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 Th6:
  ind Af = -1 iff Af is empty
proof
  not -1 in dom Seq_of_ind T;
  then
A1: not Af in (Seq_of_ind T).(-1) by FUNCT_1:def 2;
A2: (Seq_of_ind T).0={{}T} by Def1;
  hereby
    assume ind Af=-1;
    then Af in (Seq_of_ind T).(-1+1) by Def5;
    hence Af is empty by A2,TARSKI:def 1;
  end;
  assume Af is empty;
  then
A3: Af in (Seq_of_ind T).0 by A2,TARSKI:def 1;
  -1+1=0;
  hence thesis by A1,A3,Def5;
end;
