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 Th30:
  for T being non empty TopSpace,x being Element of the carrier of
  Pervin_quasi_uniformity T, b being Element of
  FinMeetCl(subbasis_Pervin_quasi_uniformity(T)) holds
  {y where y is Element of T:[x,y] in b} in the topology of T
  proof
    let T be non empty TopSpace, x be Element of the carrier of
    Pervin_quasi_uniformity T, b be Element of
    FinMeetCl(subbasis_Pervin_quasi_uniformity(T));
    consider Y being Subset-Family of
      [:the carrier of Pervin_quasi_uniformity T,
        the carrier of Pervin_quasi_uniformity T:] such that
A1: Y c= subbasis_Pervin_quasi_uniformity(T) and
A2: Y is finite and
A3: b = Intersect Y by CANTOR_1:def 3;
    per cases;
    suppose Y is empty;
      then
A4:   b = [:the carrier of T,the carrier of T:] by A3,SETFAM_1:def 9;
      {y where y is Element of T:[x,y] in b} = the carrier of T
      proof
        thus {y where y is Element of T:[x,y] in b} c=
          the carrier of T
        proof
          let a be object;
          assume a in {y where y is Element of T:[x,y] in b};
          then ex y be Element of T st a = y & [x,y] in b;
          hence thesis;
        end;
        let a be object;
        assume a in the carrier of T;
        then reconsider y = a as Element of T;
        [x,y] in [:the carrier of T,the carrier of T:];
        hence thesis by A4;
      end;
      hence thesis by PRE_TOPC:def 1;
    end;
    suppose
A7:   Y is non empty; then
A8:   b = meet Y by A3,SETFAM_1:def 9;
      for Y be non empty Subset-Family of
        [:the carrier of Pervin_quasi_uniformity T,
        the carrier of Pervin_quasi_uniformity T:] st Y c=
        subbasis_Pervin_quasi_uniformity(T) & Y is finite
        holds {y where y is Element of T:[x,y] in meet Y}
        in the topology of T
      proof
        let Y be non empty Subset-Family of
          [:the carrier of Pervin_quasi_uniformity T,
            the carrier of Pervin_quasi_uniformity T:];
        assume that
A9:     Y c= subbasis_Pervin_quasi_uniformity(T) and
A10:    Y is finite;
        defpred P[Nat] means for Z be non empty Subset-Family of
          [:the carrier of Pervin_quasi_uniformity T,
            the carrier of Pervin_quasi_uniformity T:] st
          Z c= subbasis_Pervin_quasi_uniformity(T) &
          card Z = $1 holds {y where y is Element of T:[x,y]
          in meet Z} in the topology of T;
        for Z be non empty Subset-Family of
          [:the carrier of Pervin_quasi_uniformity T,
            the carrier of Pervin_quasi_uniformity T:] st
          Z c= subbasis_Pervin_quasi_uniformity(T) &
          card Z = 1 holds {y where y is Element of T:[x,y]
          in meet Z} in the topology of T
        proof
          let Z be non empty Subset-Family of
            [:the carrier of Pervin_quasi_uniformity T,
              the carrier of Pervin_quasi_uniformity T:];
          assume that
A11:      Z c= subbasis_Pervin_quasi_uniformity(T) and
A12:      card Z = 1;
          consider t be object such that
A13:      Z = {t} by A12,CARD_2:42;
          reconsider y = t as set by TARSKI:1;
A14:      meet Z = y by A13,SETFAM_1:10;
          t in subbasis_Pervin_quasi_uniformity(T) by A11,A13,ZFMISC_1:31;
          then consider O be Element of the topology of T such that
A15:      t = block_Pervin_quasi_uniformity(O);
          per cases;
          suppose
A16:        x in O;
            {y where y is Element of T:[x,y] in meet Z}
            in the topology of T
            proof
A17:          {z where z is Element of T:[x,z] in
              block_Pervin_quasi_uniformity(O)} = O
              proof
A18:            {z where z is Element of T:[x,z] in
                block_Pervin_quasi_uniformity(O)} c= O
                proof
                  let a be object;
                  assume a in {z where z is Element of T:
                  [x,z] in block_Pervin_quasi_uniformity(O)};
                  then ex z be Element of T st a = z &
                    [x,z] in block_Pervin_quasi_uniformity(O);
                  then [x,a] in [:(the carrier of T) \ O,the carrier of T:] or
                    [x,a] in [:the carrier of T,O:] by XBOOLE_0:def 3;
                  then (x in (the carrier of T) \ O & a in the carrier of T)
                    or (x in the carrier of T & a in O) by ZFMISC_1:87;
                  hence thesis by A16,XBOOLE_0:def 5;
                end;
                O c= {z where z is Element of T:[x,z]
                  in block_Pervin_quasi_uniformity(O)}
                proof
                  let a be object;
                  assume
A19:              a in O;
                  then reconsider b = a as Element of T;
                  [x,a] in [:the carrier of T,O:] by A19,ZFMISC_1:87;
                  then [x,b] in block_Pervin_quasi_uniformity(O)
                    by XBOOLE_0:def 3;
                  hence thesis;
                end;
                hence thesis by A18;
              end;
              thus thesis by A15,A14,A17;
            end;
            hence thesis;
          end;
          suppose
A20:        not x in O;
            {z where z is Element of T:[x,z] in
              block_Pervin_quasi_uniformity(O)} = the carrier of T
            proof
              thus {z where z is Element of T:[x,z] in
                block_Pervin_quasi_uniformity(O)} c= the carrier of T
              proof
                let a be object;
                assume a in {z where z is Element of T:
                  [x,z] in block_Pervin_quasi_uniformity(O)};
                then ex z be Element of T st a = z & [x,z]
                  in block_Pervin_quasi_uniformity(O);
                hence thesis;
              end;
              let a be object;
              assume a in the carrier of T;
              then reconsider b = a as Element of T;
              x in (the carrier of T)\O by A20,XBOOLE_0:def 5;
              then [x,b] in [:(the carrier of T) \ O,the carrier of T:] or
              [x,b] in [:the carrier of T,O:] by ZFMISC_1:87;
              then [x,b] in block_Pervin_quasi_uniformity(O)
              by XBOOLE_0:def 3;
              hence thesis;
            end;
            hence thesis by A15,A14,PRE_TOPC:def 1;
          end;
        end;
        then
A22:    P[1];
A23:    for k being non zero Nat st P[k] holds P[k+1]
        proof
          let k be non zero Nat;
          assume
A24:      P[k];
          now
            let Z be non empty Subset-Family of
              [:the carrier of Pervin_quasi_uniformity T,
                the carrier of Pervin_quasi_uniformity T:];
            assume that
A25:        Z c= subbasis_Pervin_quasi_uniformity(T) and
A26:        card Z = k + 1;
            set z = the Element of Z;
A27:        card (Z \ {z}) = k by A26,STIRL2_1:55; then
A28:        Z \ {z} is non empty;
            Z \ {z} c= Z by XBOOLE_1:36; then
A29:        Z \ {z} c= subbasis_Pervin_quasi_uniformity(T) by A25;
            Z \ {z} is non empty Subset-Family of
              [:the carrier of Pervin_quasi_uniformity T,
                the carrier of Pervin_quasi_uniformity T:] by A27; then
A30:        {y where y is Element of T:[x,y] in
               meet (Z \ {z})} in the topology of T by A24,A27,A29;
A31:        {y where y is Element of T:[x,y] in
              meet (Z \ {z})} /\ {y where y is Element of T:
                [x,y] in z}
                = {y where y is Element of T:[x,y] in meet Z}
            proof
              set M1 = {y where y is Element of T:[x,y] in
                meet (Z \ {z})},
                  M2 = {y where y is Element of T:[x,y] in z},
                  M3 = {y where y is Element of T:[x,y] in
                meet Z};
A32:          M1 /\ M2 c= M3
              proof
                let a be object;
                assume a in M1 /\ M2;then
A33:            a in M1 & a in M2 by XBOOLE_0:def 4;
                then consider b be Element of T such that
A34:            a = b and
A35:            [x,b] in meet(Z \ {z});
                consider c be Element of T such that
A36:            a = c and
A37:            [x,c] in z by A33;
                now
                  let Y be set;
                  assume Y in Z;
                  then per cases;
                  suppose Y in Z & not Y in {z};
                    then Y in Z \{z} by XBOOLE_0:def 5;
                    hence [x,b] in Y by A35,SETFAM_1:def 1;
                  end;
                  suppose Y in {z};
                    hence [x,b] in Y by A34,A36,A37,TARSKI:def 1;
                  end;
                end;
                then [x,b] in meet Z by SETFAM_1:def 1;
                hence thesis by A34;
              end;
              M3 c= M1 /\ M2
              proof
                let a be object;
                assume a in M3;
                then consider b be Element of T such that
A38:            a = b and
A39:            [x,b] in meet Z;
                now
                  let Y be set;
                  assume
A40:              Y in Z \ {z};
                  Z \ {z} c= Z by XBOOLE_1:36;
                  hence [x,b] in Y by A40,A39,SETFAM_1:def 1;
                end;
                then [x,b] in meet (Z \ {z}) by A28,SETFAM_1:def 1; then
A41:            b in M1;
                [x,b] in z by A39,SETFAM_1:def 1;
                then a in M2 by A38;
                hence thesis by A41,A38,XBOOLE_0:def 4;
              end;
              hence thesis by A32;
            end;
            z in subbasis_Pervin_quasi_uniformity(T) by A25;
            then {y where y is Element of T:[x,y] in z} in
              the topology of T by Th29;
            hence {y where y is Element of T:[x,y] in meet Z}
              in the topology of T by A31,A30,PRE_TOPC:def 1;
          end;
          hence thesis;
        end;
A42:    for k being non zero Nat holds P[k] from NAT_1:sch 10(A22,A23);
        ex n be Nat st card Y = n by A10;
        hence thesis by A9,A42;
      end;
      hence thesis by A8,A1,A2,A7;
    end;
  end;
