End MoreGroupAction.
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(* Helper lemmas for quotient actions.                                        *)
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Section MoreQuotientAction.
Variables (aT rT : finGroupType).
Variables (A : {group aT})(D : {group rT}).
Variable to : groupAction A D.

(* Specifications: *)
Lemma qact_dom_doms (H : {group rT}) : H \subset D -> qact_dom to H \subset A.
Lemma acts_qact_doms (H : {group rT}) :
H \subset D -> [acts A, on H | to] -> qact_dom to H :=: A.
Lemma qacts_cosetpre (H : {group rT}) (K' : {group coset_of H}) :
H \subset D -> [acts A, on H | to] ->
[acts qact_dom to H, on K' | to / H] ->
[acts A, on coset H @*^-1 K' | to].
Lemma qacts_coset (H K : {group rT}) :
H \subset D -> [acts A, on K | to] ->
[acts qact_dom to H, on (coset H) @* K | to / H].