
theorem Th28:
  for T being non empty TopSpace for x,y being Element of
  InclPoset the topology of T st x << y for F being proper Filter of BoolePoset
  the carrier of T st x in F ex p being Element of T st p in y & p
  is_a_cluster_point_of F,T
proof
  defpred P[object,object] means ex A,B being set st
    A = $1 & B = $2 & A misses B;
  let T be non empty TopSpace;
  set L = InclPoset the topology of T;
  set B = BoolePoset the carrier of T;
  let x,y be Element of L such that
A1: x << y;
  B = InclPoset bool the carrier of T by YELLOW_1:4;
  then
A2: B = RelStr(#bool the carrier of T, RelIncl bool the carrier of T#) by
YELLOW_1:def 1;
  x in the carrier of L & L = RelStr(#the topology of T, RelIncl the
    topology of T#) by YELLOW_1:def 1;
  then reconsider x9 = x as Element of B by A2;
  let F be proper Filter of B such that
A3: x in F and
A4: for x being Element of T st x in y holds not x is_a_cluster_point_of F,T;
  set V = {A where A is Subset of T: A is open & A meets y & ex B being set st
  B in F & A misses B};
  V c= bool the carrier of T
  proof
    let a be object;
    assume a in V;
    then ex C being Subset of T st a = C & C is open & C meets y & ex B being
    set st B in F & C misses B;
    hence thesis;
  end;
  then reconsider V as Subset-Family of T;
  reconsider V as Subset-Family of T;
A5: y c= union V
  proof
    let x be object;
    assume
A6: x in y;
    L = RelStr(#the topology of T, RelIncl the topology of T#) by
YELLOW_1:def 1;
    then y in the topology of T;
    then not x is_a_cluster_point_of F,T by A4,A6;
    then consider A being Subset of T such that
A7: A is open and
A8: x in A and
A9: ex B being set st B in F & A misses B;
    A meets y by A6,A8,XBOOLE_0:3;
    then A in V by A7,A9;
    hence thesis by A8,TARSKI:def 4;
  end;
  V is open
  proof
    let a be Subset of T;
    assume a in V;
    then ex C being Subset of T st a = C & C is open & C meets y & ex B being
    set st B in F & C misses B;
    hence thesis;
  end;
  then consider W being finite Subset of V such that
A10: x c= union W by A1,A5,WAYBEL_3:34;
A11: now
    let A be object;
    assume A in W;
    then A in V;
    then ex C being Subset of T st A = C & C is open & C meets y & ex B being
    set st B in F & C misses B;
    hence ex B being object st B in F & P[A,B];
  end;
  consider f being Function such that
A12: dom f = W & rng f c= F & for A being object st A in W holds P[A,f.A]
  from FUNCT_1:sch 6(A11);
  rng f is finite by A12,FINSET_1:8;
  then consider z being Element of BoolePoset the carrier of T such that
A13: z in F and
A14: z is_<=_than rng f by A12,WAYBEL_0:2;
  set a = the Element of x9"/\"z;
  x9"/\"z in F by A3,A13,WAYBEL_0:41;
  then x9"/\"z <> Bottom B by Th4;
  then x9"/\"z <> {} by YELLOW_1:18;
  then
A15: a in x9"/\"z;
A16: x9"/\"z c= x9 /\ z by YELLOW_1:17;
  then a in x by A15,XBOOLE_0:def 4;
  then consider A being set such that
A17: a in A and
A18: A in W by A10,TARSKI:def 4;
A19: a in z by A15,A16,XBOOLE_0:def 4;
A20: f.A in rng f by A12,A18,FUNCT_1:def 3;
  then f.A in F by A12;
  then reconsider b = f.A as Element of B;
  z <= b by A14,A20,LATTICE3:def 8;
  then
A21: z c= b by YELLOW_1:2;
  P[A,f.A] by A12,A18;
  then A misses b;
  hence contradiction by A19,A17,A21,XBOOLE_0:3;
end;
