Section Cprodm.
Variables H K G : {group gT}.
Variables (fH : {morphism H >-> rT}) (fK : {morphism K >-> rT}).
Hypothesis eqHK_G : H \* K = G.
Hypothesis cfHK : fK @* K \subset 'C(fH @* H).
Hypothesis eqfHK : {in H :&: K, fH =1 fK}.
Definition cprodm := restrm cprodm_sub (pprodm cprodm_norm cprodm_actf eqfHK).
Canonical cprodm_morphism := Eval hnf in [morphism of cprodm].

Lemma ker_cprodm : 'ker cprodm = [set a * b^-1 | a in H, b in K & fH a == fK b].