This is taken from "Subgroup Theorems for Groups Presented by Generators and Relations" by H.W. Kuhn.
I do not really understand the result $H=\{U;\bar{w}r\bar{w}^{-1}=1\}$.
Isn't $\bar{w}r\bar{w}^{-1}=1$ equivalent to $r=1$, and kind of "redundant"? I mean, don't we already know $r=1$ at the beginning? What I understand is that the result of this theorem is that $H$ basically has the same relations as $G$?
I think I am missing something important. Thanks for any enlightenment.