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 Th8:
  for TM st TM is second-countable &
    ex F st F is Cover of TM & F is finite-ind & ind F<=0 & card F <= I+1
  holds TM is finite-ind & ind TM <= I
proof
  defpred P[Nat] means
  for TM st TM is second-countable & ex F be finite Subset-Family of TM st F
  is Cover of TM & F is finite-ind & ind F<=0 & card F<=$1 holds TM is
  finite-ind & ind TM<=$1-1;
  let TM  such that
A1: TM is second-countable;
  given F be finite Subset-Family of TM such that
A2: F is Cover of TM & F is finite-ind & ind F<=0 and
A3: card F<=I+1;
  reconsider i1=I+1 as Element of NAT by A3,INT_1:3;
A4: P[0]
  proof
    let TM such that
    TM is second-countable;
    given F be finite Subset-Family of TM such that
A5: F is Cover of TM and
    F is finite-ind and
    ind F<=0 and
A6: card F<=0;
    F={} by A6;
    then [#]TM={} by A5,SETFAM_1:45,ZFMISC_1:2;
    hence thesis by TOPDIM_1:6;
  end;
A7: for n st P[n] holds P[n+1]
  proof
    let n such that
A8: P[n];
    let TM such that
A9: TM is second-countable;
    given F be finite Subset-Family of TM such that
A10: F is Cover of TM and
A11: F is finite-ind and
A12: ind F<=0 and
A13: card F<=n+1;
    per cases;
    suppose F={};
      then card F=0;
      hence thesis by A4,A9,A10,A11,A12;
    end;
    suppose F<>{};
      then consider A be object such that
A14:  A in F by XBOOLE_0:def 1;
      reconsider A as Subset of TM by A14;
      set AA={A};
      set FA=F\AA;
A15:  FA\/AA=F by A14,ZFMISC_1:116;
A16:  [#]TM=union F by A10,SETFAM_1:45
        .=union FA\/union AA by A15,ZFMISC_1:78
        .=union FA\/A by ZFMISC_1:25;
A17:  FA c=F by XBOOLE_1:36;
      then
A18:  ind FA<=0 by A11,A12,TOPDIM_1:12;
      not A in FA by ZFMISC_1:56;
      then FA c<F by A14,A17;
      then card FA<n+1 by A13,CARD_2:48,XXREAL_0:2;
      then
A19:  card FA<=n by NAT_1:13;
      reconsider uFA=union FA as Subset of TM;
      set Tu=TM|uFA;
A20:  [#]Tu=uFA by PRE_TOPC:def 5;
      then reconsider FA9=FA as Subset-Family of Tu by ZFMISC_1:82;
A21:  TM|A is second-countable by A9;
      FA is finite-ind by A11,A17,TOPDIM_1:12;
      then
A22:  FA9 is finite-ind & ind FA=ind FA9 by TOPDIM_1:30;
A23:  FA9 is Cover of Tu by A20,SETFAM_1:def 11;
      then
A24:  Tu is finite-ind by A8,A9,A18,A19,A22;
      then
A25:  ind Tu=ind uFA by A20,TOPDIM_1:22;
      ind Tu<=n-1 by A8,A9,A18,A19,A22,A23;
      then
A26:  ind uFA+1<=n-1+1 by A25,XREAL_1:6;
A27:  uFA is finite-ind by A24,TOPDIM_1:18;
      A is finite-ind & ind A<=0 by A11,A12,A14,TOPDIM_1:11;
      then [#]TM is finite-ind & ind[#]TM<=ind uFA+1 by A16,A21,A27,TOPDIM_1:40
;
      hence thesis by A26,XXREAL_0:2;
    end;
  end;
  for n holds P[n] from NAT_1:sch 2(A4,A7);
  then P[i1];
  hence thesis by A1,A2,A3;
end;
