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 QU being non void Quasi-UniformSpace st
  the carrier of T = the carrier of QU &
  subbasis_Pervin_quasi_uniformity(T) c= the entourages of QU
  holds the entourages of Pervin_quasi_uniformity T c= the entourages of QU
  proof
    let QU be non void Quasi-UniformSpace;
    assume that
A2: the carrier of T = the carrier of QU and
A3: subbasis_Pervin_quasi_uniformity(T) c= the entourages of QU;
    the entourages of Pervin_quasi_uniformity T c= the entourages of QU
    proof
      let x be object;
      assume
A4:   x in the entourages of Pervin_quasi_uniformity T;
      then reconsider y = x as Subset of [:the carrier of T,the carrier of T:];
      consider b be Element of basis_Pervin_quasi_uniformity(T) such that
A5:   b c= y by A4,CARDFIL2:def 8;
      b in FinMeetCl(subbasis_Pervin_quasi_uniformity(T));
      then consider Y be Subset-Family of
        [:the carrier of T,the carrier of T:] such that
A6:   Y c= subbasis_Pervin_quasi_uniformity(T) and
A7:   Y is finite and
A8:   b = Intersect Y by CANTOR_1:def 3;
      reconsider Z = Y as finite Subset-Family of
        [:the carrier of QU,the carrier of QU:] by A2,A7;
      reconsider B = the entourages of QU as set;
A9:   now
        thus B is Subset-Family of [:the carrier of QU,the carrier of QU:];
        QU is cap-closed;
        hence B is cap-closed;
        thus [:the carrier of QU,the carrier of QU:] in B by Th27;
        thus Z c= B by A3,A6;
      end;
      QU is upper; then
A10:  the entourages of QU is upper;
      b is Subset of [:the carrier of QU,the carrier of QU:] &
      y is Subset of [:the carrier of QU,the carrier of QU:] & b c= y &
      b in the entourages of QU by A5,A9,A2,A8,ARMSTRNG:1;
      hence thesis by A10;
    end;
    hence thesis;
  end;
