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 Th32:
  for T st T is T_1 & 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 holds T is finite-ind & ind T <= 0
proof
  let T such that
A1: T is T_1 & 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;
A2: now
    let p be Point of T,U be open Subset of T such that
A3: p in U;
    reconsider P={p} as Subset of T by A3,ZFMISC_1:31;
    not p in U` by A3,XBOOLE_0:def 5;
    then
A4: P misses U` by ZFMISC_1:50;
    T is non empty by A3;
    then consider A9,B9 be closed Subset of T such that
A5: A9 misses B9 and
A6: A9\/B9=[#]T and
A7: P c=A9 and
A8: U`c=B9 by A1,A4;
    reconsider W=B9` as open Subset of T;
    take W;
    p in P by TARSKI:def 1;
    then U``=U & not p in B9 by A5,A7,XBOOLE_0:3;
    hence p in W & W c=U by A3,A8,SUBSET_1:12,XBOOLE_0:def 5;
    B9=A9` by A5,A6,PRE_TOPC:5;
    then Fr B9={}T by TOPGEN_1:14;
    hence Fr W is finite-ind & ind Fr W<=0-1 by Th6;
  end;
  then T is finite-ind by Th15;
  hence thesis by A2,Th16;
end;
