\overline {z_0}h_1 = k_1z_1, \overline {z_1}h_2 = k_2z_2, \ldots, \overline {z_{m-1}}h_m = k_mg$$ where $z_0, \ldots z_{m-1}$ lie in $L$ or $U$ according as ${\varepsilon_1}, \ldots, {\varepsilon_m}$ are $\pm 1$ and $\overline {z_{i-1}}$ represents a “downshift” or “upshift” of subscripts according as ${\varepsilon_i} = \pm 1$. When we have such a sequence, $gh_0b^{\varepsilon_1}h_1 \ldots b^{\varepsilon_i}h_i = k_0b^{\varepsilon_1}k_1 \ldots b^{\varepsilon_i}k_iz_i$ and the fact that $g \notin F(B,C)F(A,B)$ means that $z_i\notin F(B,C)F(A,B)$ and in particular is nontrivial. Moreover, when we use the Normal Form Equalities in standardised form, as described on pages 275-276 of [@C], the elements $z_i$ will always be nontrivial and of type $(A^*:C^*)$. In fact if $z_i \in U$ then $z_i$ must actually be of type $(A^*_+:C^*_+)$ in which case both $A^*_+$, and $C^*_+$ are nonempty, and therefore both $A^*_-$ and $C^*_-$ are nonempty. A similar remark applies if $z_i \in L$. This means that our applications of the results of Sections 5–6 of [@C] and of are applied under the hypotheses that there is no hidden “collapsing” of the terms denoted by the notation.

We shall use these observations throughout this section without further reference. We shall establish a series of claims which, cumulatively, will demonstrate that there 