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;
reserve u for Point of Euclid 1,
  U for Point of TOP-REAL 1,
  r,u1 for Real,
  s for Real;

theorem
  for TM st TM is finite-ind for A st ind A`<=n & TM|A` is second-countable
  for A1,A2 be closed Subset of TM st A=A1\/A2
  ex X1,X2 be closed Subset of TM st[#]TM=X1\/X2 & A1 c=X1 & A2 c=X2 &
  A1/\X2=A1/\A2 & A1/\A2=X1/\A2 & ind((X1/\X2)\(A1/\A2))<=n-1
proof
  let TM such that
A1: TM is finite-ind;
  set cTM=[#]TM;
  let A such that
A2: ind A`<=n and
A3: TM|A` is second-countable;
  let A1,A2 be closed Subset of TM such that
A4: A=A1\/A2;
  set A12=A1/\A2;
  set T912=TM| (cTM\A12);
A5: [#]T912=cTM\A12 by PRE_TOPC:def 5;
  A`c=cTM\A12 by A4,XBOOLE_1:29,34;
  then reconsider A19=A1\A12,A29=A2\A12,A9=A` as Subset of T912 by A5,
XBOOLE_1:33;
  A2/\[#]T912=(A2/\cTM)\A12 by A5,XBOOLE_1:49
    .=A29 by XBOOLE_1:28;
  then reconsider A29 as closed Subset of T912 by PRE_TOPC:13;
  A1/\[#]T912=(A1/\cTM)\A12 by A5,XBOOLE_1:49
    .=A19 by XBOOLE_1:28;
  then reconsider A19 as closed Subset of T912 by PRE_TOPC:13;
  A1\A2=A19 by XBOOLE_1:47;
  then A19 misses A2 by XBOOLE_1:79;
  then
A6: A19 misses A29 by XBOOLE_1:36,63;
A7: ind A`=ind A9 by A1,TOPDIM_1:21;
  T912|A9 is second-countable by A3,METRIZTS:9;
  then consider L be Subset of T912 such that
A8: L separates A19,A29 and
A9: ind(L/\A9)<=n-1 by A1,A2,A6,A7,Th11;
  consider U,W be open Subset of T912 such that
A10: A19 c=U and
A11: A29 c=W and
A12: U misses W and
A13: L=(U\/W)` by A8,METRIZTS:def 3;
  [#]T912 c=cTM by PRE_TOPC:def 4;
  then reconsider L9=L,U9=U,W9=W as Subset of TM by XBOOLE_1:1;
A14: A1=(A1\A12)\/A12 & A2=(A2\A12)\/A12 by XBOOLE_1:17,45;
  L misses A
  proof
    assume L meets A;
    then consider x be object such that
A15: x in L and
A16: x in A by XBOOLE_0:3;
A17: x in A1 or x in A2 by A4,A16,XBOOLE_0:def 3;
    not x in A12 by A5,A15,XBOOLE_0:def 5;
    then x in A19 or x in A29 by A17,XBOOLE_0:def 5;
    then x in U\/W by A10,A11,XBOOLE_0:def 3;
    hence thesis by A13,A15,XBOOLE_0:def 5;
  end;
  then
A18: L=L9\A by XBOOLE_1:83
    .=(L/\cTM)\A by XBOOLE_1:28
    .=L/\A9 by XBOOLE_1:49;
A19: cTM\(cTM\A12)=cTM/\A12 by XBOOLE_1:48
    .=A12 by XBOOLE_1:28;
A20: A1\(A1\A12)=A1/\A12 by XBOOLE_1:48
    .=A12 by XBOOLE_1:17,28;
  A12` is open;
  then reconsider U9,W9 as open Subset of TM by A5,TSEP_1:9;
  take X2=W9`,X1=U9`;
A21: W9 c=X1 by A12,SUBSET_1:23;
A22: W\/(cTM\(U\/W))=W\/(X1/\X2) by XBOOLE_1:53
    .=(W9\/X1)/\(W\/X2) by XBOOLE_1:24
    .=(W9\/X1)/\cTM by XBOOLE_1:45
    .=X1/\cTM by A21,XBOOLE_1:12
    .=X1 by XBOOLE_1:28;
  thus X2\/X1=cTM\(U9/\W9) by XBOOLE_1:54
    .=cTM\{} by A12
    .=cTM;
A23: U9 c=X2 by A12,SUBSET_1:23;
A24: U\/(cTM\(U\/W))=U\/((X1)/\X2) by XBOOLE_1:53
    .=(U9\/X1)/\(U\/X2) by XBOOLE_1:24
    .=cTM/\(U\/X2) by XBOOLE_1:45
    .=cTM/\X2 by A23,XBOOLE_1:12
    .=X2 by XBOOLE_1:28;
  cTM\(cTM\A12)c=cTM\(U\/W) by A5,XBOOLE_1:34;
  hence
A25: A1 c=X2 & A2 c=X1 by A10,A11,A14,A19,A24,A22,XBOOLE_1:13;
  then
A26: A12 c=A1/\X1 by XBOOLE_1:26;
  A1/\X1=(cTM/\A1)\U9 by XBOOLE_1:49
    .=A1\U9 by XBOOLE_1:28;
  then A1/\X1 c=A12 by A10,A20,XBOOLE_1:34;
  hence A1/\X1=A12 by A26;
A27: A2\(A2\A12)=A2/\A12 by XBOOLE_1:48
    .=A12 by XBOOLE_1:17,28;
A28: A12 c=A2/\X2 by A25,XBOOLE_1:26;
  A2/\X2=(cTM/\A2)\W9 by XBOOLE_1:49
    .=A2\W9 by XBOOLE_1:28;
  then A2/\X2 c=A12 by A11,A27,XBOOLE_1:34;
  hence A12=X2/\A2 by A28;
  (X2/\X1)\A12=cTM\(W9\/U9)\A12 by XBOOLE_1:53
    .=cTM\((W9\/U9)\/A12) by XBOOLE_1:41
    .=L by A5,A13,XBOOLE_1:41;
  hence thesis by A1,A9,A18,TOPDIM_1:21;
end;
