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 Th18:
  T|A is finite-ind implies A is finite-ind
proof
  assume T|A is finite-ind;
  then consider n such that
A1: [#](T|A) in (Seq_of_ind(T|A)).n by Def2;
  [#](T|A)=A by PRE_TOPC:def 5;
  then A in (Seq_of_ind T).n by A1,Th3;
  hence thesis;
end;
