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 Th37:
  for A,B be closed Subset of TM st A misses B for Null be
  finite-ind Subset of TM st ind Null<=0 & TM|Null is second-countable ex L be
  Subset of TM st L separates A,B & L misses Null
proof
  let A,B be closed Subset of TM;
  assume A misses B;
  then consider U,W be open Subset of TM such that
A1: A c=U & B c=W and
A2: Cl U misses Cl W by Th2;
  let Null be finite-ind Subset of TM such that
A3: ind Null<=0 & TM|Null is second-countable;
  set TN=TM|Null;
A4: [#]TN=Null by PRE_TOPC:def 5;
  then reconsider Un=Cl U/\Null,Wn=Cl W/\Null as Subset of TN by XBOOLE_1:17;
  Un c=Cl U & Wn c=Cl W by XBOOLE_1:17;
  then
A5: Un misses Wn by A2,XBOOLE_1:64;
A6: ind TN=ind Null by Lm5;
  Un is closed & Wn is closed by A4,TSP_1:def 2;
  then {}TN separates Un,Wn by A3,A5,A6,Th35;
  then consider L be Subset of TM such that
A7: L separates A,B and
A8: Null/\L c={}TN by A1,A2,METRIZTS:26;
  take L;
  Null/\L={} by A8;
  hence thesis by A7;
end;
