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 Th7:
  for TM st TM is second-countable finite-ind & ind TM <= I
  ex F st F is Cover of TM & F is finite-ind & ind F <= 0 & card F <= I+1
         & for A,B st A in F & B in F & A meets B holds A = B
proof
  defpred P[Nat] means
  for TM st TM is second-countable finite-ind & ind TM<=$1-1
  ex F be finite Subset-Family of TM st
        F is Cover of TM &  F is finite-ind &
        ind F<=0 & card F<=$1 & for A,B be Subset of TM st
        A in F & B in F & A meets B holds A=B;
  let TM such that
A1: TM is second-countable finite-ind and
A2: ind TM<=I;
  -1<=ind[#]TM by A1,TOPDIM_1:5;
  then -1<=I by A2,XXREAL_0:2;
  then -1+1<=I+1 by XREAL_1:6;
  then reconsider i1=I+1 as Element of NAT by INT_1:3;
A3: for n st P[n] holds P[n+1]
  proof
    let n such that
A4: P[n];
    let TM such that
A5: TM is second-countable and
A6: TM is finite-ind and
A7: ind TM<=n+1-1;
    consider A,B be Subset of TM such that
A8: [#]TM=A\/B and
A9: A misses B and
A10: ind A<=n-1 and
A11: ind B<=0 by A5,A6,A7,Lm3;
    set BB={B};
    for b be Subset of TM st b in BB holds b is finite-ind & ind b<=0
    by A6,A11,TARSKI:def 1;
    then
A12: BB is finite-ind & ind BB<=0 by TOPDIM_1:11;
    set TA=TM|A;
    ind TA=ind A by A6,TOPDIM_1:17;
    then consider F be finite Subset-Family of TA such that
A13: F is Cover of TA and
A14: F is finite-ind and
A15: ind F<=0 and
A16: card F<=n and
A17: for A,B be Subset of TA st A in F & B in F & A meets B holds A=B by A4,A5
,A6,A10;
    [#]TA c=[#]TM by PRE_TOPC:def 4;
    then bool[#]TA c=bool[#]TM by ZFMISC_1:67;
    then reconsider F9=F as finite Subset-Family of TM by XBOOLE_1:1;
    take G=F9\/BB;
A18: union F9=[#]TA by A13,SETFAM_1:45
      .=A by PRE_TOPC:def 5;
    then union G=A\/union BB by ZFMISC_1:78
      .=[#]TM by A8,ZFMISC_1:25;
    hence G is Cover of TM by SETFAM_1:def 11;
    F9 is finite-ind & ind F9=ind F by A14,TOPDIM_1:29;
    hence G is finite-ind & ind G<=0 by A12,A15,TOPDIM_1:13;
    card BB=1 by CARD_1:30;
    then
A19: card G<=card F9+1 by CARD_2:43;
    card F9+1<=n+1 by A16,XREAL_1:6;
    hence card G<=n+1 by A19,XXREAL_0:2;
    let a,b be Subset of TM such that
A20: a in G & b in G and
A21: a meets b;
    per cases by A20,XBOOLE_0:def 3;
    suppose a in F & b in F;
      hence thesis by A17,A21;
    end;
    suppose
A22:  a in F & b in BB;
      then b=B by TARSKI:def 1;
      hence thesis by A9,A18,A21,A22,XBOOLE_1:63,ZFMISC_1:74;
    end;
    suppose
A23:  a in BB & b in F;
      then a=B by TARSKI:def 1;
      hence thesis by A9,A18,A21,A23,XBOOLE_1:63,ZFMISC_1:74;
    end;
    suppose
A24:  a in BB & b in BB;
      then a=B by TARSKI:def 1;
      hence thesis by A24,TARSKI:def 1;
    end;
  end;
A25: P[0]
  proof
    let TM such that
    TM is second-countable and
A26: TM is finite-ind and
A27: ind TM<=0-1;
    ind[#]TM>=-1 by A26,TOPDIM_1:5;
    then ind[#]TM=-1 by A27,XXREAL_0:1;
    then
A28: [#]TM={}TM by A26,TOPDIM_1:6;
    reconsider F={} as empty Subset-Family of TM by TOPGEN_4:18;
    take F;
    F c={{}TM};
    hence thesis by A28,SETFAM_1:def 11,TOPDIM_1:10,ZFMISC_1:2;
  end;
  for n holds P[n] from NAT_1:sch 2(A25,A3);
  then P[i1];
  hence thesis by A1,A2;
end;
