 Normal form theorem for an HNN extension

Fix right coset transversals $R_{+},R_{-}$ for $A,B$, containing $1$. In the convention $ta{t}^{-1}=\varphi (a)$, every element has a unique normal form $g_0{t}^{{ϵ}_1}{r}_1\cdots t^{{ϵ}_k}{r}_k$, with $r_i\in R_{{ϵ}_i}$ and $r_i\ne 1$ whenever ${ϵ}_{i+1}=-{ϵ}_i$. 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.