
theorem Th33:
  for L being lower-bounded continuous LATTICE, V being Open upper
Subset of L for F being Filter of L, v being Element of L st V "/\" F c= V & v
in V & ex A being non empty GeneratorSet of F st A is countable ex O being Open
  Filter of L st O c= V & v in O & F c= O
proof
  let L be lower-bounded continuous LATTICE, V be Open upper Subset of L, F be
  Filter of L, v be Element of L such that
A1: V "/\" F c= V and
A2: v in V;
  reconsider V1 = V as non empty Open upper Subset of L by A2;
  reconsider v1 = v as Element of V1 by A2;
  reconsider G = {x where x is Element of L : V "/\" {x} c= V} as Filter of L
  by Th23,Th24,Th25;
  given A being non empty GeneratorSet of F such that
A3: A is countable;
  consider f being sequence of  A such that
A4: rng f = A by A3,CARD_3:96;
  reconsider f as sequence of  the carrier of L by FUNCT_2:7;
  deffunc F(Element of NAT) = "/\"({f.m where m is Element of NAT: m <= $1},L);
  consider g being sequence of  the carrier of L such that
A5: for n being Element of NAT holds g.n = F(n) from FUNCT_2:sch 4;
  defpred P[Nat,set,set] means ex x1, y1 being Element of V1, z1
  being Element of L st x1 = $2 & y1 = $3 & z1 = g.($1+1) & y1 << x1 "/\" z1;
A6: dom g = NAT by FUNCT_2:def 1;
  then reconsider AA = rng g as non empty Subset of L by RELAT_1:42;
A7: AA is GeneratorSet of F by A4,A5,Th32;
A8: F c= G
  proof
    let q be object;
    assume
A9: q in F;
    then reconsider s = q as Element of L;
A10: V "/\" {s} = {s "/\" t where t is Element of L : t in V} by YELLOW_4:42;
    V "/\" {s} c= V
    proof
      let w be object;
      assume w in V "/\" {s};
      then consider t being Element of L such that
A11:  w = s "/\" t and
A12:  t in V by A10;
      t "/\" s in V "/\" F by A9,A12;
      hence thesis by A1,A11;
    end;
    hence thesis;
  end;
A13: for n being Nat, x being Element of V1 ex y being Element of
  V1 st P[n,x,y]
  proof
    let n be Nat, x be Element of V1;
    AA c= F by A7,Lm4;
    then
A14: AA c= G by A8;
    g.(n+1) in AA by A6,FUNCT_1:def 3;
    then g.(n+1) in G by A14;
    then consider g1 being Element of L such that
A15: g.(n+1) = g1 and
A16: V "/\" {g1} c= V;
    g1 in {g1} by TARSKI:def 1;
    then x "/\" g1 in V "/\" {g1};
    then ex y1 being Element of L st y1 in V & y1 << x "/\" g1 by A16,
WAYBEL_6:def 1;
    hence thesis by A15;
  end;
  consider h being sequence of  V1 such that
A17: h.0 = v1 and
A18: for n being Nat holds P[n,h.n,h.(n+1)] from RECDEF_1:sch
  2 (A13);
A19: dom h = NAT by FUNCT_2:def 1;
  then
A20: h.0 in rng h by FUNCT_1:def 3;
  then reconsider R = rng h as non empty Subset of L by XBOOLE_1:1;
  set O = uparrow fininfs R;
A21: R c= O by WAYBEL_0:62;
A22: for x, y being Element of L, n being Nat st h.n = x & h.(n+1
  ) = y holds y <= x
  proof
    let x, y be Element of L, n be Nat such that
A23: h.n = x & h.(n+1) = y;
    consider x1, y1 being Element of V1, z1 being Element of L such that
A24: x1 = h.n & y1 = h.(n+1) and
    z1 = g.(n+1) and
A25: y1 << x1 "/\" z1 by A18;
A26: x1 "/\" z1 <= x1 by YELLOW_0:23;
    y1 <= x1 "/\" z1 by A25,WAYBEL_3:1;
    hence thesis by A23,A24,A26,ORDERS_2:3;
  end;
A27: for x, y being Element of L, n, m being Element of NAT st h.n = x & h.m
  = y & n <= m holds y <= x
  proof
    defpred P[Nat] means for a being Element of NAT, s, t being
    Element of L st t = h.a & s = h.$1 & a <= $1 holds s <= t;
A28: for k being Nat st P[k] holds P[k + 1]
    proof
      let k be Nat such that
A29:  for j being Element of NAT, s, t being Element of L st t = h.j
      & s = h.k & j <= k holds s <= t;
      k in NAT by ORDINAL1:def 12;
      then h.k in R by A19,FUNCT_1:def 3;
      then reconsider s = h.k as Element of L;
      let a be Element of NAT, c, d be Element of L such that
A30:  d = h.a and
A31:  c = h.(k+1) and
A32:  a <= k+1;
A33:  c <= s by A22,A31;
      per cases by A32,NAT_1:8;
      suppose
        a <= k;
        then s <= d by A29,A30;
        hence thesis by A33,ORDERS_2:3;
      end;
      suppose
        a = k + 1;
        hence thesis by A30,A31;
      end;
    end;
A34: P[0] by NAT_1:3;
A35: for k being Nat holds P[k] from NAT_1:sch 2(A34,A28);
    let x, y be Element of L, n, m be Element of NAT;
    assume h.n = x & h.m = y & n <= m;
    hence thesis by A35;
  end;
A36: for x, y being Element of L st x in R & y in R holds x <= y or y <= x
  proof
    let x, y be Element of L;
    assume that
A37: x in R and
A38: y in R;
    consider m being object such that
A39: m in dom h and
A40: y = h.m by A38,FUNCT_1:def 3;
    reconsider m as Element of NAT by A39;
    consider n being object such that
A41: n in dom h and
A42: x = h.n by A37,FUNCT_1:def 3;
    reconsider n as Element of NAT by A41;
    per cases;
    suppose
      m <= n;
      hence thesis by A27,A42,A40;
    end;
    suppose
      n <= m;
      hence thesis by A27,A42,A40;
    end;
  end;
A43: O is Open
  proof
    let x be Element of L;
    assume x in O;
    then consider y being Element of L such that
A44: y <= x and
A45: y in fininfs R by WAYBEL_0:def 16;
    consider Y being finite Subset of R such that
A46: y = "/\"(Y,L) and
    ex_inf_of Y,L by A45;
    per cases;
    suppose
      Y <> {};
      then y in Y by A36,A46,Th27;
      then consider n being object such that
A47:  n in dom h and
A48:  h.n = y by FUNCT_1:def 3;
      reconsider n as Element of NAT by A47;
      consider x1, y1 being Element of V1, z1 being Element of L such that
A49:  x1 = h.n and
A50:  y1 = h.(n+1) and
      z1 = g.(n+1) and
A51:  y1 << x1 "/\" z1 by A18;
      take y1;
      y1 in R by A19,A50,FUNCT_1:def 3;
      hence y1 in O by A21;
      x1 "/\" z1 <= x1 by YELLOW_0:23;
      then y1 << x1 by A51,WAYBEL_3:2;
      hence thesis by A44,A48,A49,WAYBEL_3:2;
    end;
    suppose
A52:  Y = {};
      consider b being object such that
A53:  b in R by XBOOLE_0:def 1;
      reconsider b as Element of L by A53;
      consider n being object such that
A54:  n in dom h and
      h.n = b by A53,FUNCT_1:def 3;
      reconsider n as Element of NAT by A54;
A55:  x <= Top L by YELLOW_0:45;
      consider x1, y1 being Element of V1, z1 being Element of L such that
      x1 = h.n and
A56:  y1 = h.(n+1) and
      z1 = g.(n+1) and
A57:  y1 << x1 "/\" z1 by A18;
      y = Top L by A46,A52,YELLOW_0:def 12;
      then x = Top L by A44,A55,ORDERS_2:2;
      then
A58:  x1 <= x by YELLOW_0:45;
      take y1;
      y1 in R by A19,A56,FUNCT_1:def 3;
      hence y1 in O by A21;
      x1 "/\" z1 <= x1 by YELLOW_0:23;
      then y1 << x1 by A57,WAYBEL_3:2;
      hence thesis by A58,WAYBEL_3:2;
    end;
  end;
A59: for n being Element of NAT, a, b being Element of L st a = g.n & b = g.
  (n+1) holds b <= a
  proof
    let n be Element of NAT, a, b be Element of L such that
A60: a = g.n & b = g.(n+1);
    reconsider gn = {f.m where m is Element of NAT: m <= n}, gn1 = {f.k where
    k is Element of NAT: k <= n+1} as non empty finite Subset of L by Lm1;
A61: ex_inf_of gn,L & ex_inf_of gn1,L by YELLOW_0:55;
A62: gn c= gn1
    proof
      let i be object;
      assume i in gn;
      then consider k being Element of NAT such that
A63:  i = f.k and
A64:  k <= n;
      k <= n+1 by A64,NAT_1:12;
      hence thesis by A63;
    end;
    a = "/\"(gn,L) & b = "/\"(gn1,L) by A5,A60;
    hence thesis by A61,A62,YELLOW_0:35;
  end;
A65: AA is_coarser_than R
  proof
    let a be Element of L;
    assume a in AA;
    then consider x being object such that
A66: x in dom g and
A67: g.x = a by FUNCT_1:def 3;
    reconsider x as Element of NAT by A66;
    consider x1, y1 being Element of V1, z1 being Element of L such that
    x1 = h.x and
A68: y1 = h.(x+1) and
A69: z1 = g.(x+1) and
A70: y1 << x1 "/\" z1 by A18;
    x1 "/\" z1 <= z1 & y1 <= x1 "/\" z1 by A70,WAYBEL_3:1,YELLOW_0:23;
    then
A71: y1 <= z1 by ORDERS_2:3;
A72: h.(x+1) in R by A19,FUNCT_1:def 3;
    z1 <= a by A59,A67,A69;
    hence ex b be Element of L st b in R & b <= a by A68,A72,A71,ORDERS_2:3;
  end;
  reconsider O as Open Filter of L by A43;
  R is_coarser_than O by A21,Th16;
  then
A73: AA c= O by A65,YELLOW_4:7,8;
  take O;
  thus O c= V
  proof
    let q be object;
    assume q in O;
    then reconsider q as Element of O;
    consider y being Element of L such that
A74: y <= q and
A75: y in fininfs R by WAYBEL_0:def 16;
    consider Y being finite Subset of R such that
A76: y = "/\"(Y,L) and
    ex_inf_of Y,L by A75;
    per cases;
    suppose
      Y <> {};
      then y in Y by A36,A76,Th27;
      then consider n being object such that
A77:  n in dom h and
A78:  h.n = y by FUNCT_1:def 3;
      reconsider n as Element of NAT by A77;
      ex x1, y1 being Element of V1, z1 being Element of L st x1 = h.n &
      y1 = h.(n+1) & z1 = g.(n+1) & y1 << x1 "/\" z1 by A18;
      hence thesis by A74,A78,WAYBEL_0:def 20;
    end;
    suppose
A79:  Y = {};
A80:  q <= Top L by YELLOW_0:45;
      y = Top L by A76,A79,YELLOW_0:def 12;
      then q = Top L by A74,A80,ORDERS_2:2;
      hence thesis by Th8;
    end;
  end;
  thus v in O by A17,A20,A21;
  uparrow fininfs AA = F by A7,Def3;
  hence thesis by A73,WAYBEL_0:62;
end;
