
theorem Th39:
  for T being non empty TopSpace st T is locally-compact
  for x,y being Element of InclPoset the topology of T st x << y
  ex Z being Subset of T st x c= Z & Z c= y & Z is compact
proof
  let T be non empty TopSpace such that
A1: for x being Point of T, X being Subset of T st x in X & X is open
  ex Y being Subset of T st x in Int Y & Y c= X & Y is compact;
  set L = InclPoset the topology of T;
A2: L = RelStr(#the topology of T, RelIncl the topology of T#)
  by YELLOW_1:def 1;
  let x,y be Element of L such that
A3: x << y;
A4: x in the topology of T by A2;
  y in the topology of T by A2;
  then reconsider X = x, Y = y as Subset of T by A4;
A5: Y is open by A2;
  set VV = {Int Z where Z is Subset of T: Z c= Y & Z is compact};
  reconsider e = {}T as Subset of T;
A6: {} c= Y;
  Int {}T = {};
  then
A7: e in VV by A6;
  VV c= bool the carrier of T
  proof
    let a be object;
    assume a in VV;
    then ex Z being Subset of T st a = Int Z & Z c= Y & Z is compact;
    hence thesis;
  end;
  then reconsider VV as non empty Subset-Family of T by A7;
  set V = union VV;
  VV is open
  proof
    let a be Subset of T;
    assume a in VV;
    then ex Z being Subset of T st a = Int Z & Z c= Y & Z is compact;
    hence thesis;
  end;
  then reconsider A = VV as Subset of L by YELLOW_1:25;
A8: sup A = V by YELLOW_1:22;
  Y c= V
  proof
    let a be object;
    assume
A9: a in Y;
    then reconsider a as Point of T;
    consider Z being Subset of T such that
A10: a in Int Z and
A11: Z c= Y and
A12: Z is compact by A1,A5,A9;
    Int Z in VV by A11,A12;
    hence thesis by A10,TARSKI:def 4;
  end;
  then y <= sup A by A8,YELLOW_1:3;
  then consider B being finite Subset of L such that
A13: B c= A and
A14: x <= sup B by A3,Th18;
  defpred P[object,object] means
  ex Z being Subset of T st $2 = Z & $1 = Int Z & Z c= Y & Z is compact;
A15: now
    let z be object;
    assume z in B;
    then z in A by A13;
    then consider Z being Subset of T such that
A16: z = Int Z and
A17: Z c= Y and
A18: Z is compact;
    reconsider s = Z as object;
    take s;
    thus P[z,s] by A16,A17,A18;
  end;
  consider f being Function such that
A19: dom f = B and
A20: for z being object st z in B holds P[z, f.z] from CLASSES1:sch 1(A15);
  reconsider W = B as Subset-Family of T by A2,XBOOLE_1:1;
  sup B = union W by YELLOW_1:22;
  then
A21: X c= union W by A14,YELLOW_1:3;
  now
    let z be set;
    assume z in rng f;
    then consider a being object such that
A22: a in B and
A23: z = f.a by A19,FUNCT_1:def 3;
    ex Z being Subset of T st z = Z & a = Int Z & Z c= Y & Z is compact
    by A20,A22,A23;
    hence z c= the carrier of T;
  end;
  then reconsider Z = union rng f as Subset of T by ZFMISC_1:76;
  take Z;
  thus x c= Z
  proof
    let z be object;
    assume z in x;
    then consider a being set such that
A24: z in a and
A25: a in W by A21,TARSKI:def 4;
    consider Z being Subset of T such that
A26: f.a = Z and
A27: a = Int Z and Z c= Y
    and Z is compact by A20,A25;
A28: Int Z c= Z by TOPS_1:16;
    Z in rng f by A19,A25,A26,FUNCT_1:def 3;
    hence thesis by A24,A27,A28,TARSKI:def 4;
  end;
  thus Z c= y
  proof
    let z be object;
    assume z in Z;
    then consider a being set such that
A29: z in a and
A30: a in rng f by TARSKI:def 4;
    consider Z being object such that
A31: Z in W and
A32: a = f.Z by A19,A30,FUNCT_1:def 3;
    ex S being Subset of T st a = S & Z = Int S & S c= Y & S is compact
    by A20,A31,A32;
    hence thesis by A29;
  end;
A33: rng f is finite by A19,FINSET_1:8;
  defpred P[set] means ex A being Subset of T st A = union $1 & A is compact;
  union {} = {}T by ZFMISC_1:2;
  then
A34: P[{}];
A35: now
    let x,B be set;
    assume that
A36: x in rng f and B c= rng f;
    assume P[B];
    then consider A being Subset of T such that
A37: A = union B and
A38: A is compact;
    thus P[B \/ {x}]
    proof
      consider Z being object such that
A39:  Z in W and
A40:  x = f.Z by A19,A36,FUNCT_1:def 3;
      consider S being Subset of T such that
A41:  x = S and Z = Int S
      and S c= Y and
A42:  S is compact by A20,A39,A40;
      reconsider Bx = A \/ S as Subset of T;
      take Bx;
      thus Bx = union B \/ union {x} by A37,A41,ZFMISC_1:25
        .= union (B \/ {x}) by ZFMISC_1:78;
      thus thesis by A38,A42,COMPTS_1:10;
    end;
  end;
  P[rng f] from FINSET_1:sch 2(A33,A34,A35);
  hence thesis;
end;
