reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;
reserve US for UniformSpace;
reserve R for Relation of X;

theorem
  <. rho(R).] = rho(R)
  proof
    <. rho(R).] c= rho(R)
    proof
      let t be object;
      assume
A1:   t in <. rho(R).];
      then reconsider u = t as Subset of [:X,X:];
      consider b be Element of rho(R) such that
A2:   b c= u by A1,CARDFIL2:def 8;
      b in rho(R);
      then ex c be Subset of [:X,X:] st b = c & R c= c;
      then R c= u by A2;
      hence thesis;
    end;
    hence thesis by CARDFIL2:18;
  end;
