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

theorem Th4:
  for Y be OrthogonalFamily of X for Z be Subset of X holds Z is
  Subset of Y implies Z is OrthogonalFamily of X
proof
  let Y be OrthogonalFamily of X;
  let Z be Subset of X;
  assume Z is Subset of Y;
  then for x, y st x in Z & y in Z & x <> y holds x.|.y = 0 by BHSP_5:def 8;
  hence thesis by BHSP_5:def 8;
end;
