Section FactorTransfer.
Variable g : gT.
Hypothesis Gg : g \in G.
Let sgG : <[g]> \subset G. Proof. by rewrite cycle_subG. Qed.
Let H_g_rcosets x : {set {set gT}} := rcosets (H :* x) <[g]>.
Let n_ x := #|<[g]> : H :* x|.
Let HGg : {set {set {set gT}}} := orbit 'Rs <[g]> @: HG.
Let partHG : partition HG G := rcosets_partition sHG.
Let actsgHG : [acts <[g]>, on HG | 'Rs].
Local Notation defHgX := rcosets_cycle_transversal.
Let injHg: {in X &, injective H_g_rcosets}.

Lemma sum_index_rcosets_cycle : (\sum_(x in X) n_ x)%N = #|G : H|.