the form $G = \langle G^*, b \ |\  bLb^{-1} = U \rangle$ where the following hold.

1.  $G^* = \langle X^* \ |\ r^* \rangle$ where $X^*=\{a_{\kappa}, \ldots, a_{\lambda}, c_{\mu}, \ldots, c_{\nu}\}\cup\{x_i, i \in {\bf Z}\}, x \neq a,c$, $\{a_{\kappa}, a_{\lambda}, c_{\mu}, c_{\nu}\}$ are the respective minimal and maximal generators in $r^*$ associated with $a$ and $c$, and otherwise the subscript range is infinite.

2.  Furthermore $L= F(A^*_-,B^*,C^*_-)$ and $U = F(A^*_+, B^*, C^*_+)$ where $$\begin{matrix}A^*= \{a_{\kappa}, \ldots, a_{\lambda}\}, &A^*_+ = \{a_{\kappa+1}, \ldots, a_{\lambda}\}, &A^*_- = \{a_{\kappa}, \ldots, a_{\lambda-1}\},\\
    C^* = \{c_{\mu}, \ldots, c_{\nu}\}, &C^*_+ = \{c_{\mu+1}, \ldots, c_{\nu}\}, &C^*_- = \{c_{\mu}, \ldots, c_{\nu-1}\},\end{matrix}$$ and $B^* = \{x_i, x \in B',i \in {\bf Z}\}$ where $B' = B \setminus \{b\}$.

We allow the possibility that $\kappa = \lambda$ or $\mu = \nu$, or both. If, for example, $\kappa = \lambda$, then $A^*_+$ and $A^*_-$ are empty ; arguments which make reference to these must be interpreted suitably for this case. Also $B^*$ may be empty but as noted already in reference to [@C], nothing in an argument will be disturbed if in fact $B^*$ is empty.

We employ a subsidiary induction on $l_b(g)$. The inductive step when $l_b(g)>0$ is 