$g \notin F(B,C)F(A,B), h,h' \in F(A,B)$ and $k,k' \in F(B,C)$ such that $\{h,h'\}$ (and, necessarily, $\{k,k'\}$) constitute a free basis of the corresponding subgroup they generate. We shall refer to such a configuration as a *counterpair*.

There are three cases:

1.  Case Assumption: Either $a$ or $c$ has exponent sum zero in $r$.

2.  Case Assumption: Neither $a$ nor $c$ has exponent sum zero in $r$ but there exists $b \in B$ such that $\sigma_b(r) =0$.

3.  Case Assumption: No generator has exponent sum zero in $r$.

**Case 2.1**Without loss of generality we may assume that $a$ has exponent sum zero in $r$. We may further assume, by replacing $r$ by a cyclic permutation if necessary, that $c^{\pm 1}$ is the initial letter of $r$.

In the standard manner we can express $G$ as an HNN-extension of the form $G = \langle G^*, a \ |\  aLa^{-1} = U \rangle $ where $L$ and $U$ are Magnus subgroups of the base group $G^*$. To do this we define $C^* = \{c_{\mu},  \ldots, c_{\nu}\}$ and $B^* = \{b_i, i\in {\bf Z}, b\in B\}$ where, as usual, $b_i$ and $c_i$ denote the conjugates $a^iba^{-i}$ and $a^ica^{-i}$ with $\mu$ and $\nu$ respectively the 