Seien ${\displaystyle {}h_{1},h_{2}\in H}$. Dann gibt es ${\displaystyle {}g_{1},g_{2}\in G}$ mit ${\displaystyle {}\varphi (g_{1})=h_{1}}$ und ${\displaystyle {}\varphi (g_{2})=h_{2}}$. Dann ist

${\displaystyle {}h_{1}h_{2}=\varphi (g_{1})\varphi (g_{2})=\varphi (g_{1}g_{2})=\varphi (g_{2}g_{1})=\varphi (g_{2})\varphi (g_{1})=h_{2}h_{1}\,.}$