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

theorem
  for US being UniformSpaceStr st US is axiom_U2 holds
  for S being Element of the entourages of US holds
  (for x,y being Element of US st [x,y] in S holds
  [y,x] in union the entourages of US)
  proof
    let US be UniformSpaceStr;
    assume
A1: US is axiom_U2;
    let S be Element of the entourages of US;
    let x,y be Element of US;
    assume
A3: [x,y] in S;
    reconsider SFX = the entourages of US as non empty
    Subset-Family of [:the carrier of US,the carrier of US:] by A1;
A6: [y,x] in S~ by A3,RELAT_1:def 7;
    S~ in the entourages of US by A1;
    hence thesis by A6,TARSKI:def 4;
  end;
