We apply fact to Q = G / kern φ {\displaystyle {}Q=G/\operatorname {kern} \varphi } and the canonical projection q : G → G / kern φ {\displaystyle {}q\colon G\rightarrow G/\operatorname {kern} \varphi } . This induces a group homomorphism
fulfilling φ = φ ~ ∘ q {\displaystyle {}\varphi ={\tilde {\varphi }}\circ q} , which is surjective. Let [ x ] ∈ G / kern φ {\displaystyle {}[x]\in G/\operatorname {kern} \varphi } and [ x ] ∈ kern φ ~ {\displaystyle {}[x]\in \operatorname {kern} {\tilde {\varphi }}} . Then
Therefore, x ∈ kern φ {\displaystyle {}x\in \operatorname {kern} \varphi } . Hence, [ x ] = e Q {\displaystyle {}[x]=e_{Q}} , that is, the kernel of φ ~ {\displaystyle {}{\tilde {\varphi }}} is trivial, and so, due to fact, φ ~ {\displaystyle {}{\tilde {\varphi }}} is also injective.
![{\displaystyle {}[x]\in G/\operatorname {kern} \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc998b73a097809ff94d2e7871a4353ddbc96ca)
![{\displaystyle {}[x]\in \operatorname {kern} {\tilde {\varphi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce797dc33b13a90d6eb4318b89790233bcd363d7)
![{\displaystyle {}{\tilde {\varphi }}([x])=\varphi (x)=e_{H}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6cd67b3d9d4400ce45086a10848cfd1aa6675c8)
![{\displaystyle {}[x]=e_{Q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f4bbf1f0f77a70fa994c7ecec7942206a5442d)
