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 Th5:
  -1 <= ind Af
proof
  Af in (Seq_of_ind T).(1+ind Af) by Def5;
  then
A1: (ind Af)+1 in dom(Seq_of_ind T) by FUNCT_1:def 2;
  0=-1+1;
  hence thesis by A1,XREAL_1:6;
end;
