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
  for T being non empty TopSpace, V being Element of the entourages of
  Pervin_quasi_uniformity T ex b being Element of
  FinMeetCl(subbasis_Pervin_quasi_uniformity(T)) st b c= V
  proof
    let T be non empty TopSpace, V be Element of the entourages of
    Pervin_quasi_uniformity T;
A1: <.basis_Pervin_quasi_uniformity(T).] = {x where x is Subset of
      [:the carrier of T,the carrier of T:]: ex b be Element of
      basis_Pervin_quasi_uniformity(T) st b c= x} by CARDFIL2:14;
    V in <.basis_Pervin_quasi_uniformity(T).];
    then ex S be Subset of [:the carrier of T,the carrier of T:] st V = S &
      ex b be Element of basis_Pervin_quasi_uniformity(T) st b c= S by A1;
    hence thesis;
  end;
