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

theorem Th10:
  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 holds US is axiom_U2
  proof
    let US be non void UniformSpaceStr;
    assume
A2: for S being Element of the entourages of US ex
    R being Relation of the carrier of US st S = R & R is symmetric;
    now
      let S be Element of the entourages of US;
      consider R being Relation of the carrier of US such that
B1:   S = R and
B2:   R is symmetric by A2;
      take R;
      thus S = R by B1;
      for x,y be object st x in the carrier of US & y in the carrier of US &
      [x,y] in R holds [y,x] in R by B2,PREFER_1:20;
      hence R is_symmetric_in the carrier of US by RELAT_2:def 3;
    end;
    hence thesis by Th9;
  end;
