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
  <. {R} .] = rho(R)
  proof
    thus <. {R} .] c= rho(R)
    proof
      let x be object;
      assume
A2:   x in <. {R} .];
      then reconsider y = x as Subset of [:X,X:];
      consider b be Element of {R} such that
A3:   b c= y by A2,CARDFIL2:def 8;
      R c= y by A3,TARSKI:def 1;
      hence thesis;
    end;
    let x be object;
    assume x in rho(R);
    then consider S be Relation of X such that
A4: x = S and
A5: R c= S;
    R is Element of {R} by TARSKI:def 1;
    hence thesis by A4,A5,CARDFIL2:def 8;
  end;
