
theorem Th34:
  for L being lower-bounded continuous LATTICE, V being Open upper
Subset of L for N being non empty countable Subset of L for v being Element of
L st V "/\" N c= V & v in V ex O being Open Filter of L st {v} "/\" N c= O & O
  c= V & v in O
proof
  let L be lower-bounded continuous LATTICE, V be Open upper Subset of L, N be
  non empty countable Subset of L, v be Element of L such that
A1: V "/\" N c= V and
A2: v in V;
  reconsider G = {x where x is Element of L : V "/\" {x} c= V} as Filter of L
  by Th23,Th24,Th25;
A3: N c= G
  proof
    let q be object;
    assume
A4: q in N;
    then reconsider q1 = q as Element of L;
A5: {q1} "/\" V = {q1 "/\" y where y is Element of L: y in V} by YELLOW_4:42;
    V "/\" {q1} c= V
    proof
      let t be object;
      assume t in V "/\" {q1};
      then consider y being Element of L such that
A6:   t = q1 "/\" y and
A7:   y in V by A5;
      q1 "/\" y in N "/\" V by A4,A7;
      hence thesis by A1,A6;
    end;
    hence thesis;
  end;
  N is GeneratorSet of uparrow fininfs N by Def3;
  then consider F being Filter of L such that
A8: N is GeneratorSet of F;
  F = uparrow fininfs N by A8,Def3;
  then
A9: F c= G by A3,WAYBEL_0:62;
  V "/\" F c= V
  proof
    let q be object;
    assume q in V "/\" F;
    then consider d, f being Element of L such that
A10: q = d "/\" f and
A11: d in V and
A12: f in F;
    f in G by A9,A12;
    then consider x being Element of L such that
A13: f = x and
A14: V "/\" {x} c= V;
    x in {x} by TARSKI:def 1;
    then d "/\" f in V "/\" {x} by A11,A13;
    hence thesis by A10,A14;
  end;
  then consider O being Open Filter of L such that
A15: O c= V and
A16: v in O and
A17: F c= O by A2,A8,Th33;
  take O;
A18: {v} "/\" N = {v "/\" n where n is Element of L: n in N} by YELLOW_4:42;
  thus {v} "/\" N c= O
  proof
    let q be object;
    assume q in {v} "/\" N;
    then consider n being Element of L such that
A19: q = v "/\" n and
A20: n in N by A18;
    N c= F by A8,Lm4;
    then N c= O by A17;
    then v "/\" n in O "/\" O by A16,A20;
    hence thesis by A19,Th21;
  end;
  thus thesis by A15,A16;
end;
