Canonical xcprod_morphism := [morphism of xcprodm].

(* Specifications: *)
Lemma xcprodmEl : {in H, forall x, xcprodm (cpairg1 x) = fH x}.
Lemma im_xcprodm : xcprodm @* C = fH @* H * fK @* K.
Lemma im_xcprodmr A : xcprodm @* (cpair1g @* A) = fK @* A.
Lemma injm_xcprodm : 'injm xcprodm = 'injm fH && 'injm fK.
End ExtCprodm.
(* Uses gzZchar. *)
Lemma Aut_cprod_by_full :
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
Aut_in (Aut K) 'Z(K) \isog Aut 'Z(K) ->
Aut_in (Aut C) 'Z(C) \isog Aut 'Z(C).