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 Th20:
  rho(R) is upper & rho(R) is cap-closed
  proof
    now
      let Y1,Y2 be Subset of [:X,X:];
      assume that
A1:   Y1 in rho(R) and
A2:   Y1 c= Y2;
      consider S be Subset of [:X,X:] such that
A3:   Y1 = S and
A4:   R c= S by A1;
      R c= Y2 by A2,A3,A4;
      hence Y2 in rho(R);
    end;
    hence rho(R) is upper;
    now
      let X1,Y1 be set;
      assume that
A5:   X1 in rho(R) and
A6:   Y1 in rho(R);
      consider SX be Subset of [:X,X:] such that
A7:   X1 = SX and
A8:   R c= SX by A5;
      consider SY be Subset of [:X,X:] such that
A9:   Y1 = SY and
A10:  R c= SY by A6;
      R c= SX /\ SY by A8,A10,XBOOLE_1:19;
      hence X1 /\ Y1 in rho(R) by A7,A9;
    end;
    hence rho(R) is cap-closed;
  end;
