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 Th22:
  F = B & F is finite-ind implies B is finite-ind & ind F = ind B
proof
  assume that
A1: F=B and
A2: F is finite-ind;
A3: T|A|F=T|B by A1,METRIZTS:9;
  then
A4: B is finite-ind by A2,Th18;
  ind F=ind(T|A|F) by A2,Lm5
    .=ind(T|B) by A1,METRIZTS:9
    .=ind B by A4,Lm5;
  hence thesis by A2,A3,Th18;
end;
