reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th17:
  for B0 being Subset of REAL 0 st B0 is orthogonal_basis holds B0 = {}
proof
  let B0 be Subset of REAL 0;
  assume that
A1: B0 is orthogonal_basis and
A2: B0 <> {};
  set x = the Element of B0;
  x in B0 by A2;
  then reconsider x0=x as Element of REAL 0;
  |( (0*(len x0)),0*(len x0) )| =0 by EUCLID_2:9; then
  |. 0*(len x0) .| =0;
  hence contradiction by A1,A2,Def4;
end;
