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 Th29:
  for T being non empty TopSpace, x being Element of
  subbasis_Pervin_quasi_uniformity(T),
  y being Element of Pervin_quasi_uniformity T holds
  {z where z is Element of Pervin_quasi_uniformity T:
  [y,z] in x} in the topology of T
  proof
    let T be non empty TopSpace, x be Element of
    subbasis_Pervin_quasi_uniformity(T),
    y be Element of Pervin_quasi_uniformity T;
    x in subbasis_Pervin_quasi_uniformity(T);
    then consider O be Element of the topology of T such that
A1: x = block_Pervin_quasi_uniformity(O);
    set M = {z where z is Element of Pervin_quasi_uniformity T:
              [y,z] in x};
    per cases;
    suppose
A2:   y in O;
      M = O
      proof
        thus M c= O
        proof
          let a be object;
          assume a in M;
          then consider b be Element of
            Pervin_quasi_uniformity T such that
A4:       b = a and
A5:       [y,b] in x;
          [y,b] in [:(the carrier of T) \ O,the carrier of T:] or
            [y,b] in [:the carrier of T,O:] by A1,A5,XBOOLE_0:def 3;
          then (y in (the carrier of T) \ O & b in the carrier of T) or
            (y in the carrier of T & b in O) by ZFMISC_1:87;
          hence thesis by A4,A2,XBOOLE_0:def 5;
        end;
        let a be object;
        assume
A6:     a in O;
        then reconsider b = a as Element of
        Pervin_quasi_uniformity T;
        [y,b] in [:the carrier of T,O:] by A6,ZFMISC_1:87;
        then [y,b] in [:(the carrier of T) \ O,the carrier of T:] \/
        [:the carrier of T,O:] by XBOOLE_0:def 3;
        hence thesis by A1;
      end;
      hence thesis;
    end;
    suppose
A7:   not y in O;
      M = the carrier of Pervin_quasi_uniformity T
      proof
        thus M c= the carrier of Pervin_quasi_uniformity T
        proof
          let a be object;
          assume a in M;
          then ex b be Element of Pervin_quasi_uniformity T st
            a = b & [y,b] in x;
          hence thesis;
        end;
        let a be object;
        assume a in the carrier of Pervin_quasi_uniformity T;
        then reconsider b = a as Element of the carrier of
        Pervin_quasi_uniformity T;
        y in (the carrier of T) \ O & b in the carrier of T
          by A7,XBOOLE_0:def 5; then
A10:    [y,b] in [:(the carrier of T) \ O, the carrier of T:] by ZFMISC_1:87;
        [:(the carrier of T) \ O, the carrier of T:] c=
        [:(the carrier of T) \ O, the carrier of T:] \/
          [:the carrier of T,O:] by XBOOLE_1:7;
        hence thesis by A1,A10;
      end;
      hence thesis by PRE_TOPC:def 1;
    end;
  end;
