I was reading the book The Theory of Fields and Galois by J.S. Milne, and I have a question about a demonstration that is done in the book, in THEOREM 3.16 (FUNDAMENTAL THEOREM OF GALOIS THEORY) in part d) in the proof use the fact that $\left (E^{H} \right )^{G/H} = F$, H be a normal subgroup of G, and also $G =Gal(E/F)$ How can I justify this fact, any clue or suggestion? Thanks for the support
link of the book https://www.jmilne.org/math/CourseNotes/FT.pdf