= Normal form theorem for an HNN extension
Fix <right coset transversals> $R_+,R_-$ for $A,B$, containing $1$. In the convention $tat^{-1}=\phi(a)$, every element has a unique normal form $g_0t^{\epsilon_1}r_1\cdots t^{\epsilon_k}r_k$, with $r_i\in R_{\epsilon_i}$ and $r_i\ne1$ whenever $\epsilon_{i+1}=-\epsilon_i$. A permutation action on the set of these formal sequences proves uniqueness and embeds the base <group>. Transporting subgroup coefficients across stable letters converts general words to these normal forms.
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