Because of φ ( e G ) = e H {\displaystyle {}\varphi (e_{G})=e_{H}} , we have e G ∈ kern φ {\displaystyle {}e_{G}\in \operatorname {kern} \varphi } . Let g , g ′ ∈ kern φ {\displaystyle {}g,g'\in \operatorname {kern} \varphi } . Then
therefore, also g g ′ ∈ kern φ {\displaystyle {}gg'\in \operatorname {kern} \varphi } . Hence, the kernel is a submonoid. Now, let g ∈ kern φ {\displaystyle {}g\in \operatorname {kern} \varphi } , and consider the inverse element g − 1 {\displaystyle {}g^{-1}} . Due to fact, we have
Hence, g − 1 ∈ kern φ {\displaystyle {}g^{-1}\in \operatorname {kern} \varphi } .
,
we have
.
Let
.
Then
.
Hence, the kernel is a submonoid. Now, let
,
and consider the inverse element 
. Due to
fact,
we have
.
