
theorem
  for V being finite-dimensional RealUnitarySpace, n being Element of
  NAT st dim V < n holds n Subspaces_of V = {}
proof
  let V be finite-dimensional RealUnitarySpace;
  let n be Element of NAT;
  assume that
A1: dim V < n and
A2: n Subspaces_of V <> {};
  consider x being object such that
A3: x in n Subspaces_of V by A2,XBOOLE_0:def 1;
  ex W being strict Subspace of V st W = x & dim W = n by A3,Def3;
  hence contradiction by A1,Th8;
end;
