Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 23 / 5 / d / 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 implyotherwise $\neg \exists \bar{x}\theta$ would belong to $T_A$, contradicting consistency. Choose a tuple $\bar{b}\in M$ satisfying $\theta$. By isolation,so $\bar{b}$ satisfies every member of $p$. Therefore 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.