reserve T,T1,T2 for TopSpace,
  A,B for Subset of T,
  F for Subset of T|A,
  G,G1, G2 for Subset-Family of T,
  U,W for open Subset of T|A,
  p for Point of T|A,
  n for Nat,
  I for Integer;
reserve Af for finite-ind Subset of T,
  Tf for finite-ind TopSpace;

theorem Th34:
  for T st T is finite-ind & ind T<=0 & T is Lindelof for A,B be
closed Subset of T st A misses B ex A9,B9 be closed Subset of T st A9 misses B9
  & A9\/B9 = [#]T & A c= A9 & B c= B9
proof
  let T such that
A1: T is finite-ind & ind T<=0 and
A2: T is Lindelof;
  set CT=[#]T;
  let A,B be closed Subset of T such that
A3: A misses B;
  per cases;
  suppose
A4: CT={};
    take A,B;
    thus thesis by A3,A4;
  end;
  suppose
A5: CT<>{};
    defpred P[object,object] means
     ex D2 being set st D2 = $2 &
     $1 is Point of T & $1 in D2 & $2 is open closed
      Subset of T & (D2 c=A` or D2 c=B`);
A6: for x be object st x in CT ex y be object st y in bool CT & P[x,y]
    proof
      let x be object such that
A7:   x in CT;
      reconsider p=x as Point of T by A7;
      per cases;
      suppose
        p in A`;
        then consider W be open Subset of T such that
A8:     p in W & W c=A` and
A9:     Fr W is finite-ind and
A10:    ind Fr W<=0-1 by A1,Th16;
        take W;
        thus W in bool CT;
        take W;
        thus W = W;
        -1<=ind Fr W by A9,Th5;
        then ind Fr W=-1 by A10,XXREAL_0:1;
        then Fr W={}T by A9,Th6;
        hence thesis by A8,TOPGEN_1:14;
      end;
      suppose
A11:    not p in A`;
A12:    A c=B` by A3,SUBSET_1:23;
        p in A by A7,A11,XBOOLE_0:def 5;
        then consider W be open Subset of T such that
A13:    p in W & W c=B` and
A14:    Fr W is finite-ind and
A15:    ind Fr W<=0-1 by A1,A12,Th16;
        take W;
        thus W in bool CT;
        take W;
        -1<=ind Fr W by A14,Th5;
        then ind Fr W=-1 by A15,XXREAL_0:1;
        then Fr W={}T by A14,Th6;
        hence thesis by A13,TOPGEN_1:14;
      end;
    end;
    consider F be Function of CT,bool CT such that
A16: for x be object st x in CT holds P[x,F.x] from FUNCT_2:sch 1(A6);
    reconsider RNG=rng F as Subset-Family of T;
A17: dom F=CT by FUNCT_2:def 1;
    CT c=union RNG
    proof
      let x be object such that
A18:  x in CT;
      reconsider p=x as Point of T by A18;
      P[x,F.x] by A16,A18;
      then p in F.p & F.p in RNG by A17,FUNCT_1:def 3;
      hence thesis by TARSKI:def 4;
    end;
    then [#]T=union RNG;
    then
A19: RNG is Cover of T by SETFAM_1:45;
    RNG is open
    proof
      let U be Subset of T;
      assume U in RNG;
      then consider x be object such that
A20:     x in dom F & F.x=U by FUNCT_1:def 3;
       P[x,U] by A20,A16;
      hence thesis;
    end;
    then consider G be Subset-Family of T such that
A21: G c=RNG and
A22: G is Cover of T and
A23: G is countable by A2,A19,METRIZTS:def 2;
    T is non empty by A5;
    then
A24: G is non empty by A22,TOPS_2:3;
    then consider U be sequence of G such that
A25: rng U=G by A23,CARD_3:96;
A26: dom U=NAT by A24,FUNCT_2:def 1;
    then reconsider U as SetSequence of CT by A25,FUNCT_2:2;
    consider V be SetSequence of CT such that
A27: union rng U=union rng V and
A28: for i,j be Nat st i<>j holds V.i misses V.j and
A29: for n ex Un be finite Subset-Family of CT st Un={U.i where i is
    Element of NAT:i<n} & V.n=U.n\union Un by Th33;
    reconsider rngV=rng V as Subset-Family of T;
    set AA={V.n where n is Element of NAT:V.n misses B};
A30: AA c=rng V
    proof
      let x be object;
      assume x in AA;
      then dom V=NAT & ex n be Element of NAT st x=V.n & V.n misses B by
FUNCT_2:def 1;
      hence thesis by FUNCT_1:def 3;
    end;
    then reconsider AA as Subset-Family of T by XBOOLE_1:1;
    set BB=rngV\AA;
A31: rngV is open
    proof
      let A be Subset of T;
      assume A in rngV;
      then consider m be object such that
A32:  m in dom V and
A33:  V.m=A by FUNCT_1:def 3;
      reconsider m as Element of NAT by A32;
      consider Un be finite Subset-Family of CT such that
A34:  Un={U.i where i is Element of NAT:i<m} and
A35:  V.m=U.m\union Un by A29;
      reconsider Un as Subset-Family of T;
      U.m in rng U by A26,FUNCT_1:def 3;
      then consider x be object such that
A36:     x in dom F & F.x=U.m by A21,A25,FUNCT_1:def 3;
      P[x,U.m] by A36,A16;
      then reconsider UN=U.m as open Subset of T;
      Un is closed
      proof
        let D be Subset of T;
        assume D in Un;
        then ex i be Element of NAT st D=U.i & i<m by A34;
        then D in rng U by A26,FUNCT_1:def 3;
        then consider x be object such that
A37:  x in dom F & F.x=D by A21,A25,FUNCT_1:def 3;
         P[x,D] by A37,A16;
        hence thesis;
      end;
      then union Un is closed by TOPS_2:21;
      then UN/\(union Un)` is open;
      hence thesis by A33,A35,SUBSET_1:13;
    end;
    then union AA is open by A30,TOPS_2:11,19;
    then reconsider UAA=union AA,UBB=union BB as open Subset of T by A31,
TOPS_2:15,19;
A38: UAA misses UBB
    proof
      assume UAA meets UBB;
      then consider x be object such that
A39:  x in union AA and
A40:  x in union BB by XBOOLE_0:3;
      consider Ax be set such that
A41:  x in Ax and
A42:  Ax in AA by A39,TARSKI:def 4;
      consider n be Element of NAT such that
A43:  V.n=Ax and
A44:  V.n misses B by A42;
      consider Bx be set such that
A45:  x in Bx and
A46:  Bx in BB by A40,TARSKI:def 4;
      Bx in rngV by A46,XBOOLE_0:def 5;
      then consider m be object such that
A47:  m in dom V and
A48:  V.m=Bx by FUNCT_1:def 3;
      reconsider m as Element of NAT by A47;
      not Bx in AA by A46,XBOOLE_0:def 5;
      then m<>n by A44,A48;
      then V.n misses V.m by A28;
      hence thesis by A41,A43,A45,A48,XBOOLE_0:3;
    end;
    rngV=BB\/AA by A30,XBOOLE_1:45;
    then
A49: UAA\/UBB=union G by A25,A27,ZFMISC_1:78
      .=[#]T by A22,SETFAM_1:45;
    then
A50: UAA=UBB` by A38,PRE_TOPC:5;
    B misses UAA
    proof
      assume B meets UAA;
      then consider x be object such that
A51:  x in B and
A52:  x in union AA by XBOOLE_0:3;
      consider Ax be set such that
A53:  x in Ax and
A54:  Ax in AA by A52,TARSKI:def 4;
      ex n be Element of NAT st V.n=Ax & V.n misses B by A54;
      hence thesis by A51,A53,XBOOLE_0:3;
    end;
    then
A55: B c=UAA` by SUBSET_1:23;
    A misses UBB
    proof
      assume A meets UBB;
      then consider x be object such that
A56:  x in A and
A57:  x in union BB by XBOOLE_0:3;
      consider Bx be set such that
A58:  x in Bx and
A59:  Bx in BB by A57,TARSKI:def 4;
      Bx in rngV by A59,XBOOLE_0:def 5;
      then consider m be object such that
A60:  m in dom V and
A61:  V.m=Bx by FUNCT_1:def 3;
      reconsider m as Element of NAT by A60;
      not V.m in AA by A59,A61,XBOOLE_0:def 5;
      then V.m meets B;
      then consider b be object such that
A62:  b in V.m and
A63:  b in B by XBOOLE_0:3;
      U.m in rng U by A26,FUNCT_1:def 3;
      then consider p be object such that
A64:  p in dom F and
A65:  F.p=U.m by A21,A25,FUNCT_1:def 3;
      reconsider Fp=F.p as Subset of T by A65;
A66:  ex Un be finite Subset of bool CT st Un={U.i where i is Element of
      NAT:i<m} & V.m=U.m\union Un by A29;
      then b in U.m by A62,XBOOLE_0:def 5;
      then Fp meets B by A63,A65,XBOOLE_0:3;
      then
A67:     not Fp c=B` by SUBSET_1:23;
      P[p,F.p] by A16,A64;
      then
A68:  U.m c=A` by A65,A67;
      x in U.m by A58,A61,A66,XBOOLE_0:def 5;
      hence thesis by A56,A68,XBOOLE_0:def 5;
    end;
    then A c=UAA by A50,SUBSET_1:23;
    hence thesis by A38,A49,A50,A55;
  end;
end;
