= Solution
Expand the language by constants for $A$ and write $T_A=\operatorname{Th}(\mathcal M,a)_{a\in A}$. An <A-type over a structure> is a set of formulas in a fixed finite tuple $\bar x$, with parameters from $A$, consistent with this <complete theory>.
Let $Q=\{\varphi_1(\bar x),\ldots,\varphi_m(\bar x)\}\subseteq P$. If no tuple of $M$ realized $Q$, then
$$
\mathcal M\models\neg\exists\bar x\,\bigwedge_{j=1}^m\varphi_j(\bar x).
$$
That sentence belongs to $T_A$ and contradicts the consistency of $T_A\cup P$. Consequently
$$
\boxed{\text{every finite subset of an A-type is realized in }\mathcal M.}
$$
For $Q=\varnothing$, any tuple works because domains are nonempty. This proves finite satisfiability; realization of the entire <partial type> may require an <elementary extension>.
Back to article page
