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
  for Ga be Subset-Family of T|A st G is finite-ind & Ga = G holds Ga is
  finite-ind & ind G = ind Ga
proof
  let G1 be Subset-Family of T|A such that
A1: G is finite-ind and
A2: G1=G;
A3: for B be Subset of T|A st B in G1 holds B is finite-ind & ind B <=ind G
  proof
    let B be Subset of T|A;
    assume
A4: B in G1;
    then reconsider B9=B as Subset of T by A2;
A5: B9 is finite-ind by A1,A2,A4;
    then ind B=ind B9 by Th21;
    hence thesis by A1,A2,A4,A5,Th11,Th21;
  end;
A6: -1<=ind G by A1,Th11;
  then
A7: ind G1<=ind G by A3,Th11;
A8: G1 is finite-ind by A3,A6,Th11;
A9: for B be Subset of T st B in G holds B is finite-ind & ind B<= ind G1
  proof
    let B be Subset of T;
    assume
A10: B in G;
    then reconsider B9=B as Subset of T|A by A2;
    B9 is finite-ind by A2,A3,A10;
    then ind B=ind B9 by Th22;
    hence thesis by A1,A2,A8,A10,Th11;
  end;
  -1<=ind G1 by A8,Th11;
  then ind G<=ind G1 by A9,Th11;
  hence thesis by A3,A6,A7,Th11,XXREAL_0:1;
end;
