reserve i for Nat;
reserve R for Relation;
reserve A for set;
reserve PT for non empty TopSpace;
reserve PM for MetrSpace;
reserve FX,GX,HX for Subset-Family of PT;
reserve Y,V,W for Subset of PT;
reserve Mn for Relation;
reserve n,k,l,q,p,q1 for Nat;

theorem Th6:
  PT is metrizable implies for FX being Subset-Family of PT st FX
is Cover of PT & FX is open ex GX being Subset-Family of PT st GX is open & GX
  is Cover of PT & GX is_finer_than FX & GX is locally_finite
proof
  assume PT is metrizable;
  then consider
  metr being Function of [:the carrier of PT,the carrier of PT:],REAL
  such that
A1: metr is_metric_of (the carrier of PT) and
A2: Family_open_set( SpaceMetr (the carrier of PT,metr) ) = the topology
  of PT by PCOMPS_1:def 8;
  let FX;
  consider R such that
A3: R well_orders FX by WELLORD2:17;
  defpred P1[set] means $1 in FX;
  consider Mn such that
A4: Mn = R |_2 FX;
  assume that
A5: FX is Cover of PT and
A6: FX is open;
  consider PM being MetrSpace such that
A7: PM = SpaceMetr(the carrier of PT,metr);
  reconsider PM as non empty MetrSpace by A1,A7,PCOMPS_1:36;
  deffunc F1(Element of PM,Nat) = Ball($1,1/(2 |^($2+1)));
  set UB = {union {Ball(c,jd) where c is Element of PM: c in V\PartUnion(V,Mn
  ) & Ball(c,3/2) c= V} where V is Subset of PM: V in FX};
  UB is Subset-Family of PM
  proof
    reconsider UB as set;
    UB c= bool the carrier of PM
    proof
      let x be object;
       reconsider xx=x as set by TARSKI:1;
      assume x in UB;
      then consider V be Subset of PM such that
A8:   x = union {Ball(c,jd) where c is Element of PM: c in V\
      PartUnion(V,Mn) & Ball(c,3/2) c= V} and
      V in FX;
      xx c= the carrier of PM
      proof
        let y be object;
        assume y in xx;
        then consider W be set such that
A9:     y in W and
A10:    W in {Ball(c,jd) where c is Element of PM: c in V\PartUnion(
        V,Mn) & Ball(c,3/2) c= V} by A8,TARSKI:def 4;
        ex c be Element of PM st W = Ball(c,jd) & c in V\ PartUnion(V,Mn)
        & Ball(c,3/2) c= V by A10;
        hence thesis by A9;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
  then reconsider UB as Subset-Family of PM;
  defpred Q1[Element of PM, Subset of PM,Nat] means $1 in $2\
  PartUnion($2,Mn) & Ball($1,3/(2 |^ ($3+1))) c= $2;
  consider f being sequence of bool bool the carrier of PM such that
A11: f.0 = UB and
A12: for n being Nat
  holds f.(n+1) = {union { F1(c,n) where c is Element of PM: Q1
[c,V,n] & not c in union{union (f.q): q <= n } } where V is Subset of PM: P1[V]
  } from XXX;
  defpred P2[set] means ex n being Nat st $1 in f.n;
  consider GX being Subset-Family of PM such that
A13: for X being Subset of PM holds X in GX iff P2[X] from SUBSET_1:sch
  3;
  reconsider GX as Subset-Family of PT by A1,A7,Th4;
  take GX;
A14: for X being Subset of PT st X in GX holds X is open
  proof
    let X be Subset of PT;
    assume
A15: X in GX;
    then reconsider X as Subset of PM;
    consider n  being Nat such that
A16: X in f.n by A13,A15;
    now
      per cases;
      suppose
A17:    n=0;
        thus X in the topology of PT
        proof
          consider V be Subset of PM such that
A18:      X = union {Ball(c,jd) where c is Element of PM: c in V\
          PartUnion(V,Mn) & Ball(c,3/2) c= V} and
          V in FX by A11,A16,A17;
          set NF = {Ball(c,jd) where c is Element of PM: c in V\PartUnion(V,
          Mn) & Ball(c,3/2) c= V};
          NF c= bool the carrier of PM
          proof
            let a be object;
            assume a in NF;
            then
            ex c be Element of PM st a = Ball(c,jd) & c in V\ PartUnion(V
            ,Mn) & Ball(c,3/2) c= V;
            hence thesis;
          end;
          then reconsider NF as Subset-Family of PM;
          NF c= Family_open_set(PM)
          proof
            let a be object;
            assume a in NF;
            then
            ex c be Element of PM st a = Ball(c,jd) & c in V\ PartUnion(V
            ,Mn) & Ball(c,3/2) c= V;
            hence thesis by PCOMPS_1:29;
          end;
          hence thesis by A2,A7,A18,PCOMPS_1:32;
        end;
      end;
      suppose
        n>0;
        then consider k being Nat such that
A19:    n = k + 1 by NAT_1:6;
        reconsider k as Element of NAT by ORDINAL1:def 12;
        thus X in the topology of PT
        proof
          X in {union { Ball(c,1/(2 |^ (k+1))) where c is Element of PM:
c in V\PartUnion(V,Mn) & Ball(c,3/(2 |^ (k+1))) c= V & not c in union{union(f.q
) where q is Nat: q <= k } } where V is Subset of PM: V in FX} by
A12,A16,A19;
          then consider V be Subset of PM such that
A20:      X = union { Ball(c,1/(2 |^ (k+1))) where c is Element of PM
: c in V\PartUnion(V,Mn) & Ball(c,3/(2 |^ (k+1))) c= V & not c in union{union(f
          .q) where q is Nat: q <= k } } and
          V in FX;
          reconsider NF = { Ball(c,1/(2 |^ (k+1))) where c is Element of PM: c
in V\PartUnion(V,Mn) & Ball(c,3/(2 |^ (k+1))) c= V & not c in union { union(f.q
          ) where q is Nat: q <= k } } as set;
          NF c= bool the carrier of PM
          proof
            let a be object;
            assume a in NF;
            then ex c be Element of PM st a = Ball(c,1/(2 |^ (k+1))) & c in V\
PartUnion(V,Mn) & Ball(c,3/(2 |^ (k+1))) c= V & not c in union { union(f.l ): l
            <= k};
            hence thesis;
          end;
          then reconsider NF as Subset-Family of PM;
          NF c= Family_open_set(PM)
          proof
            let a be object;
            assume a in NF;
            then ex c be Element of PM st a = Ball(c,1/(2 |^ (k+1))) & c in V\
PartUnion(V,Mn) & Ball(c,3/(2 |^ (k+1))) c= V & not c in union{union(f.l): l <=
            k};
            hence thesis by PCOMPS_1:29;
          end;
          hence thesis by A2,A7,A20,PCOMPS_1:32;
        end;
      end;
    end;
    hence thesis by PRE_TOPC:def 2;
  end;
  hence GX is open by TOPS_2:def 1;
A21: Mn well_orders FX by A3,A4,Th1;
  the carrier of PT c= union GX
  proof
    let x be object;
    defpred P1[set] means x in $1;
    assume
A22: x in the carrier of PT;
    then reconsider x9=x as Element of PM by A1,A7,Th4;
    ex G be Subset of PT st x in G & G in FX by A5,A22,PCOMPS_1:3;
    then
A23: ex G be set st G in FX & P1[G];
    consider X be set such that
A24: X in FX and
A25: P1[X] and
A26: for Y be set st Y in FX & P1[Y] holds [X,Y] in Mn from MinSet(A21,A23 );
    reconsider X as Subset of PT by A24;
    X is open by A6,A24,TOPS_2:def 1;
    then
A27: X in Family_open_set(PM) by A2,A7,PRE_TOPC:def 2;
    reconsider X as Subset of PM by A1,A7,Th4;
    consider r be Real such that
A28: r>0 and
A29: Ball(x9,r) c= X by A25,A27,PCOMPS_1:def 4;
    defpred P2[Nat] means 3/(2 |^ $1) <= r;
    ex k be Nat st P2[k] by A28,PREPOWER:92;
    then
A30: ex k be Nat st P2[k];
    consider k be Nat such that
A31: P2[k] and
    for l be Nat st P2[l] holds k <= l from NAT_1:sch 5(A30 );
    2 |^ (k+1) = 2 |^ k * 2 by NEWTON:6;
    then 2 |^ k > 0 & 2 |^ (k+1) >= 2 |^ k by PREPOWER:6,XREAL_1:151;
    then
A32: 3/2 |^ (k+1) <= 3/2 |^ k by XREAL_1:118;
    assume
A33: not x in union GX;
A34: for V be set,n st V in f.n holds not x in V
    proof
      let V be set;
      let n;
      reconsider m = n as Element of NAT by ORDINAL1:def 12;
A35:  f.m in bool bool the carrier of PM;
      assume V in f.n;
      then V in GX by A13,A35;
      hence thesis by A33,TARSKI:def 4;
    end;
A36: for n holds not x in union (f.n)
    proof
      let n;
      assume x in union (f.n);
      then ex V be set st x in V & V in f.n by TARSKI:def 4;
      hence contradiction by A34;
    end;
    now
      set W = union{ Ball(y,1/(2 |^ (k+1))) where y is Element of PM: y in X\
PartUnion(X,Mn) & Ball(y,3/(2 |^ (k+1))) c= X & not y in union{ union(f.q)
      where q is Nat: q <= k} };
A37:  x in W
      proof
A38:    not x9 in union { union(f.q) where q is Nat: q <= k}
        proof
          assume x9 in union { union(f.q) where q is Nat: q <= k};
          then consider D be set such that
A39:      x9 in D and
A40:      D in { union(f.q) where q is Nat: q <= k} by TARSKI:def 4;
          ex q be Nat st D = union (f.q) & q <= k by A40;
          hence contradiction by A36,A39;
        end;
        not x9 in PartUnion(X,Mn)
        proof
          reconsider FX as set;
          assume x9 in PartUnion(X,Mn);
          then consider M be set such that
A41:      x9 in M and
A42:      M in Mn-Seg(X) by TARSKI:def 4;
A43:      M <> X by A42,WELLORD1:1;
A44:      Mn is_antisymmetric_in FX by A21;
A45:      [M,X] in Mn by A42,WELLORD1:1;
          then M in field Mn by RELAT_1:15;
          then
A46:      M in FX by A3,A4,Th1;
          then [X,M] in Mn by A26,A41;
          hence contradiction by A24,A43,A45,A46,A44;
        end;
        then
A47:    x9 in X\PartUnion(X,Mn) by A25,XBOOLE_0:def 5;
        consider A be Subset of PM such that
A48:    A = Ball(x9,1/(2 |^ (k+1)));
        0 < 2 |^ (k+1) by PREPOWER:6;
        then
A49:    x in A by A48,TBSP_1:11,XREAL_1:139;
        Ball(x9,3/(2 |^ (k+1))) c= Ball(x9,r) by A31,A32,PCOMPS_1:1,XXREAL_0:2;
        then Ball(x9,3/(2 |^ (k+1))) c= X by A29;
        then A in { Ball(y,1/(2 |^ (k+1))) where y is Element of PM: y in X\
PartUnion(X,Mn) & Ball(y,3/(2 |^ (k+1))) c= X & not y in union { union(f.q)
        where q is Nat: q <= k}} by A48,A47,A38;
        hence thesis by A49,TARSKI:def 4;
      end;
      reconsider W as set;
      W in {union{ Ball(y,1/(2 |^ (k+1))) where y is Element of PM: y in
V\PartUnion(V,Mn) & Ball(y,3/(2 |^ (k+1))) c= V & not y in union { union(f.q)
where q is Nat: q <= k}} where V is Subset of PM: V in FX} by A24;
      then W in f.(k+1) by A12;
      hence ex W be set,l be Element of NAT st W in f.l & x in W by A37;
    end;
    hence contradiction by A34;
  end;
  hence
A50: GX is Cover of PT by SETFAM_1:def 11;
  for X be set st X in GX ex Y be set st Y in FX & X c= Y
  proof
    let X be set;
    assume
A51: X in GX;
    then reconsider X as Subset of PM;
    consider n being Nat such that
A52: X in f.n by A13,A51;
    now
      per cases;
      suppose
A53:    n=0;
        thus ex Y be set st Y in FX & X c= Y
        proof
          consider V be Subset of PM such that
A54:      X = union {Ball(c,1/2) where c is Element of PM: c in V\
          PartUnion(V,Mn) & Ball(c,3/2) c= V} and
A55:      V in FX by A11,A52,A53;
          set NF = {Ball(c,1/2) where c is Element of PM: c in V\PartUnion(V,
          Mn) & Ball(c,3/2) c= V};
          NF c= bool the carrier of PM
          proof
            let a be object;
            assume a in NF;
            then ex c be Element of PM st a = Ball(c,1/2) & c in V\ PartUnion(
            V,Mn) & Ball(c,3/2) c= V;
            hence thesis;
          end;
          then reconsider NF as Subset-Family of PM;
A56:      for W be set st W in NF holds W c= V
          proof
            let W be set;
            assume W in NF;
            then consider c be Element of PM such that
A57:        W = Ball(c,1/2) and
            c in V\PartUnion(V,Mn) and
A58:        Ball(c,3/2) c= V;
            Ball(c,1/2) c= Ball(c,3/2) by PCOMPS_1:1;
            hence thesis by A57,A58;
          end;
          reconsider V as set;
          take Y = V;
          thus Y in FX by A55;
          thus thesis by A54,A56,ZFMISC_1:76;
        end;
      end;
      suppose
        n>0;
        then consider k being Nat such that
A59:    n = k + 1 by NAT_1:6;
        reconsider k as Element of NAT by ORDINAL1:def 12;
        thus ex Y be set st Y in FX & X c= Y
        proof
          X in {union { Ball(c,1/(2 |^ (k+1))) where c is Element of PM:
c in V\PartUnion(V,Mn) & Ball(c,3/(2 |^ (k+1))) c= V & not c in union { union(f
.q) where q is Nat: q <= k}} where V is Subset of PM: V in FX} by
A12,A52,A59;
          then consider V be Subset of PM such that
A60:      X = union { Ball(c,1/(2 |^ (k+1))) where c is Element of PM:       c
 in V\PartUnion(V,Mn) & Ball(c,3/(2 |^ (k+1))) c= V & not c in union{union
          (f.q) where q is Nat: q <= k}} and
A61:      V in FX;
          reconsider NF = { Ball(c,1/(2 |^ (k+1))) where c is Element of PM: c
in V\PartUnion(V,Mn) & Ball(c,3/(2 |^ (k+1))) c= V & not c in union{union(f.q)
          where q is Nat: q <= k}} as set;
          NF c= bool the carrier of PM
          proof
            let a be object;
            assume a in NF;
            then ex c be Element of PM st a = Ball(c,1/(2 |^ (k+1))) & c in V\
PartUnion(V,Mn) & Ball(c,3/(2 |^ (k+1))) c= V & not c in union { union(f .q)
            where q is Nat: q <= k};
            hence thesis;
          end;
          then reconsider NF as Subset-Family of PM;
A62:      for W be set st W in NF holds W c= V
          proof
            let W be set;
            assume W in NF;
            then consider c be Element of PM such that
A63:        W = Ball(c,1/(2 |^ (k+1))) and
            c in V\PartUnion(V,Mn) and
A64:        Ball(c,3/(2 |^ (k+1))) c= V and
            not c in union { union(f.q) where q is Nat: q <= k};
            Ball(c,1/(2 |^ (k+1))) c= Ball(c,3/(2 |^ (k+1))) by PCOMPS_1:1
,XREAL_1:72;
            hence thesis by A63,A64;
          end;
          reconsider V as set;
          take Y = V;
          thus Y in FX by A61;
          thus thesis by A60,A62,ZFMISC_1:76;
        end;
      end;
    end;
    hence thesis;
  end;
  hence GX is_finer_than FX;
  for x be Point of PT ex W be Subset of PT st x in W & W is open & { V
  : V in GX & V meets W } is finite
  proof
    let x be Point of PT;
    reconsider x9=x as Element of PM by A1,A7,Th4;
    consider X be Subset of PT such that
A65: x in X and
A66: X in GX by A50,PCOMPS_1:3;
    reconsider X as Subset of PT;
    X is open by A14,A66;
    then X in Family_open_set(PM) by A2,A7,PRE_TOPC:def 2;
    then consider r be Real such that
A67: r>0 and
A68: Ball(x9,r) c= X by A65,PCOMPS_1:def 4;
    consider m be Nat such that
A69: 1/(2 |^ m) <= r by A67,PREPOWER:92;
    defpred P3[set] means X in f.$1;
    ex n be Nat st P3[n] by A13,A66;
    then
A70: ex n be Nat st P3[n];
    consider k be Nat such that
A71: P3[k] and
    for l be Nat st P3[l] holds k <= l from NAT_1:sch 5(A70 );
    consider W be Subset of PM such that
A72: W = Ball(x9,1/(2 |^ (m+1+k+1)));
    reconsider W as Subset of PT by A1,A7,Th4;
A73: { V : V in GX & V meets W } is finite
    proof
      defpred P4[object,set] means $1 in f.$2;
      set NZ={ V : V in GX & V meets W };
A74:  for p be object st p in NZ ex n being Nat st P4[p,n]
      proof
        let p be object;
        assume p in NZ;
        then ex V be Subset of PT st p = V & V in GX & V meets W;
        hence thesis by A13;
      end;
      consider g be Function such that
A75:  dom g = NZ and
A76:  for y be object st y in NZ
      ex n being Nat st g.y=n & P4[y,n] &
      for t be Nat st P4[y,t] holds n <=t from TREES_2:sch 4(A74);
A77:  rng g c= {i: i < (m+1+k+1)}
      proof
        let t be object;
        assume t in rng g;
        then consider a be object such that
A78:    a in dom g and
A79:    t = g.a by FUNCT_1:def 3;
        assume
A80:    not t in {i: i < (m+1+k+1)};
A81:    ex n be Nat st g.a = n & a in f.n &
      for p be Nat st a in f.p holds n <= p by A75,A76,A78;
        then reconsider t as Element of NAT by A79,ORDINAL1:def 12;
        consider V such that
A82:    a=V and
        V in GX and
A83:    V meets W by A75,A78;
        consider y being object such that
A84:    y in V and
A85:    y in W by A83,XBOOLE_0:3;
A86:    t >= (m+1+k+1) by A80;
        now
          per cases;
          suppose
            t=0;
            hence contradiction by A80;
          end;
          suppose
            t>0;
            then consider l be Nat such that
A87:        t=l+1 by NAT_1:6;
            reconsider l as Element of NAT by ORDINAL1:def 12;
A88:        V in {union { Ball(c,1/(2 |^ (l+1))) where c is Element of
PM:         c
 in Y\PartUnion(Y,Mn) & Ball(c,3/(2 |^ (l+1))) c= Y & not c in union{union
(f.q) where q is Nat: q <= l}} where Y is Subset of PM: Y in FX} by
A12,A79,A81,A82,A87;
            2 |^ t >= 2 |^ (m+1+k+1) & 2 |^ (m+1+k+1) > 0 by A86,PREPOWER:6,93;
            then
A89:        1/(2 |^ (m+1+k+1)) >= 1/(2 |^ t) by XREAL_1:118;
            consider Y be Subset of PM such that
A90:        V = union { Ball(c,1/(2 |^ (l+1))) where c is Element of
PM:         c
 in Y\PartUnion(Y,Mn) & Ball(c,3/(2 |^ (l+1))) c= Y & not c in union{union
            (f.q) where q is Nat: q <= l}} and
            Y in FX by A88;
            reconsider NF = { Ball(c,1/(2 |^ (l+1) ) ) where c is Element of
PM:         c
 in Y\PartUnion(Y,Mn) & Ball(c,3/(2 |^ (l+1))) c= Y & not c in union{union
            (f.q) where q is Nat: q <= l}} as set;
            consider T be set such that
A91:        y in T and
A92:        T in NF by A84,A90,TARSKI:def 4;
            reconsider y as Element of PM by A85;
            consider c be Element of PM such that
A93:        T = Ball(c,1/(2 |^ (l+1))) and
            c in Y\PartUnion(Y,Mn) and
            Ball(c,3/(2 |^ (l+1))) c= Y and
A94:        not c in union{union (f.q) where q is Nat: q
            <= l} by A92;
            dist(c,y) < 1/(2 |^ t) by A87,A91,A93,METRIC_1:11;
            then dist(c,y) < 1/(2 |^ (m+1+k+1)) by A89,XXREAL_0:2;
            then
A95:        dist(c,y) + dist(y,x9) < 1/(2 |^ (m+1+k+1)) + dist(y,x9) by
XREAL_1:6;
A96:        for t be Element of NAT st t < l holds not c in union(f.t)
            proof
              let t be Element of NAT;
              assume t < l;
              then
A97:          union(f.t) in {union(f.q) where q is Nat: q <= l};
              assume c in union(f.t);
              hence contradiction by A94,A97,TARSKI:def 4;
            end;
A98:        dist(c,x9) >= 1/(2 |^ m)
            proof
              assume not dist(c,x9) >= 1/(2 |^ m);
              then dist(x9,c) < r by A69,XXREAL_0:2;
              then c in Ball(x9,r) by METRIC_1:11;
              then
A99:          c in union (f.k) by A71,A68,TARSKI:def 4;
A100:         k <> l
              proof
                assume k=l;
                then union (f.k) in {union(f.q) where q is Nat: q <= l};
                hence contradiction by A94,A99,TARSKI:def 4;
              end;
              l >= k+(m+1) & k+(m+1)>=k by A86,A87,NAT_1:11,XREAL_1:6;
              then k <= l by XXREAL_0:2;
              then k in NAT & k < l by A100,ORDINAL1:def 12,XXREAL_0:1;
              hence contradiction by A96,A99;
            end;
            dist(c,y) + dist(y,x9) >= dist(c,x9) by METRIC_1:4;
            then dist(c,y) + dist(y,x9) >= 1/(2 |^ m) by A98,XXREAL_0:2;
            then 1/(2 |^ (m+1+k+1)) + dist(y,x9) > 1/(2 |^ m) by A95,XXREAL_0:2
;
            then
A101:       dist(y,x9) >= 1/(2 |^ m) - 1/(2 |^ (m+1+k+1)) by XREAL_1:19;
            (2 |^ (1+k))*2>=2 by PREPOWER:11,XREAL_1:151;
            then
A102:       ((2 |^ (1+k))*2)-1>=2-1 by XREAL_1:9;
            2 |^ (1+k+1) <> 0 by PREPOWER:5;
            then 1/(2 |^ m) - 1/(2 |^ (m+1+k+1)) = (1*(2 |^ (1+k+1)))/((2 |^
            m)*(2 |^ (1+k+1))) - 1/(2 |^ (m+1+k+1)) by XCMPLX_1:91
              .= (1*(2 |^ (1+k+1)))/(2 |^ (m+((1+k)+1))) - 1/(2 |^ (m+1+k+1)
            ) by NEWTON:8
              .= ((2 |^ (1+k))*2)/(2 |^ (m+1+k+1)) - 1/(2 |^ (m+1+k+1)) by
NEWTON:6
              .= (((2 |^ (1+k))*2)-1)/(2 |^ (m+1+k+1)) by XCMPLX_1:120;
            then 1 /(2 |^ m) - 1/(2 |^ (m+1+k+1)) >= 1/(2 |^ (m+1+k+1)) by A102
,XREAL_1:72;
            then dist(y,x9) >= 1/(2 |^ (m+1+k+1)) by A101,XXREAL_0:2;
            hence contradiction by A72,A85,METRIC_1:11;
          end;
        end;
        hence contradiction;
      end;
      for x1,x2 be object st x1 in dom g & x2 in dom g & g.x1 = g.x2
holds x1 = x2
      proof
        let x1,x2 be object;
        assume that
A103:   x1 in dom g and
A104:   x2 in dom g and
A105:   g.x1 = g.x2;
        assume
A106:   x1<>x2;
        reconsider x1,x2 as set by TARSKI:1;
        ex V2 be Subset of PT st x2=V2 & V2 in GX & V2 meets W by A75,A104;
        then consider w2 being object such that
A107:   w2 in W and
A108:   w2 in x2 by XBOOLE_0:3;
        consider n1 be Nat such that
A109:   g.x1 = n1 and
A110:   x1 in f.n1 and
        for t be Nat st x1 in f.t holds n1 <= t by A75,A76,A103;
        ex V1 be Subset of PT st x1=V1 & V1 in GX & V1 meets W by A75,A103;
        then consider w1 being object such that
A111:   w1 in W and
A112:   w1 in x1 by XBOOLE_0:3;
        reconsider w1, w2 as Element of PM by A111,A107;
A113:   ex n2 be Nat st g.x2 = n2 & x2 in f.n2 & for t be
        Nat st x2 in f.t holds n2 <= t by A75,A76,A104;
        now
          per cases;
          suppose
A114:       n1=0;
            m+k+1 >= 1 by NAT_1:11;
            then 2 |^ (m+1+k) >= 2 |^ 1 by PREPOWER:93;
            then 2 |^ (m+1+k) >= 2;
            then
A115:       1/(2 |^ (m+1+k)) <= 1/2 by XREAL_1:118;
A116:       2/(2 |^ (m+1+k+1)) =(1*2)/(2 |^ (m+1+k)*2) by NEWTON:6
              .= 1/(2 |^ (m+1+k)) by XCMPLX_1:91;
            (1/2 + 1/(2 |^ (m+1+k+1))) + (1/(2 |^ (m+1+k+1)) + 1/2) = (1
            +1)/2 + (1/(2 |^ (m+1+k+1)) + 1/(2 |^ (m+1+k+1)))
              .= 2/2 + 2/(2 |^ (m+1+k+1)) by XCMPLX_1:62;
            then (1/2 + 1/(2 |^ (m+1+k+1))) + (1/(2 |^ (m+1+k+1)) + 1/2) - 2/
            2 = 1/(2 |^ (m+1+k)) by A116;
            then
A117:       (1/2 + 1/(2 |^ (m+1+k+1))) + (1/(2 |^ (m+1+k+1)) + 1/2) <= 2
            /2 + 1/2 by A115,XREAL_1:20;
A118:       Mn is_connected_in FX by A21;
A119:       dist(x9,w2) < 1/(2 |^ (m+1+k+1)) by A72,A107,METRIC_1:11;
            consider M1 be Subset of PM such that
A120:       x1 = union {Ball(c,1/2) where c is Element of PM: c in
            M1\PartUnion(M1,Mn) & Ball(c,3/2) c= M1} and
A121:       M1 in FX by A11,A110,A114;
            consider k1 be set such that
A122:       w1 in k1 and
A123:       k1 in {Ball(c,1/2) where c is Element of PM: c in M1\
            PartUnion(M1,Mn) & Ball(c,3/2) c= M1} by A112,A120,TARSKI:def 4;
            consider c1 be Element of PM such that
A124:       k1 = Ball(c1,1/2) and
A125:       c1 in M1\PartUnion(M1,Mn) and
A126:       Ball(c1,3/2) c= M1 by A123;
A127:       dist(c1,w1) < 1/2 by A122,A124,METRIC_1:11;
            consider M2 be Subset of PM such that
A128:       x2 = union {Ball(c,1/2) where c is Element of PM: c in
            M2\PartUnion(M2,Mn) & Ball(c,3/2) c= M2} and
A129:       M2 in FX by A11,A105,A109,A113,A114;
            consider k2 be set such that
A130:       w2 in k2 and
A131:       k2 in {Ball(c,1/2) where c is Element of PM: c in M2\
            PartUnion(M2,Mn) & Ball(c,3/2) c= M2} by A108,A128,TARSKI:def 4;
            consider c2 be Element of PM such that
A132:       k2 = Ball(c2,1/2) and
A133:       c2 in M2\PartUnion(M2,Mn) and
A134:       Ball(c2,3/2) c= M2 by A131;
            dist(x9,c2) <= dist(x9,w2) + dist(w2,c2) & dist(c1,x9) <=
            dist(c1,w1) + dist (w1,x9) by METRIC_1:4;
            then
A135:       dist(c1,x9) + dist(x9,c2) <= (dist(c1,w1) + dist(w1,x9)) + (
            dist(x9,w2) + dist(w2,c2)) by XREAL_1:7;
            dist(c1,c2) <= dist(c1,x9) + dist(x9,c2) by METRIC_1:4;
            then dist(c1,c2) <= dist(c1,w1) + (dist(w1,x9) + (dist(x9,w2) +
            dist(w2,c2))) by A135,XXREAL_0:2;
            then dist(c1,c2) - (dist(w1,x9) + (dist(x9,w2) + dist(w2,c2))) <=
            dist(c1,w1) by XREAL_1:20;
            then dist(c1,c2) - (dist(w1,x9) + (dist(x9,w2) + dist(w2,c2))) <
            1/2 by A127,XXREAL_0:2;
            then dist(c1,c2) < 1/2 + (dist(w1,x9) + (dist(x9,w2) + dist(w2,c2
            ))) by XREAL_1:19;
            then dist(c1,c2) < dist(w1,x9) + (1/2 + (dist(x9,w2) + dist(w2,c2
            )));
            then
A136:       dist(c1,c2) - (1/2 + (dist(x9,w2) + dist(w2,c2))) < dist(w1,
            x9) by XREAL_1:19;
            dist(x9,w1) < 1/(2 |^ (m+1+k+1)) by A72,A111,METRIC_1:11;
            then dist(c1,c2) - (1/2 + (dist(x9,w2) + dist(w2,c2))) < 1/(2 |^
            (m+1+k+1)) by A136,XXREAL_0:2;
            then dist(c1,c2) < 1/(2 |^ (m+1+k+1)) + (1/2 + (dist(x9,w2) +
            dist(w2,c2))) by XREAL_1:19;
            then dist(c1,c2) < dist(x9,w2) + (dist(w2,c2) + (1/(2 |^ (m+1+k+1
            )) + 1/2));
            then dist(c1,c2) - (dist(w2,c2) + (1/(2 |^ (m+1+k+1)) + 1/2)) <
            dist(x9,w2) by XREAL_1:19;
            then dist(c1,c2) - (dist(w2,c2) + (1/(2 |^ (m+1+k+1)) + 1/2)) < 1
            /(2 |^ (m+1+k+1)) by A119,XXREAL_0:2;
            then dist(c1,c2) < 1/(2 |^ (m+1+k+1)) + (dist(w2,c2) + (1/(2 |^ (
            m+1+k+1)) + 1/2)) by XREAL_1:19;
            then dist(c1,c2) < dist(w2,c2) + (1/(2 |^ (m+1+k+1)) + (1/(2 |^ (
            m+1+k+1)) + 1/2));
            then
A137:       dist(c1,c2) - (1/(2 |^ (m+1+k+1)) + (1/(2 |^ (m+1+k+1)) + 1/
            2)) < dist(w2,c2) by XREAL_1:19;
            dist(c2,w2) < 1/2 by A130,A132,METRIC_1:11;
            then dist(c1,c2) - (1/(2 |^ (m+1+k+1)) + (1/(2 |^ (m+1+k+1)) + 1/
            2)) < 1/2 by A137,XXREAL_0:2;
            then dist(c1,c2) < 1/2 + (1/(2 |^ (m+1+k+1)) + (1/(2 |^ (m+1+k+1)
            ) + 1/2)) by XREAL_1:19;
            then
A138:       dist(c1,c2) < 3/2 by A117,XXREAL_0:2;
            then
A139:       c1 in Ball(c2,3/2) by METRIC_1:11;
A140:       M1 <> M2 by A106,A120,A128;
A141:       c2 in Ball(c1,3/2) by A138,METRIC_1:11;
            now
              per cases by A121,A129,A118,A140;
              suppose
                [M1,M2] in Mn;
                then M1 in Mn-Seg(M2) by A140,WELLORD1:1;
                then c2 in PartUnion(M2,Mn) by A126,A141,TARSKI:def 4;
                hence contradiction by A133,XBOOLE_0:def 5;
              end;
              suppose
                [M2,M1] in Mn;
                then M2 in Mn-Seg(M1) by A140,WELLORD1:1;
                then c1 in PartUnion(M1,Mn) by A134,A139,TARSKI:def 4;
                hence contradiction by A125,XBOOLE_0:def 5;
              end;
            end;
            hence contradiction;
          end;
          suppose
            n1>0;
            then consider l be Nat such that
A142:       n1 = l+1 by NAT_1:6;
            reconsider l as Element of NAT by ORDINAL1:def 12;
A143:       x1 in {union {Ball(c,1/(2 |^ (l+1))) where c is Element of
PM:         c
 in M1\PartUnion(M1,Mn) & Ball(c,3/(2 |^ (l+1))) c= M1 & not c in union {
union(f.q) where q is Nat: q <= l}} where M1 is Subset of PM: M1 in
            FX} by A12,A110,A142;
A144:       dist(x9,w2) < 1/(2 |^ (m+1+k+1)) by A72,A107,METRIC_1:11;
A145:       x2 in {union {Ball(c,1/(2 |^ (l+1))) where c is Element of
PM:         c
 in M2\PartUnion(M2,Mn) & Ball(c,3/(2 |^ (l+1))) c= M2 & not c in union {
union(f.q) where q is Nat: q <= l}} where M2 is Subset of PM: M2 in
            FX} by A12,A105,A109,A113,A142;
A146:       (1/(2 |^ (l+1)) + 1/(2 |^ (m+1+k+1))) + (1/(2 |^ (m+1+k+1))
+ 1/(2 |^ (l+1))) = (1/(2 |^ (l+1)) + 1/(2 |^ (l+1))) + (1/(2 |^ (m+1+k+1)) + 1
            /(2 |^ (m+1+k+1)))
              .= (1+1)/(2 |^ (l+1)) + (1/(2 |^ (m+1+k+1)) + 1/(2 |^ (m+1+k+1
            ))) by XCMPLX_1:62
              .= 2/(2 |^ (l+1)) + 2/(2 |^ (m+1+k+1)) by XCMPLX_1:62;
            n1 in rng g by A103,A109,FUNCT_1:def 3;
            then n1 in {i: i<m+1+k+1} by A77;
            then
A147:       ex i st n1=i & i < m+1+k+1;
            then consider h be Nat such that
A148:       m+1+k+1 = (l+1) + h by A142,NAT_1:10;
            h <> 0 by A142,A147,A148;
            then consider i be Nat such that
A149:       h = i + 1 by NAT_1:6;
            (l+1)+i >= l+1 by NAT_1:11;
            then 2 |^ (l+1) > 0 & 2 |^ ((l+1)+i) >= 2 |^ (l+1) by PREPOWER:6,93
;
            then
A150:       1/(2 |^ ((l+1)+i)) <= 1/(2 |^ (l+1)) by XREAL_1:118;
            2/(2 |^ (m+1+k+1)) = (1*2)/(2 |^ ((l+1)+i)*2) by A148,A149,NEWTON:6
              .= 1/(2 |^ ((l+1)+i)) by XCMPLX_1:91;
            then (1/(2 |^ (l+1)) + 1/(2 |^ (m+1+k+1))) + (1/(2 |^ (m+1+k+1))
            + 1/(2 |^ (l+1))) - 2/(2 |^ (l+1)) = 1/(2 |^ ((l+1)+i)) by A146;
            then (1/(2 |^ (l+1)) + 1/(2 |^ (m+1+k+1))) + (1/(2 |^ (m+1+k+1))
+ 1/(2 |^ (l+1))) <= 2/(2 |^ (l+1)) + 1/(2 |^ (l+1)) by A150,XREAL_1:20;
            then
A151:       (1/(2 |^ (l+1)) + 1/(2 |^ (m+1+k+1))) + (1/(2 |^ (m+1+k+1))
            + 1/(2 |^ (l+1))) <= (2+1)/(2 |^ (l+1)) by XCMPLX_1:62;
A152:       Mn is_connected_in FX by A21;
            consider M1 be Subset of PM such that
A153:       x1 = union {Ball(c,1/(2 |^ (l+1))) where c is Element of
PM:         c
 in M1\PartUnion(M1,Mn) & Ball(c,3/(2 |^ (l+1))) c= M1 & not c in union {
            union(f.q) where q is Nat: q <= l}} and
A154:       M1 in FX by A143;
            reconsider NF = {Ball(c,1/(2 |^ (l+1))) where c is Element of PM:
c in M1\PartUnion(M1,Mn) & Ball(c,3/(2 |^ (l+1))) c= M1 & not c in union {
            union(f.q) where q is Nat: q <= l}} as set;
            consider k1 be set such that
A155:       w1 in k1 and
A156:       k1 in NF by A112,A153,TARSKI:def 4;
            consider c1 be Element of PM such that
A157:       k1 = Ball(c1,1/(2 |^ (l+1))) and
A158:       c1 in M1\PartUnion(M1,Mn) and
A159:       Ball(c1,3/(2 |^ (l+1))) c= M1 and
            not c1 in union { union(f.q) where q is Nat: q <= l
            } by A156;
A160:       dist(c1,w1) < 1/(2 |^ (l+1)) by A155,A157,METRIC_1:11;
            consider M2 be Subset of PM such that
A161:       x2 = union {Ball(c,1/(2 |^ (l+1))) where c is Element of
PM:         c
 in M2\PartUnion(M2,Mn) & Ball(c,3/(2 |^ (l+1))) c= M2 & not c in union {
            union(f.q) where q is Nat: q <= l}} and
A162:       M2 in FX by A145;
A163:       M1 <> M2 by A106,A153,A161;
            reconsider NF = {Ball(c,1/(2 |^ (l+1))) where c is Element of PM:
c in M2\PartUnion(M2,Mn) & Ball(c,3/(2 |^ (l+1))) c= M2 & not c in union {
            union(f.q) where q is Nat: q <= l}} as set;
            consider k2 be set such that
A164:       w2 in k2 and
A165:       k2 in NF by A108,A161,TARSKI:def 4;
            consider c2 be Element of PM such that
A166:       k2 = Ball(c2,1/(2 |^ (l+1))) and
A167:       c2 in M2\PartUnion(M2,Mn) and
A168:       Ball(c2,3/(2 |^ (l+1))) c= M2 and
            not c2 in union { union(f.q) where q is Nat: q <= l
            } by A165;
            dist(x9,c2) <= dist(x9,w2) + dist(w2,c2) & dist(c1,x9) <=
            dist(c1,w1) + dist (w1,x9) by METRIC_1:4;
            then
A169:       dist(c1,x9) + dist(x9,c2) <= (dist(c1,w1) + dist(w1,x9)) + (
            dist(x9,w2) + dist(w2,c2)) by XREAL_1:7;
            dist(c1,c2) <= dist(c1,x9) + dist(x9,c2) by METRIC_1:4;
            then dist(c1,c2) <= dist(c1,w1) + (dist(w1,x9) + (dist(x9,w2) +
            dist(w2,c2))) by A169,XXREAL_0:2;
            then dist(c1,c2) - (dist(w1,x9) + (dist(x9,w2) + dist(w2,c2))) <=
            dist(c1,w1) by XREAL_1:20;
            then dist(c1,c2) - (dist(w1,x9) + (dist(x9,w2) + dist(w2,c2))) <
            1/(2 |^ (l+1)) by A160,XXREAL_0:2;
            then dist(c1,c2) < 1/(2 |^ (l+1)) + (dist(w1,x9) + (dist(x9,w2) +
            dist(w2,c2))) by XREAL_1:19;
            then dist(c1,c2) < dist(w1,x9) + (1/(2 |^ (l+1)) + (dist(x9,w2) +
            dist(w2,c2)));
            then
A170:       dist(c1,c2) - (1/(2 |^ (l+1)) + (dist(x9,w2) + dist(w2,c2)))
            < dist(w1,x9) by XREAL_1:19;
            dist(x9,w1) < 1/(2 |^ (m+1+k+1)) by A72,A111,METRIC_1:11;
            then dist(c1,c2) - (1/(2 |^ (l+1)) + (dist(x9,w2) + dist(w2,c2)))
            < 1/(2 |^ (m+1+k+1)) by A170,XXREAL_0:2;
            then dist(c1,c2) < 1/(2 |^ (m+1+k+1)) + (1/(2 |^ (l+1)) + (dist(
            x9,w2) + dist(w2,c2))) by XREAL_1:19;
            then dist(c1,c2) < dist(x9,w2) + (dist(w2,c2) + (1/(2 |^ (m+1+k+1
            )) + 1/(2 |^ (l+1))));
            then dist(c1,c2) - (dist(w2,c2) + (1/(2 |^ (m+1+k+1)) + 1/(2 |^ (
            l+1)))) < dist(x9,w2) by XREAL_1:19;
            then dist(c1,c2) - (dist(w2,c2) + (1/(2 |^ (m+1+k+1)) + 1/(2 |^ (
            l+1)))) < 1/(2 |^ (m+1+k+1)) by A144,XXREAL_0:2;
            then dist(c1,c2) < 1/(2 |^ (m+1+k+1)) + (dist(w2,c2) + (1/(2 |^ (
            m+1+k+1)) + 1/(2 |^ (l+1)))) by XREAL_1:19;
            then dist(c1,c2) < dist(w2,c2) + (1/(2 |^ (m+1+k+1)) + (1/(2 |^ (
            m+1+k+1)) + 1/(2 |^ (l+1))));
            then
A171:       dist(c1,c2) - (1/(2 |^ (m+1+k+1)) + (1/(2 |^ (m+1+k+1)) + 1/
            (2 |^ (l+1)))) < dist(w2,c2) by XREAL_1:19;
            dist(c2,w2) < 1/(2 |^ (l+1)) by A164,A166,METRIC_1:11;
            then dist(c1,c2) - (1/(2 |^ (m+1+k+1)) + (1/(2 |^ (m+1+k+1)) + 1/
            (2 |^ (l+1)))) < 1/(2 |^ (l+1)) by A171,XXREAL_0:2;
            then dist(c1,c2) < 1/(2 |^ (l+1)) + (1/(2 |^ (m+1+k+1)) + (1/(2
            |^ (m+1+k+1)) + 1/(2 |^ (l+1)))) by XREAL_1:19;
            then
A172:       dist(c1,c2) < 3/(2 |^ (l+1)) by A151,XXREAL_0:2;
            then
A173:       c1 in Ball(c2,3/(2 |^ (l+1))) by METRIC_1:11;
A174:       c2 in Ball(c1,3/(2 |^ (l+1))) by A172,METRIC_1:11;
            now
              per cases by A154,A162,A152,A163;
              suppose
                [M1,M2] in Mn;
                then M1 in Mn-Seg(M2) by A163,WELLORD1:1;
                then c2 in PartUnion(M2,Mn) by A159,A174,TARSKI:def 4;
                hence contradiction by A167,XBOOLE_0:def 5;
              end;
              suppose
                [M2,M1] in Mn;
                then M2 in Mn-Seg(M1) by A163,WELLORD1:1;
                then c1 in PartUnion(M1,Mn) by A168,A173,TARSKI:def 4;
                hence contradiction by A158,XBOOLE_0:def 5;
              end;
            end;
            hence contradiction;
          end;
        end;
        hence contradiction;
      end;
      then g is one-to-one by FUNCT_1:def 4;
      then
A175: NZ,rng g are_equipotent by A75,WELLORD2:def 4;
      {i: i < m+1+k+1 } is finite
      proof
        {i: i < m+1+k+1 } c= {0} \/ Seg(m+1+k+1)
        proof
          let x be object;
          assume x in {i: i < m+1+k+1};
          then
A176:     ex i be Nat st x = i & i < (m+1+k+1);
          then reconsider x1=x as Nat;
          now
            per cases;
            suppose
              x1 = 0;
              hence x1 in {0} or x1 in Seg(m+1+k+1) by TARSKI:def 1;
            end;
            suppose
              x1 > 0;
              then ex l be Nat st x1 = l +1 by NAT_1:6;
              then x1 >= 1 by NAT_1:11;
              hence x1 in {0} or x1 in Seg(m+1+k+1) by A176,FINSEQ_1:1;
            end;
          end;
          hence thesis by XBOOLE_0:def 3;
        end;
        hence thesis;
      end;
      hence thesis by A77,A175,CARD_1:38;
    end;
    2 |^ (m+1+k+1) > 0 by PREPOWER:6;
    then
A177: 1/(2 |^ (m+1+k+1)) > 0 by XREAL_1:139;
    W in the topology of PT by A2,A7,A72,PCOMPS_1:29;
    then W is open by PRE_TOPC:def 2;
    hence thesis by A177,A72,A73,TBSP_1:11;
  end;
  hence GX is locally_finite by PCOMPS_1:def 1;
end;
