proposition: Lemma coset_splitting_field gT (H : {set gT}) : group_closure_field gT -> group_closure_field (coset_of H).
proof: Proof. move=> split_gT Gbar; have ->: Gbar = (coset H @*^-1 Gbar / H)%G. by apply: val_inj; rewrite /= /quotient morphpreK ?sub_im_coset. by apply: quotient_splitting_field; [apply: subsetIl | apply: split_gT]. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime) mathcomp(ssralg poly polydiv finset fingroup morphism) mathcomp(perm automorphism quotient finalg action zmodp) mathcomp(commutator cyclic center pgroup matrix mxalgebra) mathcomp(mxpoly)