reserve n for Nat,
        X for set,
        Fx,Gx for Subset-Family of X;
reserve TM for metrizable TopSpace,
        TM1,TM2 for finite-ind second-countable metrizable TopSpace,
        A,B,L,H for Subset of TM,
        U,W for open Subset of TM,
        p for Point of TM,

        F,G for finite Subset-Family of TM,
        I for Integer;

theorem
  for TM st TM is finite-ind second-countable & ind TM<=n
  for A,B st A is closed & B is closed & A misses B
  ex L st L separates A,B & ind L<=n-1
proof
  let TM such that
A1: TM is finite-ind second-countable and
A2: ind TM<=n;
A3: TM| ([#]TM) is second-countable by A1;
  let A,B such that
A4: A is closed & B is closed and
A5: A misses B;
  consider L be Subset of TM such that
A6: L separates A,B & ind(L/\[#]TM)<=n-1 by A4,A1,A2,A3,A5,Th11;
  take L;
  thus thesis by A6,XBOOLE_1:28;
end;
