
theorem
  for V being RealUnitarySpace, W being Subspace of V holds 0.V in Ort_Comp W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  for w being VECTOR of V st w in W holds w,0.V are_orthogonal
  proof
    let w be VECTOR of V;
    assume w in W;
    w .|. 0.V = 0 by BHSP_1:15;
    hence thesis by BHSP_1:def 3;
  end;
  then 0.V in {v where v is VECTOR of V : for w being VECTOR of V st w in W
  holds w,v are_orthogonal};
  then 0.V in the carrier of Ort_Comp W by Def3;
  hence thesis;
end;
