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

theorem Th26:
for X being RealHilbertSpace,
    M be strict Subspace of X
st
X is strict
&
the carrier of M is closed Subset of TopSpaceNorm RUSp2RNSp X
holds M = Ort_Comp Ort_Comp M
proof
  let X be RealHilbertSpace,
      M be strict Subspace of X;
  assume A1: X is strict &
  the carrier of M is closed Subset of TopSpaceNorm RUSp2RNSp X;
  reconsider N = the carrier of M as Subset of X by RUSUB_1:def 1;
  reconsider N as non empty Subset of X;
  N is Subset of Ort_Comp Ort_Comp N by Th23; then
N c= the carrier of Ort_Comp Ort_Comp N; then
N c= the carrier of Ort_Comp Ort_Comp M by Lm6; then
A2: M is Subspace of Ort_Comp Ort_Comp M by RUSUB_1:22;
the carrier of Ort_Comp(Ort_Comp(M)) c= N
proof
let z be object;
assume A3: z in the carrier of Ort_Comp Ort_Comp M;
the carrier of Ort_Comp Ort_Comp M
  c= the carrier of X by RUSUB_1:def 1; then
reconsider x = z as Point of X by A3;
X is_the_direct_sum_of M,Ort_Comp(M) by Th25,A1; then
A4:the carrier of X
= { (v + u) where v, u is VECTOR of X :
( v in M & u in Ort_Comp(M) ) } by RUSUB_2:def 1;
x in the carrier of X; then
consider m,n be Point of X such that
A5: x=m+n & m in M & n in Ort_Comp(M) by A4;
A6: x-m =n by RLVECT_4:1,A5;
A7: x in Ort_Comp(Ort_Comp(M)) by A3,STRUCT_0:def 5;
m in Ort_Comp(Ort_Comp(M)) by A5,A2,RUSUB_1:2; then
A8: n in Ort_Comp(Ort_Comp(M)) by A6,RUSUB_1:17,A7;
 reconsider K = the carrier of Ort_Comp(M)
     as non empty Subset of X by RUSUB_1:def 1;
A9: n in K by A5,STRUCT_0:def 5;
 n in Ort_Comp(K) by A8,Lm6;
 then n in the carrier of Ort_Comp(K) by STRUCT_0:def 5;
 then n in Ort_CompSet(K) by Lm5;
 then n.|.n = 0 by Def1,A9;
 then ||.n.|| = 0;
 then n=0.X by BHSP_1:26;
 hence thesis by A5,STRUCT_0:def 5;
end; then
Ort_Comp Ort_Comp M is Subspace of M by RUSUB_1:22;
hence thesis by RUSUB_1:20,A2;
end;
