Section Automorphism.
End Automorphism.
Definition fixedField V (A : {set gal_of V}) :=
(V :&: \bigcap_(x in A) fixedSpace x)%VS.
Fact fixedField_is_aspace E (A : {set gal_of E}) : is_aspace (fixedField A).
Canonical fixedField_aspace E A : {subfield L} :=
ASpace (@fixedField_is_aspace E A).

Lemma galois_connection K E (A : {set gal_of E}):
(K <= E)%VS -> (A \subset 'Gal(E / K)) = (K <= fixedField A)%VS.