proposition: Lemma sub_orthonormal S1 S2 : {subset S1 <= S2} -> uniq S1 -> orthonormal S2 -> orthonormal S1.
proof: Proof. move=> sS12 uniqS1 /orthonormalP[_ oS1]. by apply/orthonormalP; split; last apply: sub_in2 sS12 _ _. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime order) mathcomp(ssralg poly finset fingroup morphism perm) mathcomp(automorphism quotient finalg action gproduct) mathcomp(zmodp commutator cyclic center pgroup sylow) mathcomp(matrix vector falgebra ssrnum algC algnum) mathcomp(archimedean)