case $$g = \tilde gb^{-1}g_m = \tilde gb^{-1}z_0h_0 = \tilde g{\overleftarrow {z_0}} h_0\in F(B,C)F(A,B)$$ and we have the required contradiction when $l_b(g) >0$.

Case 2.2: the case when $l_b(g) = 0$ {#sec4}
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Our standpoint is, again, that we have an overall inductive hypothesis which asserts that the theorem holds for relators of shorter length and, arguing by contradiction, we are assuming that there exist counterpairs $ghg^{-1}=k$ and $gh'g^{-1}=k'$. This time, however, we assume that there is a counterpair where $l_b(g)=0$, ie $g \in G^*$.

Since $l_b(g) = 0$, $h$ and $k$ have the same signature pattern in $b$ and the same is true for $h'$ and $k'$. Possibly $l_b(h)=0$ and $h\in G^*$ while if $l_b(h) > 0$ then we have a sequence of Normal Form Equalities derived from the equality $$ghg^{-1} \equiv gh_0b^{\varepsilon_1}h_1 \ldots b^{\varepsilon_m}h_mg^{-1} = k_0b^{\varepsilon_1}k_1 \ldots b^{\varepsilon_m}k_m \equiv k$$ in which $h,k$ are expressed in reduced form in terms of the HNN extension $G = \langle G^*, b\ |\  bLb^{-1} = U \rangle$. A similar observation applies to $h'$.

By Lemma 5.1 of [@C], $h_0, \ldots h_m \in F(A^*,B^*)$ and $k_0, \ldots k_m \in F(B^*,C^*)$. The Normal Form Equalities for the $ghg^{-1}=k$ are then $$gh_0 = k_0z_0, 