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 Th22:
  for R being symmetric Relation of X holds rho(R) is axiom_UP2
  proof
    let R be symmetric Relation of X;
    let B1 be Element of rho(R);
    B1 in rho(R);
    then consider C1 be Subset of [:X,X:] such that
A1: B1 = C1 and
A2: R c= C1;
    reconsider R1 = C1 as Relation of X;
A3: R~ c= R1~ by A2,SYSREL:11;
    R in rho(R);
    then reconsider B2 = R as Element of rho(R);
    B2 c= B1~ by A3,A1,RELAT_2:13;
    hence ex B2 be Element of rho(R) st B2 c= B1~;
  end;
