
theorem
  for V being RealUnitarySpace, M being non empty Subset of V holds
  Ort_Comp M = Ort_Comp (Ort_Comp (Ort_Comp M))
proof
  let V be RealUnitarySpace;
  let M be non empty Subset of V;
  set N = the carrier of Ort_Comp M;
  reconsider N as Subset of V by RUSUB_1:def 1;
  reconsider N as non empty Subset of V;
  set L = the carrier of Ort_Comp (Ort_Comp M);
  reconsider L as Subset of V by RUSUB_1:def 1;
  reconsider L as non empty Subset of V;
  N c= the carrier of Ort_Comp (Ort_Comp N) by Th25;
  then
A1: N c= the carrier of Ort_Comp (Ort_Comp (Ort_Comp M)) by Th27;
  the carrier of Ort_Comp L c= the carrier of Ort_Comp M by Th25,Th26;
  then the carrier of Ort_Comp (Ort_Comp (Ort_Comp M)) c= the carrier of
  Ort_Comp M by Th27;
  then the carrier of Ort_Comp (Ort_Comp (Ort_Comp M)) = the carrier of
  Ort_Comp M by A1;
  hence thesis by RUSUB_1:24;
end;
