reserve X for set,
        A for Subset of X,
        R,S for Relation of X;
reserve QUS for Quasi-UniformSpace;

theorem
  the entourages of QUS is empty implies the entourages of QUS[~] = {{}}
  proof
    assume
A1: the entourages of QUS is empty;
    reconsider EQUS = the entourages of QUS as Subset-Family of
    [:the carrier of QUS,the carrier of QUS:];
    set X = the set of all U~ where U is Element of the entourages of QUS;
    X = {{}}
    proof
      thus X c= {{}}
      proof
        let x be object;
        assume x in X;
        then consider U be Element of the entourages of QUS such that
A11:    x = U~;
        U = {} by A1,SUBSET_1:def 1;
        then x = {} by A11;
        hence thesis by TARSKI:def 1;
      end;
      let x be object;
      assume x in {{}}; then
A14:  x = {} by TARSKI:def 1;
      then reconsider y = x as Element of the entourages of QUS
        by A1,SUBSET_1:def 1;
      y~ = {} by A14;
      hence thesis by A14;
    end;
    hence thesis;
  end;
