Proof of lemma 15.40 in Jech

Question

I'm stuck in the following proof:

First, should it say $G_\alpha=G\cap X_\alpha$ instead of $G\cap\overline{X_\alpha}$?

Also, in the inductive clause for $G_\alpha$, does Jech mean $$ G_\alpha=\left\{ a\in X_\alpha: \exists Z\ \exists \beta<\alpha \left[a=\sum Z\wedge Z\cap (G_\beta\cup\overline{G_\beta})\neq\emptyset\right]\right\}? $$ If so, I'm not sure how to get that $G_\alpha$ actually equals that. Namely, if $a=\sum Z$ for some $Z\subset\bigcup_{\beta<\alpha}X_\beta\cup\overline{X_\beta}$, how can we modify $Z$ to get it to meet some $G_\beta\cup\overline{G_\beta}$?