
theorem
  for X be RealUnitarySpace, M,N be Subspace of X
    st the carrier of M c= the carrier of N
   holds the carrier of Ort_Comp N c= the carrier of Ort_Comp M
proof
  let X be RealUnitarySpace;
  let M,N be Subspace of X;
  assume A1: the carrier of M c= the carrier of N;
  let x be object;
  assume x in the carrier of Ort_Comp N; then
  x in {v where v is VECTOR of X : for w being VECTOR of X st w in N
         holds w, v are_orthogonal} by RUSUB_5:def 3; then
A2: ex v1 being VECTOR of X st
      x = v1 &
  for w being VECTOR of X st w in N holds w,v1 are_orthogonal; then
  reconsider x as VECTOR of X;
  for y being VECTOR of X st y in M holds y,x are_orthogonal
  proof
    let y being VECTOR of X;
    assume y in M; then
    y in N by A1;
    hence y,x are_orthogonal by A2;
  end; then
  x in {v where v is VECTOR of X : for w being VECTOR of X st w in M
         holds w,v are_orthogonal};
  hence thesis by RUSUB_5:def 3;
end;
