Section NormalComplement.
Variables K H G : {group gT}.
Hypothesis complH_K : H \in [complements to K in G].
End NormalComplement.
(* Semi-direct product *)
(* Central product *)
Canonical cprod_law := Monoid.Law cprodA cprod1g cprodg1.
Canonical cprod_abelaw := Monoid.ComLaw cprodC.

Lemma reindex_bigcprod (I J : finType) (h : J -> I) P (A : I -> {set gT}) G x :
{on SimplPred P, bijective h} -> \big[cprod/1]_(i | P i) A i = G ->
{in SimplPred P, forall i, x i \in A i} ->
\prod_(i | P i) x i = \prod_(j | P (h j)) x (h j).