Section BuildIsometries.
Variable (gT : finGroupType) (L G : {group gT}).
Implicit Types (phi psi xi : 'CF(L)) (R S : seq 'CF(L)).
Implicit Types (U : pred 'CF(L)) (W : pred 'CF(G)).

Lemma isometry_of_free S f :
free S -> {in S &, isometry f} ->
{tau : {linear 'CF(L) -> 'CF(G)} |
{in S, tau =1 f} & {in <<S>>%VS &, isometry tau}}.