= Solution
Let $\vec a\in A^n$, with $A$ finite, and put $T=\operatorname{Th}(\mathcal A)$. There are only finitely many competing <tuples> $\vec b\in A^n$. For each whose <complete type> differs from that of $\vec a$, choose a <logical formula> $\phi_{\vec b}$ true at $\vec a$ and false at $\vec b$. Such a <logical formula> exists by the inequality of their <complete types>, negating a distinguishing <logical formula> if necessary.
Let $\theta$ be the <conjunction> of this finite list, or <logical truth> if the list is empty. It is true at $\vec a$, and every <tuple> satisfying it has the same <partial type> as $\vec a$. For each $\psi\in\operatorname{tp}(\vec a)$, the <first-order sentence>
$$
\forall\vec x\,(\theta(\vec x)\to\psi(\vec x))
$$
is therefore true in $\mathcal A$ and belongs to $T$. The <logical formula> $\theta$ is <first-order consistent> with $T$ because it is realized at $\vec a$. It is consequently a <principal formula of a type> of the <partial type>, proving \b[every <partial type> realized in a finite <first-order structure> is principal]. Finiteness of the <first-order structure>, rather than finiteness of the <first-order language>, is what makes the <conjunction> finite.
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