End CfDotOrthonormal.

(* Specifications: *)
Lemma cfnorm_orthonormal S :
orthonormal S -> '[\sum_(xi <- S) xi] = (size S)%:R.
Lemma vchar_norm1P phi :
phi \in 'Z[irr G] -> '[phi] = 1 ->
exists b : bool, exists i : Iirr G, phi = (-1) ^+ b *: 'chi_i.
Lemma zchar_small_norm phi n :
phi \in 'Z[irr G] -> '[phi] = n%:R -> (n < 4)%N ->
{S : n.-tuple 'CF(G) |
[/\ orthonormal S, {subset S <= 'Z[irr G]} & phi = \sum_(xi <- S) xi]}.
Lemma vchar_norm2 phi :
phi \in 'Z[irr G, G^#] -> '[phi] = 2%:R ->
exists i, exists2 j, j != i & phi = 'chi_i - 'chi_j.