reserve X for RealUnitarySpace;
reserve x, y, y1, y2 for Point of X;

theorem Th5:
  for Y be OrthonormalFamily of X for Z be Subset of X holds Z is
  Subset of Y implies Z is OrthonormalFamily of X
proof
  let Y be OrthonormalFamily of X;
  let Z be Subset of X;
  assume
A1: Z is Subset of Y;
  then
A2: for x st x in Z holds x.|.x = 1 by BHSP_5:def 9;
  Y is OrthogonalFamily of X by BHSP_5:def 9;
  then Z is OrthogonalFamily of X by A1,Th4;
  hence thesis by A2,BHSP_5:def 9;
end;
