– however since the role of $B^*$ throughout the argument is essentially passive, it is clear that nothing is disturbed if $B^*$ is empty.

$\mspace{-14mu}$ [to [@C]]{}\[add2\] In Lemma 6.1 on page 286 of [@C], the notation $L$ is used with two distinct meanings, only one of which is explained in the text. The meaning explained is the one that occurs right throughout the whole of [@C], namely that $L$ denotes the “lower” edge group in the representation of our one-relator group $G$ as an HNN-extension, for example as $G= \langle G^*, b \ |\ bLb^{-1}=U \rangle$, where $U$ is the upper edge group. The second meaning, which is used throughout §6 of [@C] and in , is to denote by $L(z)$ the syllable length, as defined on page 283 of [@C] of an element $z$ of, for instance, $F(A^*_+,B^*,C^*)$.

Structure and simple cases in the proof of  {#sec2}
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The proof of proceeds, as is usual, by induction on the length of the relator. We make various initial reductions and then address three separate cases at the inductive step. Of these, the first is straightforward and the third reduces easily to the second. However, the second case is complicated 