reserve X for set,
        A for Subset of X,
        R,S for Relation of X;

theorem Th9:
  for US being non void UniformSpaceStr st
  for S being Element of the entourages of US holds
  ex R being Relation of the carrier of US st S = R &
  R is_symmetric_in the carrier of US holds US is axiom_U2
  proof
    let US be non void UniformSpaceStr;
    assume
A2: for S being Element of the entourages of US holds
    ex R being Relation of the carrier of US st S = R &
    R is_symmetric_in the carrier of US;
A1: US is non void;
    US is axiom_U2
    proof
      let S be Element of the entourages of US;
      consider R being Relation of the carrier of US such that
A3:   S = R and
A4:   R is_symmetric_in the carrier of US by A2;
      thus S~ in the entourages of US
      proof
        R~ = R
        proof
          thus R~ c= R
          proof
            let x be object;
            assume
A7:         x in R~;
            then consider a,b be object such that
A8:         a in the carrier of US and
A9:         b in the carrier of US and
A10:        x = [a,b] by ZFMISC_1:def 2;
            [b,a] in R by A7,A10,RELAT_1:def 7;
            hence thesis by A10,A4,A8,A9,RELAT_2:def 3;
          end;
          let x be object;
          assume
A11:      x in R;
          then consider a,b be object such that
A12:      a in the carrier of US and
A13:      b in the carrier of US and
A14:      x = [a,b] by ZFMISC_1:def 2;
          [b,a] in R~ by A11,A14,RELAT_1:def 7;
          hence thesis by Th4,A4,A12,A13,A14,RELAT_2:def 3;
        end;
        hence thesis by A1,A3;
      end;
    end;
    hence thesis;
  end;
