Definition cprod_by_def := subFinGroupType [group of setX H K / kerHK].
Definition cprod_by := locked_with cprod_by_key cprod_by_def.
Local Notation C := [set: FinGroup.arg_sort (FinGroup.base cprod_by)].
Definition in_cprod : gTH * gTK -> cprod_by :=
let: tt as k := cprod_by_key return _ -> locked_with k cprod_by_def in
subg _ \o coset kerHK.

Lemma ker_in_cprod : 'ker in_cprod = kerHK.