0PA2 Lemma 7.3.14. Let γ,γ′\gamma,\gamma^{\prime} be two admissible paths in ZZ. If [f(γ)]=[f(γ′)]≠id[f(\gamma)]=[f(\gamma^{\prime})]\neq\operatorname{id}\nolimits, then [γ]=[γ′][\gamma]=[\gamma^{\prime}]. The functor f:𝒮∙(Z,1)→𝒮∙(Z′,1)f:{\mathcal{S}}^{\bullet}(Z,1)\to{\mathcal{S}}^{\bullet}(Z^{\prime},1) is faithful.