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_cfnorm S tauS :
pairwise_orthogonal S -> pairwise_orthogonal tauS ->
map cfnorm tauS = map cfnorm S ->
{tau : {linear 'CF(L) -> 'CF(G)} | map tau S = tauS
& {in <<S>>%VS &, isometry tau}}.