
theorem
  for V being RealUnitarySpace holds Ort_Comp (Omega).V = (0).V
proof
  let V be RealUnitarySpace;
  for x being object st x in the carrier of Ort_Comp (Omega).V
   holds x in the carrier of (0).V
  proof
    let x be object;
    assume x in the carrier of Ort_Comp (Omega).V;
    then x in {v where v is VECTOR of V : for w being VECTOR of V st w in
    (Omega).V holds w,v are_orthogonal} by Def3;
    then
A1: ex v being VECTOR of V st x = v & for w being VECTOR of V st w in
    (Omega).V holds w,v are_orthogonal;
    then reconsider x as VECTOR of V;
    x in the UNITSTR of V;
    then x in (Omega).V by RUSUB_1:def 3;
    then x,x are_orthogonal by A1;
    then 0 = x .|. x by BHSP_1:def 3;
    then x = 0.V by BHSP_1:def 2;
    then x in {0.V} by TARSKI:def 1;
    hence thesis by RUSUB_1:def 2;
  end;
  then
A2: the carrier of Ort_Comp (Omega).V c= the carrier of (0).V;
  for x being object st x in the carrier of (0).V holds x in the carrier of
  Ort_Comp (Omega).V
  proof
    let x be object;
    assume x in the carrier of (0).V;
    then
A3: x in {0.V} by RUSUB_1:def 2;
    then reconsider x as VECTOR of V;
    for w being VECTOR of V st w in (Omega).V holds w,x are_orthogonal
    proof
      let w be VECTOR of V;
      assume w in (Omega).V;
      w .|. x = w .|. 0.V by A3,TARSKI:def 1
        .= 0 by BHSP_1:15;
      hence thesis by BHSP_1:def 3;
    end;
    then x in {v where v is VECTOR of V : for w being VECTOR of V st w in
    (Omega).V holds w,v are_orthogonal};
    hence thesis by Def3;
  end;
  then the carrier of (0).V c= the carrier of Ort_Comp (Omega).V;
  then the carrier of Ort_Comp (Omega).V = the carrier of (0).V by A2;
  hence thesis by RUSUB_1:24;
end;
