If $H \vartriangleleft G$ and $K \vartriangleleft G$ then $K/H \vartriangleleft G/H$. Then : $(G/H) / (K/H)$ is isomorphic to $G/H$ .
I know this is the statement of the theorem but would it be correct to state the theorem as "If $H \vartriangleleft K \vartriangleleft G$...."?
When you write $H\vartriangleleft K \vartriangleleft G$, you say "$H$ is normal in $K$, and $K$ is normal in $G$". This does not imply that $H$ is normal in $G$.
As a quick counterexample, one can consider the dihedral group $D_8$. This has a (noncyclic) subgroup $K$ of order $4$, so it is normal in $D_8$ (since it has index 2). However, $K$ has a subgroup $H$ that is normal in $K$ but not normal in $D_8$.