Theorem 3.19 (The Third Isomorphism Theorem) from Dummit's Abstract Algebra

Question

I'm reading the proof of this Theorem 3.19 (The Third Isomorphism Theorem) from Dummit's Abstract Algebra page 98:

I could understand the proof. What I cannot understand is how could we use an undefined $K/H$ in the theorem?

If you need to read the proof, it goes as following:

Answer 1

Since $H\unlhd G$, we have $gHg^{-1}=H$ for all $g\in G$. In particular, $kHk^{-1}=H$ for all $k\in K\unlhd G$. Hence $H\unlhd K$.