
theorem Th30:
  for T being compact Hausdorff non empty TopSpace, N being net
  of T ex c being Point of T st c is_a_cluster_point_of N
proof
  let T be compact Hausdorff non empty TopSpace, N be net of T;
  defpred P[set,set] means ex X being Subset of T, a being Element of N st a =
  $1 & X = {N.j where j is Element of N : a <= j} & $2 = Cl X;
A1: for e being Element of N ex u being Subset of T st P[e,u]
  proof
    let e be Element of N;
    reconsider a = e as Element of N;
    {N.j where j is Element of N : a <= j} c= the carrier of T
    proof
      let q be object;
      assume q in {N.j where j is Element of N : a <= j};
      then ex j being Element of N st q = N.j & a <= j;
      hence thesis;
    end;
    then reconsider
    X = {N.j where j is Element of N : a <= j} as Subset of the
    carrier of T;
    take Cl X, X, a;
    thus thesis;
  end;
  consider F being Function of the carrier of N, bool the carrier of T such
  that
A2: for e being Element of N holds P[e,F.e] from FUNCT_2:sch 3(A1);
  reconsider RF = rng F as Subset-Family of T;
A3: dom F = the carrier of N by FUNCT_2:def 1;
A4: RF is centered
  proof
    thus RF <> {} by A3,RELAT_1:42;
    defpred P[set,set] means ex i being Element of N, h, Ch being Subset of T
st i = $2 & Ch = $1 & h = {N.j where j is Element of N : i <= j} & Ch = Cl h;
    set J = {i where i is Element of N : ex h, Ch being Subset of T st h = {N.
    j where j is Element of N : i <= j} & Ch = Cl h};
    let H be set such that
A5: H <> {} and
A6: H c= RF and
A7: H is finite;
    reconsider H1 = H as non empty set by A5;
    set e = the Element of H1;
    e in RF by A6;
    then consider x being object such that
A8: x in dom F and
    e = F.x by FUNCT_1:def 3;
    reconsider x as Element of N by A8;
    consider X being Subset of T, a being Element of N such that
    a = x and
A9: X = {N.j where j is Element of N : a <= j} & F.x = Cl X by A2;
    a in J by A9;
    then reconsider J as non empty set;
A10: for e being Element of H1 ex u being Element of J st P[e,u]
    proof
      let e be Element of H1;
      e in RF by A6;
      then consider x being object such that
A11:  x in dom F and
A12:  e = F.x by FUNCT_1:def 3;
      reconsider x as Element of N by A11;
      consider X being Subset of T, a being Element of N such that
A13:  a = x and
A14:  X = {N.j where j is Element of N : a <= j} and
A15:  F.x = Cl X by A2;
      a in J by A14,A15;
      then reconsider a as Element of J;
      take u = a, i = x, h = X, Ch = Cl X;
      thus i = u by A13;
      thus Ch = e by A12,A15;
      thus h = {N.j where j is Element of N : i <= j} by A13,A14;
      thus thesis;
    end;
    consider Q being Function of H1, J such that
A16: for e being Element of H1 holds P[e,Q.e] from FUNCT_2:sch 3(A10);
    rng Q c= [#]N
    proof
      let q be object;
      assume q in rng Q;
      then consider x being object such that
A17:  x in dom Q and
A18:  Q.x = q by FUNCT_1:def 3;
      reconsider x as Element of H1 by A17;
      ex i being Element of N, h, Ch being Subset of T st i = Q .x & Ch =
      x & h = {N.j where j is Element of N : i <= j} & Ch = Cl h by A16;
      hence thesis by A18;
    end;
    then reconsider RQ = rng Q as Subset of [#]N;
A19: [#]N is non empty directed by WAYBEL_0:def 6;
    dom Q = H by FUNCT_2:def 1;
    then rng Q is finite by A7,FINSET_1:8;
    then consider i0 being Element of N such that
    i0 in [#]N and
A20: i0 is_>=_than RQ by A19,WAYBEL_0:1;
    for Y being set holds Y in H implies N.i0 in Y
    proof
      let Y be set;
      assume
A21:  Y in H;
      then consider i being Element of N, h, Ch being Subset of T such that
A22:  i = Q.Y and
A23:  Ch = Y and
A24:  h = {N.j where j is Element of N : i <= j} and
A25:  Ch = Cl h by A16;
      Y in dom Q by A21,FUNCT_2:def 1;
      then i in rng Q by A22,FUNCT_1:def 3;
      then i <= i0 by A20;
      then
A26:  N.i0 in h by A24;
      h c= Cl h by PRE_TOPC:18;
      hence thesis by A23,A25,A26;
    end;
    hence thesis by A5,SETFAM_1:def 1;
  end;
  RF is closed
  proof
    let P be Subset of T;
    assume P in RF;
    then consider x being object such that
A27: x in dom F and
A28: F.x = P by FUNCT_1:def 3;
    reconsider x as Element of N by A27;
    ex X being Subset of T, a being Element of N st a = x & X = {N.j where
    j is Element of N : a <= j} & F.x = Cl X by A2;
    then P = Cl P by A28;
    hence thesis;
  end;
  then meet RF <> {} by A4,COMPTS_1:4;
  then consider c being object such that
A29: c in meet RF by XBOOLE_0:def 1;
  reconsider c as Point of T by A29;
  take c;
  assume not c is_a_cluster_point_of N;
  then consider O being a_neighborhood of c such that
A30: not N is_often_in O;
  N is_eventually_in (the carrier of T) \ O by A30,WAYBEL_0:10;
  then consider s0 being Element of N such that
A31: for j being Element of N st s0 <= j holds N.j in (the carrier of T)
  \ O;
  consider Y being Subset of T, a being Element of N such that
A32: a = s0 & Y = {N.j where j is Element of N : a <= j} and
A33: F.s0 = Cl Y by A2;
  Cl Y in RF by A3,A33,FUNCT_1:def 3;
  then
A34: c in Cl Y by A29,SETFAM_1:def 1;
  {} = O /\ Y
  proof
    thus {} c= O /\ Y;
    let q be object such that
A35: q in O /\ Y;
    q in Y by A35,XBOOLE_0:def 4;
    then consider j being Element of N such that
A36: q = N.j and
A37: s0 <= j by A32;
    assume not q in {};
    N.j in (the carrier of T) \ O by A31,A37;
    then not N.j in O by XBOOLE_0:def 5;
    hence contradiction by A35,A36,XBOOLE_0:def 4;
  end;
  then O misses Y;
  then
A38: Int O misses Y by TOPS_1:16,XBOOLE_1:63;
  Int O is open & c in Int O by CONNSP_2:def 1;
  hence contradiction by A34,A38,PRE_TOPC:def 7;
end;
