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 Th15:
  (ex n st for p be Point of T,U be open Subset of T st p in U ex
W be open Subset of T st p in W & W c= U & Fr W is finite-ind & ind Fr W <= n-1
  ) implies T is finite-ind
proof
  given n such that
A1: for p be Point of T,U be open Subset of T st p in U ex W be open
  Subset of T st p in W & W c=U & Fr W is finite-ind & ind Fr W<=n-1;
  set CT=[#]T;
  set TT=T|CT;
A2: [#]TT=CT by PRE_TOPC:def 5;
  T is SubSpace of T by TSEP_1:2;
  then
A3: the TopStruct of T=the TopStruct of TT by A2,TSEP_1:5;
  now
    let p9 be Point of TT,U9 be open Subset of TT such that
A4: p9 in U9;
    reconsider p=p9 as Point of T by A2;
    U9 in the topology of TT by PRE_TOPC:def 2;
    then reconsider U=U9 as open Subset of T by A3,PRE_TOPC:def 2;
    consider W be open Subset of T such that
A5: p in W and
A6: W c=U & Fr W is finite-ind & ind Fr W<=n-1 by A1,A4;
    W in the topology of T by PRE_TOPC:def 2;
    then reconsider W9=W as open Subset of TT by A3,PRE_TOPC:def 2;
    take W9;
    T is non empty by A5;
    then Cl W=Cl W9 & Int W=Int W9 by A2,TOPS_3:54;
    then Fr W=Cl W9\Int W9 by TOPGEN_1:8
      .=Fr W9 by TOPGEN_1:8;
    hence p9 in W9 & W9 c=U9 & Fr W9 in (Seq_of_ind T).n by A5,A6,Th7;
  end;
  then CT in (Seq_of_ind T).(n+1) by Def1;
  then CT is finite-ind;
  hence thesis;
end;
