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 Subset of
  [:the carrier of T,the carrier of T:] st ex b being Element of
  FinMeetCl(subbasis_Pervin_quasi_uniformity(T)) st b c= V
  holds V is Element of the entourages of Pervin_quasi_uniformity T
  proof
    let T be non empty TopSpace, V be Subset of
    [:the carrier of T,the carrier of T:];
    given b being Element of FinMeetCl(subbasis_Pervin_quasi_uniformity(T))
    such that
A1: b c= V;
A2: <.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).] by A1,A2;
    hence thesis;
  end;
