
theorem
  for V being RealUnitarySpace, W being Subspace of V holds Up(W) is
  Subspace-like
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  0.V in W by RUSUB_1:11;
  hence 0.V in Up(W);
  thus for x,y being Element of V, a being Real
    st x in Up(W) & y in Up(W)
  holds x + y in Up(W) & a * x in Up(W)
  proof
    let x,y be Element of V;
    let a be Real;
    assume that
A1: x in Up(W) and
A2: y in Up(W);
    reconsider x,y as Element of V;
A3: x in W by A1;
    then
A4: a * x in W by RUSUB_1:15;
    y in W by A2;
    then x + y in W by A3,RUSUB_1:14;
    hence thesis by A4;
  end;
end;
