proposition: Lemma cosetpre_cent1 x : 'C_('N(H))[x] \subset coset H @*^-1 'C[coset H x].
proof: Proof. case Nx: (x \in 'N(H)); first by rewrite morphpre_cent1. by rewrite coset_default // cent11T morphpreT subsetIl. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq div) mathcomp(choice fintype prime finset fingroup morphism) mathcomp(automorphism)