 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem
for X being RealHilbertSpace,
    K be strict Subspace of X,
    M be non empty Subset of X
st X is strict
    &
    the carrier of K is closed Subset of TopSpaceNorm RUSp2RNSp X
    &
    Lin M is Subspace of K holds
  Ort_Comp Ort_Comp M is Subspace of K
proof
  let X be RealHilbertSpace,
      K be strict Subspace of X,
      M be non empty Subset of X;
  assume that
A1:  X is strict and
A2: the carrier of K is closed Subset of TopSpaceNorm RUSp2RNSp X
    and
A3: Lin M is Subspace of K;
  reconsider L = the carrier of Lin M
    as non empty Subset of X by RUSUB_1:def 1;
  reconsider N = the carrier of K as non empty Subset of X
    by RUSUB_1:def 1;
  reconsider TL = L as Subset of TopSpaceNorm RUSp2RNSp X;
A4: the carrier of Ort_Comp(Ort_Comp(M)) = Cl(TL) by A1,Lm7;
  the carrier of Lin M c= the carrier of K
    by RUSUB_1:def 1,A3;
  hence Ort_Comp(Ort_Comp(M)) is Subspace of K
    by RUSUB_1:22,A2,TOPS_1:5,A4;
end;
