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;
