Lemma 1

a) For every vertex $a$ of Ker($\varphi , \psi$ ), $((a,1))$ belongs to $Z'_1$.

b) For every vertex $(a, b)$ of $H\times_GH$, $((a,1),(b,1),(\mu(a, b),-1))$ belongs to $Z'_2$.

a) Let $A'_1$ be the set of vertices $a$ of Ker($\varphi , \psi$ ) such that $((a,1))$ belongs to $Z'_1$. We have $A_1\subset A'_1$, by definition of $Z'_1$. Let $f$ be an arrow of Ker($\varphi , \psi$ ), let $a$ be its origin and $b$ its end. It follows from property (iv) in the definition of a distinguished subset (II, p. 197, definition 1) applied to the sequence $((f,1))$ that $a\in A'_1$ is equivalent to $b\in A'_1$. Since $A_1$ meets every connected component of Ker($\varphi , \psi$ ), we have $A'_1=$ Som(Ker($\varphi , \psi$ )), which was to be proved.

b) Let $A'_2$ be the set of vertices $(a, b)$ of $H\times_GH$ such that the triple $((a,1),(b,1),(\mu(a, b),-1))$ belongs to $Z'_2$. By hypothesis, we have $A_2\subset A'_2$. Since $A_2$ meets each connected component of $H\times_GH$, it is enough to establish that, if there exists an arrow $(f, f')$ joining a vertex $(a', b')$ to a vertex $(a, b)$, then $(a, b)$ belongs to $A'_2$ if and only if analogously for $(a', b')$.

Put $f''=\mu(f, f')$; this is an arrow joining $\mu(a', b')$ to $\mu(a, b)$, and one has $\varphi (f'') =\varphi (f)$ and $\psi (f'') =\psi (f')$. By condition (iv) in the definition of a distinguished subset of $\Omega (\widetilde{\Gamma})$ (loc. cit.) applied to the sequence $((f,1),(f',1),(f'',-1))$, the two conditions

(i) the triple $((a,1),(b,1),(\mu(a, b),-1))$ belongs to $Z'$;

(ii) the triple $((a',1),(b',1),(\mu(a', b'),-1))$ belongs to $Z'$;

are equivalent. This shows that $(a, b)$ belongs to $A'_2$ if and only if $(a', b')$ belongs to $A'_2$ and completes the proof of the lemma.

Let us now prove by induction on the integer $n\geqslant 1$ that every element $((a_1, \varepsilon_1),(a_2, \varepsilon_2), . . . ,(a_n, \varepsilon_n))$ of $\Omega (\widetilde{\Gamma})$ belongs to $Z'$.

A) Case in which $n= 1$.

Let $((a, \varepsilon ))$ be an element of $\Omega (\widetilde{\Gamma})$ of length 1. We have $\varphi (a) =\psi (a)$, whence $((a,1))\in Z'$ by lemma 1. Condition (ii) in the definition of a distinguished subset then implies that $((a,-1))$ belongs to $Z'$.

B) Case in which $n\geqslant 2$.

By virtue of condition (ii) for a distinguished subset, it is enough to treat the case in which $\varepsilon_2= 1$. Suppose that $\varepsilon_1= 1$; then $(a_1, a_2)$ is a vertex of $H\times_GH$; put $a=\mu(a_1, a_2)$; by lemma 1, the triple $((a_1,1),(a_2,1),(a,-1))$ therefore belongs to $Z'$. In the case in which $\varepsilon_1=-1$, we have $\varphi (a_1) =\varphi (a_2)$; we can therefore choose a vertex $a$ of H such that $\mu(a_2, a) =a_1$ and the triple $((a_2,1),(a,1),(a_1,-1))$ belongs to $Z'$, hence the triple $((a_1,-1),(a_2,1),(a,1))$ also, by virtue of condition (ii) in the definition of a distinguished subset.

Then $((a, \varepsilon_1),(a_3, \varepsilon_3), . . . ,(a_n, \varepsilon_n))$ belongs to $\Omega (\widetilde{\Gamma})$ and is of length $n-1$. By induction, it is an element of $Z'$. By virtue of conditions (i), (ii) and (iii) in the definition of a distinguished subset, we deduce successively that the elements

belong to $Z'$. This completes the proof of the proposition.