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
  for USS being upper UniformSpaceStr st meet the entourages of USS in
  the entourages of USS
  holds ex R being Relation of the carrier of USS st
  meet the entourages of USS = R & the entourages of USS = rho(R)
  proof
    let USS be upper UniformSpaceStr;
    assume
A1: meet the entourages of USS in the entourages of USS;
    reconsider R = meet the entourages of USS as
    Relation of the carrier of USS;
    take R;
A2: rho(R) c= the entourages of USS
    proof
      let x be object;
      assume x in rho(R);
      then consider S be Subset of
        [:the carrier of USS,the carrier of USS:] such that
A3:   x = S and
A4:   R c= S;
      the entourages of USS is upper by UNIFORM2:def 7;
      hence thesis by A1,A3,A4;
    end;
    the entourages of USS c= rho(R)
    proof
      let x be object;
      assume x in the entourages of USS;
      then consider S be Subset of
      [:the carrier of USS,the carrier of USS:] such that
A5:   x = S and
A6:   S in the entourages of USS;
      R c= S by A6,SETFAM_1:3;
      hence thesis by A5;
    end;
    hence thesis by A2;
end;
