
theorem
  for V being RealUnitarySpace holds Ort_Comp (0).V = (Omega).V
proof
  let V be RealUnitarySpace;
  for x being object st x in the carrier of (Omega).V holds x in the carrier
  of Ort_Comp (0).V
  proof
    let x be object;
    assume x in the carrier of (Omega).V;
    then x in (Omega).V;
    then x in the UNITSTR of V by RUSUB_1:def 3;
    then reconsider x as Element of V;
    for w being VECTOR of V st w in (0).V holds w,x are_orthogonal
    proof
      let w be VECTOR of V;
      assume w in (0).V;
      then w in the carrier of (0).V;
      then w in {0.V} by RUSUB_1:def 2;
      then w .|. x = 0.V .|. x by TARSKI:def 1
        .= 0 by BHSP_1:14;
      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 (0).
    V holds w,v are_orthogonal};
    hence thesis by Def3;
  end;
  then
A1: the carrier of (Omega).V c= the carrier of Ort_Comp (0).V;
  for x being object st x in the carrier of Ort_Comp (0).V holds x in the
  carrier of (Omega).V
  proof
    let x be object;
    assume x in the carrier of Ort_Comp (0).V;
    then x in Ort_Comp (0).V;
    then x in V by RUSUB_1:2;
    then x in the carrier of V;
    then x in the UNITSTR of V;
    then x in (Omega).V by RUSUB_1:def 3;
    hence thesis;
  end;
  then the carrier of Ort_Comp (0).V c= the carrier of (Omega).V;
  then the carrier of (Omega).V = the carrier of Ort_Comp (0).V by A1;
  hence thesis by RUSUB_1:24;
end;
