
theorem Lm89B:
  for V being RealUnitarySpace, W being Subspace of V,
      v being VECTOR of V st v <> 0.V holds
    v in Ort_Comp W implies not v in W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let v be VECTOR of V;
  assume
A1: v <> 0.V;
  v in Ort_Comp W implies not v in W
  proof
    assume
A2: v in Ort_Comp W;
    assume A3: v in W;
    v in {v1 where v1 is VECTOR of V : for w being VECTOR of V st w in W
    holds w,v1 are_orthogonal} by A2,RUSUB_5:def 3;
    then ex v1 being VECTOR of V st v = v1 & for w being VECTOR of V st w in W
    holds w,v1 are_orthogonal;
    then v,v are_orthogonal by A3;
    hence contradiction by A1,BHSP_1:def 2;
  end;
  hence thesis;
end;
