$H,K$ are normal in $G$, then $HK$ is normal in $G$ (product of normal subgroups is normal) This is a proof I couldn't find anywhere. Could somebody give me a help?
I need this to show that $$\frac{H}{H\cap K}\cong \frac{HK}{K}$$
but to form the quotient group I need first to show that $H\cap K$ is normal in $H$ and $K$ is normal in $HK$.

We have $g(hk)g^{-1}=(ghg^{-1})(gkg^{-1})\in HK$
