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 Th26:
  for X being set,R being symmetric Relation of X holds
  uniformity_induced_by(R) is axiom_U2
  proof
    let X be set, R be symmetric Relation of X;
A1: rho(R) is axiom_UP2 by Th22;
    now
      let S be Element of the entourages of uniformity_induced_by(R);
      reconsider S1 = S as Element of rho(R);
      consider T be Element of rho(R) such that
A2:   T c= S1~ by A1;
      T in rho(R);
      then consider S2 be Subset of [:X,X:] such that
A3:   T = S2 and
A4:   R c= S2;
      R c= S[~] by A2,A3,A4;
      hence S[~] in the entourages of uniformity_induced_by(R);
    end;
    hence thesis;
  end;
