proposition: Lemma ker_in_cprod : 'ker in_cprod = kerHK.
proof: Proof. transitivity ('ker (subg [group of setX H K / kerHK] \o coset kerHK)). rewrite /ker /morphpre /= /in_cprod /cprod_by; case: cprod_by_key => /=. by rewrite ['N(_) :&: _]quotientGK ?sub_center_normal ?ker_cprod_by_central. by rewrite ker_comp ker_subg -kerE ker_coset. Qed.
imports: mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq div) mathcomp(fintype bigop finset fingroup morphism perm) mathcomp(automorphism quotient action gproduct gfunctor) mathcomp(cyclic)