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 Th19:
  A c= Af implies A is finite-ind & ind A <= ind Af
proof
  assume
A1: A c=Af;
  [#](T|Af)=Af by PRE_TOPC:def 5;
  then reconsider A9=A as Subset of T|Af by A1;
A2: ind T|Af=ind Af by Lm5;
A3: T|Af|A9=T|A by METRIZTS:9;
  hence A is finite-ind by Th18;
  then ind T|A=ind A by Lm5;
  hence thesis by A2,A3,Lm6;
end;
