Trouble understanding Lindenbaum's lemma's proof I'm stuck on the section (b) of the proof of the Lindenbaum's lemma in Geoffrey Hunter's Metalogic (part 32.12). Can't these two derivations ($\Gamma ' \vdash_{PS} A $ and $\Gamma ' \vdash _{PS}\sim A $) have no formulas in common at all? And even if they are guaranteed to have some wff in common, how does the author infer $\Gamma _n \vdash _{PS}A $ and $\Gamma _n \vdash _{PS} \sim A $? I cannot understand what his line of reasoning is. I thought reading further will help, but it's not the case. Any help is appreciated :)
 A: Recap: $\Gamma'$ is the union of the all the sets $\Gamma_n$ that are constructed in stages starting from a given consistent set $\Gamma$  as $\Gamma_0$ by walking along a list of all prop. calculus wff, and adding the $n$-th wff at the $n$-th stage if it keeps things consistent, and leaving it out otherwise.
Then (a) each $\Gamma_n$ is consistent by construction.
So we now need to prove that (b) their union $\Gamma'$ is consistent too. 
Suppose otherwise. 
In other words, suppose we have both $\Gamma' \vdash A$ and  $\Gamma' \vdash\ \sim\! A$ for some $A$. Now, both derivations are, by definition of a derivation, finite. So for some number $k$, any wff that appears in either derivation will appear earlier than at number $k$ in the list of prop. calculus wffs that we earlier walked along. Note there is NO assumption that the proofs have any wffs in common at all. All they have in common is that their constituent wffs appear somewhere or other in the first $k$ wffs in our imagined list of all the prop. calculus wffs, if we make $k$ big enough.
Now we just note that since the proof that shows  $\Gamma' \vdash A$ involves no wff which appears later than the $k$-th wff in our enumeration, then the same sequence of wffs shows that $\Gamma_k \vdash A$ (for note that any axiom appealed to will either be in $\Gamma$ or will be one of those added wffs from the first $k$ wffs on our list and so be available by the time we get to $\Gamma_k$). Likewise since the proof that shows  $\Gamma' \vdash\ \sim\! A$ involves no wff which appears later than the $k$-th wff in our enumeration, then the same sequence of wffs shows that $\Gamma_k \vdash\ \sim\! A$. But that makes $\Gamma_k$ inconsistent, contrary to result (a).
