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
  subbasis_Pervin_quasi_uniformity(T) c=
    the entourages of Pervin_quasi_uniformity T
  proof
    now
      let x be object;
      assume
A1:   x in subbasis_Pervin_quasi_uniformity(T);
A2:   subbasis_Pervin_quasi_uniformity(T) c=
        basis_Pervin_quasi_uniformity(T) by CANTOR_1:4;
      basis_Pervin_quasi_uniformity(T) c= <.basis_Pervin_quasi_uniformity(T).]
        by CARDFIL2:18;
      hence x in the entourages of Pervin_quasi_uniformity T by A1,A2;
    end;
    hence thesis;
  end;
