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[~] = {{}} & the entourages of QUS[~] is upper
  implies the carrier of QUS is empty
  proof
    assume that
A1: the entourages of QUS[~] = {{}} and
A2: the entourages of QUS[~] is upper;
    reconsider X = the carrier of QUS as set;
    [:X,X:] c= [:the carrier of QUS[~],the carrier of QUS[~]:];
    then reconsider XX =
      [:X,X:] as Subset of [:the carrier of QUS[~],the carrier of QUS[~]:];
    {} c= [:the carrier of QUS[~],the carrier of QUS[~]:];
    then reconsider Y =
      {} as Subset of [:the carrier of QUS[~],the carrier of QUS[~]:];
    Y in the entourages of QUS[~] & Y c= XX by A1,TARSKI:def 1;
    then XX in the entourages of QUS[~] by A2;
    hence thesis by A1,TARSKI:def 1;
  end;
