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 Th7:
  ind Af <= n-1 iff Af in (Seq_of_ind T).n
proof
  set I=ind Af;
A1: Af in (Seq_of_ind T).(I+1) by Def5;
A2: I+1 is Nat by Lm3;
  hereby
    assume I<=n-1;
    then
A3: I+1<=n-1+1 by XREAL_1:6;
    I+1 in NAT & n in NAT by A2,ORDINAL1:def 12;
    then (Seq_of_ind T).(I+1)c=(Seq_of_ind T).n by A3,PROB_1:def 5;
    hence Af in (Seq_of_ind T).n by A1;
  end;
  assume
A4: Af in (Seq_of_ind T).n;
  assume I>n-1;
  then
A5: I>=n-1+1 by INT_1:7;
  then n in NAT & I in NAT by INT_1:3;
  then (Seq_of_ind T).n c=(Seq_of_ind T).I by A5,PROB_1:def 5;
  hence contradiction by A4,Def5;
end;
