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 A1 be Subset of T1,A2 be Subset of T2 st A1,A2 are_homeomorphic &
  A1 is finite-ind holds ind A1 = ind A2
proof
  let A1 be Subset of T1,A2 be Subset of T2 such that
A1: A1,A2 are_homeomorphic and
A2: A1 is finite-ind;
  T1|A1,T2|A2 are_homeomorphic by A1,METRIZTS:def 1;
  then
A3: ind T1|A1=ind T2|A2 by A2,Lm9;
  A2 is finite-ind & ind T1|A1=ind A1 by A1,A2,Lm5,Th26;
  hence thesis by A3,Lm5;
end;
