
theorem
  for X be RealUnitarySpace, M be Subspace of X
    holds the carrier of M c= the carrier of Ort_Comp (Ort_Comp M)
proof
  let X be RealUnitarySpace;
  let M be Subspace of X;
  let x be object;
  assume AS: x in the carrier of M; then
A1: x in M;
  the carrier of M c= the carrier of X by RUSUB_1:def 1; then
  reconsider x as VECTOR of X by AS;
  for y being VECTOR of X st y in Ort_Comp M holds x,y are_orthogonal
  proof
    let y be VECTOR of X;
    assume y in Ort_Comp M; then
    y in {v where v is VECTOR of X : for w being VECTOR of X st w in M
           holds w, v are_orthogonal} by RUSUB_5:def 3; then
    ex v being VECTOR of X st
      y = v & for w being VECTOR of X st w in M holds w, v are_orthogonal;
    hence thesis by A1;
  end; then
  x in {v where v is VECTOR of X : for w being VECTOR of X st w in Ort_Comp M
         holds w, v are_orthogonal};
  hence thesis by RUSUB_5:def 3;
end;
