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 Th24:
  for R being total transitive Relation of X holds rho(R) is axiom_UP3
  proof
    let R be total transitive Relation of X;
    let B1 be Element of rho(R);
    B1 in rho(R);
    then consider C be Subset of [:X,X:] such that
A1: B1 = C and
A2: R c= C;
    R in rho(R);
    then reconsider B2 = R as Element of rho(R);
    R * R c= R by RELAT_2:27;
    then B2 * B2 c= B1 by A1,A2;
    hence ex B2 being Element of rho(R) st B2 * B2 c= B1;
  end;
