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 Th21:
  for R being total reflexive Relation of X holds
  rho(R) is axiom_UP1
  proof
    let R be total reflexive Relation of X;
    now
      let B be Element of rho(R);
      B in rho(R);
      then consider C be Subset of [:X,X:] such that
A1:   B = C and
A2:   R c= C;
A3:   field R = X by ORDERS_1:12;
      id X c= R
      proof
        let t be object;
        assume
A4:     t in id X;
        then consider a,b be object such that
        a in X and
A5:     b in X and
A6:     t = [a,b] by ZFMISC_1:def 2;
        a = b by A4,A6,RELAT_1:def 10;
        hence thesis by A5,A3,A6,RELAT_2:def 9,RELAT_2:def 1;
      end;
      hence id X c= B by A1,A2;
    end;
    hence thesis;
  end;
