Lemma 7.5:

(i)

$\gamma \ast \gamma^{-1} \sim \varepsilon_{\gamma(0)}$. Thus $[ \gamma ] \cdot [ \gamma^{-1} ]=[\varepsilon_{\gamma(0)}]$.

(ii)

$\gamma \ast \varepsilon_{\gamma(1)} \sim \gamma.$ Thus $[\gamma][\varepsilon_{\gamma(1)}]=[\gamma]$

(iii)

$\varepsilon_{\gamma(0)} \ast \gamma \sim \gamma.$ Thus $[\varepsilon_{\gamma(0)}][\gamma]=[\gamma]$.