proposition: Lemma galois_sub (K E : {subfield L}) (A : {group gal_of E}) : galois K E -> (('Gal(E / K))%g \subset A) = (fixedField A <= K)%VS.
proof: Proof. move=> galKE; apply/idP/idP; first exact: fixedField_sub. move=> /galS-/(_ E)/=/subset_trans->//. by apply/subsetP => u; rewrite gal_fixedField. Qed.
imports: HB(structures) mathcomp(all_ssreflect all_fingroup all_algebra) mathcomp(all_solvable all_field) Abel(various char0)