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 Th5:
  for A,B be finite-ind Subset of TM st A is closed &
  TM| (A\/B) is second-countable & ind A<=I & ind B<=I holds
  ind(A\/B)<=I & A\/B is finite-ind
proof
  let A,B be finite-ind Subset of TM such that
A1: A is closed and
A2: TM| (A\/B) is second-countable and
A3: ind A<=I and
A4: ind B<=I;
  -1<=ind A by TOPDIM_1:5; then
A5: -1<=I by A3,XXREAL_0:2;
  per cases;
  suppose A\/B is empty;
    hence thesis by A5,TOPDIM_1:6;
  end;
  suppose
A6: A\/B is non empty; then
A7: TM is non empty;
    A is non empty or B is non empty by A6;
    then 0<=ind A or 0<=ind B by A7;
    then
A8: I in NAT by A3,A4,INT_1:3;
    reconsider AB=A\/B as Subset of TM;
    set Tab=TM|AB;
A9: [#]Tab=AB by PRE_TOPC:def 5;
    then reconsider a=A,b=B as Subset of Tab by XBOOLE_1:7;
    A/\[#]Tab=a by XBOOLE_1:28;
    then
A10: a is closed by A1,TSP_1:def 2;
    then consider F be closed countable Subset-Family of Tab such that
A11: a`=union F by TOPGEN_4:def 6;
    reconsider a,b as finite-ind Subset of Tab by TOPDIM_1:21;
    reconsider AA={a} as Subset-Family of Tab;
    union(AA\/F)=union AA\/union F by ZFMISC_1:78
      .=a\/a` by A11,ZFMISC_1:25
      .=[#]Tab by PRE_TOPC:2;
    then
A12: AA\/F is Cover of Tab by SETFAM_1:def 11;
    AA is closed
    by A10,TARSKI:def 1;
    then
A13: AA\/F is closed by TOPS_2:16;
    for D be Subset of Tab st D in AA\/F holds D is finite-ind&ind D<=I
    proof
      let D be Subset of Tab such that
A14:  D in AA\/F;
      per cases by A14,XBOOLE_0:def 3;
      suppose D in AA;
        then D=a by TARSKI:def 1;
        hence thesis by A3,TOPDIM_1:21;
      end;
      suppose
A15:    D in F;
        a`=b\a by A9,XBOOLE_1:40;
        then
A16:    a`c=b by XBOOLE_1:36;
        D c=a` by A11,A15,ZFMISC_1:74;
        then
A17:    D c=b by A16;
        then ind b=ind B & ind D<=ind b by TOPDIM_1:19,21;
        hence thesis by A4,A17,TOPDIM_1:19,XXREAL_0:2;
      end;
    end;
    then
A18: AA\/F is finite-ind & ind(AA\/F)<=I by A5,TOPDIM_1:11;
A19: AA\/F is countable by CARD_2:85;
    then Tab is finite-ind by A2,A8,A12,A13,A18,Lm3;
    then
A20: A\/B is finite-ind by TOPDIM_1:18;
    ind Tab<=I by A2,A8,A12,A13,A18,A19,Lm3;
    hence thesis by A20,TOPDIM_1:17;
  end;
end;
