Solution (source code)

= Solution

Suppose $a\in\operatorname{acl}(A)$ in the sense of <model-theoretic algebraic closure>. Some formula $\varphi(x,\bar c)$ over $A$ has exactly $n<\infty$ realizations in the <monster model> and includes $a$. Every model $M$ containing $A$ is an <elementary substructure> of the monster and therefore contains $n$ distinct realizations. Since there are only $n$ in the monster, it contains all of them, including $a$. Hence
$$
\boxed{a\in\operatorname{acl}(A)\Longrightarrow a\in M\text{ for every model }M\supseteq A.}
$$