ over $A$. How would I go about proving this?
I tried to assume that the statement failed: then the failure should be witnessed by a formula. But I can't really see any way to proceed after that. The other option seems to be to treat $N$ itself as a monster model and argue that since it's a monster model that the theorems of stability apply to prove the statement (or some variant thereof). But this seems to be asking for too much of $N$....