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
  G1 is finite-ind & G2 c= G1 implies G2 is finite-ind & ind G2 <= ind G1
proof
  assume that
A1: G1 is finite-ind and
A2: G2 c=G1;
A3: -1<=ind G1 by A1,Th11;
  then for A st A in G2 holds A is finite-ind & ind A<=ind G1 by A1,A2,Th11;
  hence thesis by A3,Th11;
end;
