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 Th11:
  for A,B st A is closed & B is closed & A misses B
  for H st ind H<=n & TM|H is second-countable finite-ind
    ex L st L separates A,B & ind(L/\H) <= n-1
proof
  let A,B such that
A1: A is closed & B is closed and
A2: A misses B;
  let H such that
A3: ind H<=n and
A4: TM|H is second-countable finite-ind;
  H is finite-ind by A4,TOPDIM_1:18;
  then ind H=ind(TM|H) by TOPDIM_1:17;
  then consider a,b be Subset of TM|H such that
A5: [#](TM|H)=a\/b and
  a misses b and
A6: ind a<=n-1 and
A7: ind b<=0 by A3,A4,Lm3;
  [#](TM|H)c=[#]TM by PRE_TOPC:def 4;
  then reconsider aa=a,bb=b as Subset of TM by XBOOLE_1:1;
A8: bb is finite-ind by A4,TOPDIM_1:22;
A9: H=aa\/bb by A5,PRE_TOPC:def 5;
  then
A10: bb/\H=bb by XBOOLE_1:7,28;
  ind b=ind bb & TM|H|b=TM|bb by A4,METRIZTS:9,TOPDIM_1:22;
  then consider L be Subset of TM such that
A11: L separates A,B and
A12: L misses bb by A1,A2,A4,A7,A8,TOPDIM_1:37;
  take L;
  L/\H misses bb & L/\H c= H by A10,A12,XBOOLE_1:17,76;
  then aa is finite-ind & L/\H c=aa by A4,A9,TOPDIM_1:22,XBOOLE_1:73;
  then ind a=ind aa & ind(L/\H)<=ind aa by TOPDIM_1:19,21;
  hence thesis by A6,A11,XXREAL_0:2;
end;
