Section ExternalSDirProd.
Variables (aT rT : finGroupType) (D : {group aT}) (R : {group rT}).
(* The pair (a, x) denotes the product sdpair2 a * sdpair1 x *)
Inductive sdprod_by (to : groupAction D R) : predArgType :=
SdPair (ax : aT * rT) of ax \in setX D R.
Coercion pair_of_sd to (u : sdprod_by to) := let: SdPair ax _ := u in ax.
Variable to : groupAction D R.
Notation sdT := (sdprod_by to).
Notation sdval := (@pair_of_sd to).
Canonical sdprod_subType := Eval hnf in [subType for sdval].
Definition sdprod_eqMixin := Eval hnf in [eqMixin of sdT by <:].
Canonical sdprod_eqType := Eval hnf in EqType sdT sdprod_eqMixin.
Definition sdprod_choiceMixin := [choiceMixin of sdT by <:].
Canonical sdprod_choiceType := ChoiceType sdT sdprod_choiceMixin.
Definition sdprod_countMixin := [countMixin of sdT by <:].
Canonical sdprod_countType := CountType sdT sdprod_countMixin.
Canonical sdprod_subCountType := Eval hnf in [subCountType of sdT].
Definition sdprod_finMixin := [finMixin of sdT by <:].
Canonical sdprod_finType := FinType sdT sdprod_finMixin.
Canonical sdprod_subFinType := Eval hnf in [subFinType of sdT].
Definition sdprod_one := SdPair to (group1 _).
Definition sdprod_inv u := SdPair to (sdprod_inv_proof u).
Definition sdprod_mul u v := SdPair to (sdprod_mul_proof u v).
Canonical sdprod_groupMixin :=
FinGroup.Mixin sdprod_mulgA sdprod_mul1g sdprod_mulVg.
Canonical sdprod_baseFinGroupType :=
Eval hnf in BaseFinGroupType sdT sdprod_groupMixin.
Canonical sdprod_groupType := FinGroupType sdprod_mulVg.
Definition sdpair1 x := insubd sdprod_one (1, x) : sdT.
Definition sdpair2 a := insubd sdprod_one (a, 1) : sdT.
Canonical sdpair1_morphism := Morphism sdpair1_morphM.
Canonical sdpair2_morphism := Morphism sdpair2_morphM.

Lemma sdpair_setact (G : {set rT}) a : G \subset R -> a \in D ->
sdpair1 @* (to^~ a @: G) = (sdpair1 @* G) :^ sdpair2 a.