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

theorem
for X being RealHilbertSpace,
    M be non empty Subset of X
st X is strict
holds
  the carrier of Ort_Comp Ort_Comp M
                 is closed Subset of TopSpaceNorm RUSp2RNSp X
    &
   ( ex L be Subset of TopSpaceNorm RUSp2RNSp X
       st L = the carrier of Lin M
         &
         the carrier of Ort_Comp(Ort_Comp(M)) = Cl(L) )
    &
    Lin M is Subspace of Ort_Comp(Ort_Comp(M))
proof
  let X be RealHilbertSpace,
      M be non empty Subset of X;
  assume A1:X is strict;
  reconsider L = the carrier of Lin M
    as non empty Subset of X by RUSUB_1:def 1;
  reconsider TL = L
    as Subset of TopSpaceNorm RUSp2RNSp X;
A2:the carrier of Ort_Comp(Ort_Comp(M))
= Cl(TL) by A1,Lm7;
hence the carrier of Ort_Comp(Ort_Comp(M))
                 is closed Subset of TopSpaceNorm RUSp2RNSp X;
thus ex TL be Subset of TopSpaceNorm RUSp2RNSp X st
TL = the carrier of Lin M &
         the carrier of Ort_Comp(Ort_Comp(M)) = Cl(TL) by A2;
thus Lin M is Subspace of Ort_Comp(Ort_Comp(M))
  by RUSUB_1:22, PRE_TOPC:18,A2;
end;
