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

theorem Th27:
  for QU being non void Quasi-UniformSpace holds
  [:the carrier of QU,the carrier of QU:] in the entourages of QU
  proof
    let QU be non void Quasi-UniformSpace;
A1: QU is non void;
    set U = the Element of the entourages of QU;
    U in the entourages of QU by A1;
    then reconsider U as Subset of [:the carrier of QU,the carrier of QU:];
    QU is upper; then
A2: the entourages of QU is upper;
    [:the carrier of QU,the carrier of QU:] c=
      [:the carrier of QU,the carrier of QU:];
    then reconsider Y = [:the carrier of QU,the carrier of QU:] as
      Subset of [:the carrier of QU,the carrier of QU:];
    the entourages of QU is non empty by A1;
    then Y in the entourages of QU by A2;
    hence thesis;
  end;
