Canonical sdprodm_morphism := Eval hnf in [morphism of sdprodm].

(* Specifications: *)
Lemma sdprodmE a b : a \in H -> b \in K -> sdprodm (a * b) = fH a * fK b.
Lemma im_sdprodm : sdprodm @* G = fH @* H * fK @* K.
Lemma morphim_sdprodmr B : B \subset K -> sdprodm @* B = fK @* B.
Lemma ker_sdprodm :
'ker sdprodm = [set a * b^-1 | a in H, b in K & fH a == fK b].
Lemma injm_sdprodm :
'injm sdprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1].