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

theorem Th23:
  for X being RealUnitarySpace,
      M be non empty Subset of X holds
    M is Subset of Ort_Comp Ort_Comp M
proof
  let X be RealUnitarySpace,
      M be non empty Subset of X;
  reconsider N = the carrier of Ort_Comp(M)
    as non empty Subset of X by RUSUB_1:def 1;
A2: the carrier of Ort_Comp(Ort_Comp(M))
    = the carrier of Ort_Comp(N) by Lm6
   .= Ort_CompSet(N) by Lm5;
  M c= Ort_CompSet(Ort_CompSet(M))
  proof let x0 be object;
    assume A3:x0 in M; then
    reconsider x=x0 as Point of X;
    for y be Point of X st y in Ort_CompSet(M)
      holds y .|. x = 0 by A3,Def1;
    hence x0 in Ort_CompSet(Ort_CompSet(M)) by Def1;
  end;
  hence thesis by A2,Lm5;
end;
