
theorem
  for V being RealUnitarySpace, W being Subspace of V, v being VECTOR of
  V st v <> 0.V holds v in W implies not v in Ort_Comp 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 W implies not v in Ort_Comp W
  proof
    assume
A2: v in W;
    assume v in Ort_Comp W;
    then v in the carrier of Ort_Comp W;
    then 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 Def3;
    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 A2;
    then 0 = v .|. v by BHSP_1:def 3;
    hence contradiction by A1,BHSP_1:def 2;
  end;
  hence thesis;
end;
