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In Lemma 4.4.13 (page 161) of D.Marker- Model Theory book it is proved that $D(F,T)$ (the set of all possible F-diagrams of models of T) is a Borel set.

Can someone explain to me the proof given a little bit more detailed?

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    I´m a little bit confused about the sets that he sais are Borel, like the $E_1$, $E_2$ etc. Also when he proved that $D \subseteq D(F,T)$.2012-11-27

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