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;
reserve TM for metrizable TopSpace;

theorem
  for Null,A be Subset of TM st TM|Null is second-countable & Null is
finite-ind & A is finite-ind & ind Null<=0 holds A\/Null is finite-ind & ind(A
  \/Null)<=ind A+1
proof
  let Null,A be Subset of TM such that
A1: TM|Null is second-countable and
A2: Null is finite-ind and
A3: A is finite-ind and
A4: ind Null<=0;
  set TAN=TM|(A\/Null);
A5: [#]TAN=A\/Null by PRE_TOPC:def 5;
  then reconsider N9=Null,A9=A as Subset of TAN by XBOOLE_1:7;
A6: ind N9=ind Null by A2,Th21;
  N9 is finite-ind & TAN|N9 is second-countable by A1,A2,Th21,METRIZTS:9;
  then consider B be Basis of TAN such that
A7: for b be Subset of TAN st b in B holds N9 misses Fr b by A4,A6,Th39;
  set i=ind A;
  -1<=i by A3,Th5;
  then -1+1<=i+1 by XREAL_1:6;
  then reconsider i1=i+1 as Element of NAT by INT_1:3;
A8: A9 is finite-ind & ind A9=ind A by A3,Th21;
A9: for b be Subset of TAN st b in B holds Fr b is finite-ind & ind Fr b<= i1-1
  proof
    let b be Subset of TAN;
    assume b in B;
    then N9 misses Fr b by A7;
    then Fr b c=A9 by A5,XBOOLE_1:73;
    hence thesis by A8,Th19;
  end;
  then TAN is finite-ind by Th31;
  then
A10: A\/Null is finite-ind by Th18;
  ind TAN<=i1 by A9,Th31;
  hence thesis by A10,Lm5;
end;
