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 second-countable finite-ind & ind TM <= n
  for F st F is open & F is Cover of TM
  ex G st G is open & G is Cover of TM & G is_finer_than F &
  card G <= card F*(n+1) & order G <= n
proof
  let TM such that
A1: TM is second-countable finite-ind and
A2: ind TM<=n;
  reconsider I=n as Integer;
  consider G be finite Subset-Family of TM such that
A3: G is Cover of TM and
A4: G is finite-ind & ind G<=0 and
A5: card G<=I+1 and
  for a,b be Subset of TM st a in G & b in G & a meets b holds a=b by A1,A2,Th7
;
  set cTM=the carrier of TM;
  let F be finite Subset-Family of TM such that
A6: F is open and
A7: F is Cover of TM;
A8: card((n+1)*card F)=(n+1)*card F;
  defpred P[object,object] means
  for A be Subset of TM st A=$1ex g be Function of F,bool cTM st$2=rng g &
     rng g is open & rng g is Cover of A &
    (for a be set st a in F holds g.a c=a) &
    for a,b be set st a in F & b in F & a<>b holds g.a misses g.b;
A9: for x be object st x in G
ex y be object st y in bool bool cTM & P[x,y]
  proof
    let x be object;
    assume
A10: x in G;
    then reconsider A=x as Subset of TM;
A11: TM|A is second-countable by A1;
    [#]TM c=union F by A7,SETFAM_1:def 11;
    then A c=union F;
    then
A12: F is Cover of A by SETFAM_1:def 11;
    A is finite-ind & ind A<=0 by A4,A10,TOPDIM_1:11;
    then consider g be Function of F,bool the carrier of TM such that
A13: rng g is open & rng g is Cover of A & ((for a be set st a in F holds g
.a c=a) & for a,b be set st a in F & b in F & a<>b holds g.a misses g.b) by A6
,A11,A12,Th22;
    take rng g;
    thus thesis by A13;
  end;
  consider GG be Function of G,bool bool cTM such that
A14: for x be object st x in G holds P[x,GG.x] from FUNCT_2:sch 1(A9);
  union rng GG c=union bool bool cTM by ZFMISC_1:77;
  then reconsider Ugg=Union GG as Subset-Family of TM;
A15: dom GG=G by FUNCT_2:def 1;
A16: for x be object st x in dom GG holds card(GG.x)c=card F
  proof
    let x be object;
    assume
A17: x in dom GG;
    then reconsider A=x as Subset of TM by A15;
    consider g be Function of F,bool cTM such that
A18: GG.x=rng g and
    rng g is open and
    rng g is Cover of A and
    for a be set st a in F holds g.a c=a and
    for a,b be set st a in F & b in F & a<>b holds g.a misses g.b by A14,A17;
    dom g=F by FUNCT_2:def 1;
    hence thesis by A18,CARD_1:12;
  end;
  Segm card dom GG c= Segm(n+1) by A5,A15,NAT_1:39;
  then
A19: card Ugg c=(n+1)*`card F by A16,CARD_2:86;
  card card F=card F & card(n+1)=n+1;
  then
A20: (n+1)*`card F=(n+1)*card F by A8,CARD_2:39;
  then reconsider Ugg as finite Subset-Family of TM by A19;
  take Ugg;
  thus Ugg is open
  proof
    let A be Subset of TM;
    assume A in Ugg;
    then consider Y be set such that
A21: A in Y and
A22: Y in rng GG by TARSKI:def 4;
    consider x be object such that
A23: x in dom GG & Y=GG.x by A22,FUNCT_1:def 3;
    reconsider x as set by TARSKI:1;
    ex g be Function of F,bool cTM st Y=rng g & rng g is open &
rng g is Cover of x &
  (for a be set st a in F holds g.a c=a) &
   for a,b be set st a in F & b in
    F & a<>b holds g.a misses g.b by A14,A15,A23;
    hence thesis by A21;
  end;
  [#]TM c=union Ugg
  proof
A24: [#]TM c=union G by A3,SETFAM_1:def 11;
    let x be object such that
A25: x in [#]TM;
    consider A be set such that
A26: x in A and
A27: A in G by A24,A25,TARSKI:def 4;
    consider g be Function of F,bool cTM such that
A28: GG.A=rng g and
    rng g is open and
A29: rng g is Cover of A and
    for a be set st a in F holds g.a c=a and
    for a,b be set st a in F & b in F & a<>b holds g.a misses g.b by A14,A27;
    A c=union rng g by A29,SETFAM_1:def 11;
    then consider y be set such that
A30: x in y and
A31: y in rng g by A26,TARSKI:def 4;
    GG.A in rng GG by A15,A27,FUNCT_1:def 3;
    then y in Ugg by A28,A31,TARSKI:def 4;
    hence thesis by A30,TARSKI:def 4;
  end;
  hence Ugg is Cover of TM by SETFAM_1:def 11;
  thus Ugg is_finer_than F
  proof
    let A be set;
    assume A in Ugg;
    then consider Y be set such that
A32: A in Y and
A33: Y in rng GG by TARSKI:def 4;
    consider x be object such that
A34: x in dom GG & Y=GG.x by A33,FUNCT_1:def 3;
    reconsider x as set by TARSKI:1;
    consider g be Function of F,bool cTM such that
A35: Y=rng g and
    rng g is open and
    rng g is Cover of x and
A36: for a be set st a in F holds g.a c=a and
    for a,b be set st a in F & b in F & a<>b holds g.a misses g.b by A14,A15
,A34;
    dom g=F by FUNCT_2:def 1;
    then ex z be object st z in F & g.z=A by A32,A35,FUNCT_1:def 3;
    hence thesis by A36;
  end;
   Segm card Ugg c= Segm(card F*(n+1)) by A19,A20;
  hence card Ugg<=card F*(n+1) by NAT_1:39;
  defpred Q[object,object] means
  $1 in GG.$2;
  now
    let H be finite Subset-Family of TM such that
A37: H c=Ugg and
A38: card H>n+1;
    H is non empty by A38;
    then consider XX be object such that
A39: XX in H;
A40: for x be object st x in H ex y be object st y in G & Q[x,y]
    proof
      let A be object;
      assume A in H;
      then consider Y be set such that
A41:  A in Y and
A42:  Y in rng GG by A37,TARSKI:def 4;
      ex x be object st x in dom GG & Y=GG.x by A42,FUNCT_1:def 3;
      hence thesis by A41;
    end;
    consider q be Function of H,G such that
A43: for x be object st x in H holds Q[x,q.x] from FUNCT_2:sch 1(A40);
    ex y be object st y in G & Q[XX,y] by A40,A39;
    then
A44: dom q=H by FUNCT_2:def 1;
    assume meet H is non empty;
    then consider x be object such that
A45: x in meet H;
    for x1,x2 be object st x1 in dom q & x2 in dom q & q.x1=q.x2 holds x1=x2
    proof
      let x1,x2 be object such that
A46:  x1 in dom q and
A47:  x2 in dom q and
A48:  q.x1=q.x2;
      reconsider x1,x2 as set by TARSKI:1;
      x in x1 & x in x2 by A45,A46,A47,SETFAM_1:def 1;
      then
A49:  x1 meets x2 by XBOOLE_0:3;
      q.x1 in rng q by A46,FUNCT_1:def 3;
      then q.x1 in G;
      then consider g be Function of F,bool cTM such that
A50:  GG.(q.x1)=rng g and
      rng g is open and
      rng g is Cover of(q.x1) and
      for a be set st a in F holds g.a c=a and
A51:  for a,b be set st a in F & b in F & a<>b holds g.a misses g.b by A14;
A52:  dom g=F by FUNCT_2:def 1;
      x2 in GG.(q.x1) by A43,A47,A48;
      then
A53:  ex y2 be object st y2 in F & x2=g.y2 by A50,A52,FUNCT_1:def 3;
      x1 in GG.(q.x1) by A43,A46;
      then ex y1 be object st y1 in F & x1=g.y1 by A50,A52,FUNCT_1:def 3;
      hence thesis by A49,A51,A53;
    end;
    then
A54: q is one-to-one by FUNCT_1:def 4;
    rng q c=G;
    then Segm card H c= Segm card G by A54,A44,CARD_1:10;
    then card H<=card G by NAT_1:39;
    hence contradiction by A5,A38,XXREAL_0:2;
  end;
  hence thesis by Th3;
end;
