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 Th9:
  ind Af <= n iff for p be Point of T|Af,U be open Subset of T|Af
st p in U ex W be open Subset of T|Af st p in W & W c= U & Fr W is finite-ind &
  ind Fr W <= n-1
proof
  set I=ind Af;
A1: [#](T|Af)c=[#]T by PRE_TOPC:def 4;
A2: Af in (Seq_of_ind T).(I+1) & not Af in (Seq_of_ind T).I by Def5;
  hereby
    assume
A3: I<=n;
    let p be Point of T|Af,U be open Subset of T|Af such that
A4: p in U;
    Af is non empty & T is non empty by A4;
    then reconsider I as Nat by TARSKI:1;
A5: I-1<=n-1 by A3,XREAL_1:9;
    consider W be open Subset of T|Af such that
A6: p in W & W c=U and
A7: Fr W in (Seq_of_ind T).I by A2,A4,Def1;
    take W;
A8: Fr W in (Seq_of_ind T|Af).I by A7,Th3;
    then Fr W is finite-ind;
    then ind Fr W<=I-1 by A8,Th7;
    hence p in W & W c=U & Fr W is finite-ind & ind Fr W<=n-1 by A5,A6,A8,
XXREAL_0:2;
  end;
  assume
A9: for p be Point of T|Af,U be open Subset of T|Af st p in U ex W be
  open Subset of T|Af st p in W & W c=U & Fr W is finite-ind & ind Fr W <=n-1;
  now
    let p be Point of T|Af,U be open Subset of T|Af;
    assume p in U;
    then consider W be open Subset of T|Af such that
A10: p in W & W c=U and
A11: Fr W is finite-ind & ind Fr W<=n-1 by A9;
A12: Fr W is Subset of T by A1,XBOOLE_1:1;
    Fr W in (Seq_of_ind T|Af).n by A11,Th7;
    then Fr W in (Seq_of_ind T).n by A12,Th3;
    hence
    ex W be open Subset of T|Af st p in W & W c=U & Fr W in (Seq_of_ind T
    ).n by A10;
  end;
  then Af in (Seq_of_ind T).(n+1) by Def1;
  then ind Af<=n+1-1 by Th7;
  hence thesis;
end;
