= Solution
Let $p(\bar x)$ be an <isolated type> over $A$, with isolating formula $\theta(\bar x)$ over $A$. Its consistency with $T_A$ and the <complete theory> property imply
$$
T_A\models\exists\bar x\,\theta(\bar x):
$$
otherwise $\neg\exists\bar x\,\theta$ would belong to $T_A$, contradicting consistency. Choose a tuple $\bar b\in M$ satisfying $\theta$. By isolation,
$$
T_A\models\forall\bar x\,(\theta\to\psi)\qquad(\psi\in p),
$$
so $\bar b$ satisfies every member of $p$. Therefore \b[every <isolated A-type> is realized in the original structure]. The same reasoning works for a <principal partial type> whose isolating formula need not itself be a member of the type.
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