
theorem Th30:
  for T being non empty TopSpace for x,y being Element of
InclPoset the topology of T st x c= y & for F being ultra Filter of BoolePoset
  the carrier of T st x in F ex p being Element of T st p in y & p
  is_a_convergence_point_of F,T holds x << y
proof
  let T be non empty TopSpace;
  set B = BoolePoset the carrier of T;
  let x,y be Element of InclPoset the topology of T such that
A1: x c= y and
A2: for F being ultra Filter of B st x in F ex p being Element of T st p
  in y & p is_a_convergence_point_of F,T;
  InclPoset the topology of T = RelStr(#the topology of T, RelIncl the
    topology of T#) by YELLOW_1:def 1;
  then x in the topology of T;
  then reconsider X = x as Subset of T;
  reconsider X as Subset of T;
  assume not x << y;
  then consider F being Subset-Family of T such that
A3: F is open and
A4: y c= union F and
A5: not ex G being finite Subset of F st x c= union G by WAYBEL_3:35;
  set xF = {x\z where z is Subset of T: z in F};
  set z = the Element of F;
A6: now
    assume
A7: F = {};
    then
 y = {} by A4,ZFMISC_1:2;
    then x = y by A1;
    hence contradiction by A4,A5,A7;
  end;
  then
A8: z in F;
  B = InclPoset bool the carrier of T by YELLOW_1:4;
  then
A9: B = RelStr(#bool the carrier of T, RelIncl bool the carrier of T#) by
YELLOW_1:def 1;
  xF c= the carrier of B
  proof
    let a be object;
     reconsider aa=a as set by TARSKI:1;
    assume a in xF;
    then ex z being Subset of T st a = x\z & z in F;
    then aa c= X;
    then aa c= the carrier of T by XBOOLE_1:1;
    hence thesis by A9;
  end;
  then reconsider xF as Subset of B;
  set FF = uparrow fininfs xF;
  now
    defpred P[object,object] means
       ex A being set st A = $2 & $1 = x\A;
    assume Bottom B in FF;
    then consider a being Element of B such that
A10: a <= Bottom B and
A11: a in fininfs xF by WAYBEL_0:def 16;
    consider s being finite Subset of xF such that
A12: a = "/\"(s,B) and
    ex_inf_of s,B by A11;
    reconsider t = s as Subset of B by XBOOLE_1:1;
A13: now
      let v be object;
      assume v in s;
      then v in xF;
      then ex z being Subset of T st v = x\z & z in F;
      hence ex z being object st z in F & P[v,z];
    end;
    consider f being Function such that
A14: dom f = s & rng f c= F & for v being object st v in s holds P[v,f.v]
    from FUNCT_1:sch 6(A13);
    reconsider G = rng f as finite Subset of F by A14,FINSET_1:8;
    Bottom B = {} by YELLOW_1:18;
    then
A15: a c= {} by A10,YELLOW_1:2;
A16: now
      assume s = {};
      then a = Top B by A12;
      hence contradiction by A15,YELLOW_1:19;
    end;
    then
A17: a = meet t by A12,YELLOW_1:20;
    x c= union G
    proof
      let c be object;
      assume that
A18:  c in x and
A19:  not c in union G;
      now
        let v be set;
        assume
A20:    v in s;
        then f.v in rng f by A14,FUNCT_1:def 3;
        then
A21:    not c in (f.v) by A19,TARSKI:def 4;
        P[v,f.v] by A14,A20;
        then v = x\(f.v);
        hence c in v by A18,A21,XBOOLE_0:def 5;
      end;
      hence contradiction by A15,A16,A17,SETFAM_1:def 1;
    end;
    hence contradiction by A5;
  end;
  then FF is proper;
  then consider GG being Filter of B such that
A22: FF c= GG and
A23: GG is ultra by Th26;
  reconsider z as Subset of T by A8;
A24: xF c= FF by WAYBEL_0:62;
  x\z in xF by A6;
  then
A25: x\z in FF by A24;
  x\z c= X;
  then x in GG by A22,A25,Th7;
  then consider p being Element of T such that
A26: p in y and
A27: p is_a_convergence_point_of GG,T by A2,A23;
  consider W being set such that
A28: p in W and
A29: W in F by A4,A26,TARSKI:def 4;
  reconsider W as Subset of T by A29;
  W is open by A3,A29;
  then
A30: W in GG by A27,A28;
A31: xF c= FF by WAYBEL_0:62;
  x\W in xF by A29;
  then x\W in FF by A31;
  then W misses (x\W) & W/\(x\W) in GG by A22,A30,Th9,XBOOLE_1:79;
  then {} in GG;
  then Bottom B in GG by YELLOW_1:18;
  hence thesis by A23,Th4;
end;
