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
  for X be non empty set holds ind 1TopSp X = 0
proof
  let X be non empty set;
  set T=1TopSp X;
  (Seq_of_ind T).0={{}T} by Def1;
  then
A1: not[#]T in (Seq_of_ind T).0 by TARSKI:def 1;
A2: [#]T in (Seq_of_ind T).(0+1) by Lm2;
  then [#]T is finite-ind;
  hence thesis by A1,A2,Def5;
end;
