Reading a proof in the book 'Set theory - on the structure of the real line' by Tomek Bartoszynski and Haim Judah, I encountered problems.
I have several questions about the proof of Lemma 7.3.7; these are beneath the images. Definition 7.3.3 is essentiell, Lemma 7.3.4 - 7.3.6 might be used in 7.3.7 (I don't know). Notice, that the forcingnotation for conditions $p$ and $q$ saying that $p$ is stronger than $q$ in this book is not as usually $p\leq q$, but $p\geq q$.
Why can we find $\tilde S_1 \geq \tilde S$ and $a\in\omega$, such that $\tilde S_1 \Vdash_{PT_{f,g}} \dot a = a$? And how do we get $t\in T$ (what is $t$ supposed to be? $t=$stem$(T)$?) and $\tilde S_2\geq\tilde S_1$, such that $\tilde S_2\geq_n T_t$?
I'm also not entirely sure, why we want to show, that stem$(T)$ belongs to $S$.
Thank you for answering!