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 Th22:
  for A st TM|A is second-countable finite-ind & ind A<=0
     for F st F is open & F is Cover of A
     ex g be Function of F,bool the carrier of TM st
       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
proof
  defpred P[Nat] means
  for A be Subset of TM st TM|A is second-countable & A is finite-ind
  & ind A<=0 for F be finite Subset-Family of TM st F is open &
  F is Cover of A & card F<=$1ex g be Function of F,bool the carrier of TM st
  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;
  let A be Subset of TM such that
A1: TM|A is second-countable finite-ind & ind A<=0;
  let F be finite Subset-Family of TM such that
A2: F is open & F is Cover of A;
A3: for n st P[n] holds P[n+1]
  proof
    let n such that
A4: P[n];
    let A be Subset of TM such that
A5: TM|A is second-countable and
A6: A is finite-ind and
A7: ind A<=0;
    let F be finite Subset-Family of TM such that
A8: F is open and
A9: F is Cover of A and
A10: card F<=n+1;
    per cases by A10,NAT_1:8;
    suppose card F<=n;
      hence thesis by A4,A5,A6,A7,A8,A9;
    end;
    suppose
A11:  card F=n+1;
      per cases;
      suppose n=0;
        then consider x be object such that
A12:    F={x} by A11,CARD_2:42;
        set g=F-->x;
        dom g=F & rng g=F by A12,FUNCOP_1:8,13;
        then reconsider g as Function of F,bool the carrier of TM by FUNCT_2:2;
        take g;
        thus rng g is open & rng g is Cover of A by A8,A9,A12,FUNCOP_1:8;
        hereby
          let a be set;
          assume
A13:      a in F;
          then g.a=x by FUNCOP_1:7;
          hence g.a c=a by A12,A13,TARSKI:def 1;
        end;
        let a,b be set such that
A14:    a in F and
A15:    b in F & a<>b;
        x=a by A12,A14,TARSKI:def 1;
        hence thesis by A12,A15,TARSKI:def 1;
      end;
      suppose n>0;
        then reconsider n1=n-1 as Element of NAT by NAT_1:20;
        F is non empty by A11;
        then consider x be object such that
A16:    x in F;
A17:    card(F\{x})=n1+1 by A11,A16,STIRL2_1:55;
        then F\{x} is non empty;
        then consider y be object such that
A18:    y in F\{x};
        y in F by A18,XBOOLE_0:def 5;
        then reconsider x,y as open Subset of TM by A8,A16;
        set X={x},xy=x\/y,Y={y},XY={xy};
A19:    (F\X)\/X=F by A16,ZFMISC_1:116;
        set Fxy=F\X\Y;
A20:    card Fxy=n1 by A17,A18,STIRL2_1:55;
        set FXY=Fxy\/XY;
        card XY=1 by CARD_1:30;
        then
A21:    card FXY<=n1+1 by A20,CARD_2:43;
        F\X is open by A8,TOPS_2:15;
        then
A22:    Fxy is open by TOPS_2:15;
A23:    ((F\X)\Y)\/Y=F\X by A18,ZFMISC_1:116;
        for A be Subset of TM st A in XY holds A is open by TARSKI:def 1;
        then
A24:    XY is open;
        per cases;
        suppose
A25:      Fxy is Cover of A;
          card Fxy<=n1+1 by A20,NAT_1:13;
          then consider g be Function of Fxy,bool the carrier of TM such that
A26:      rng g is open and
A27:      rng g is Cover of A and
A28:      for a be set st a in Fxy holds g.a c=a and
A29:      for a,b be set st a in Fxy & b in Fxy & a<>b holds g.a misses g.b
          by A4,A5,A6,A7,A22,A25;
A30:      A c=union rng g by A27,SETFAM_1:def 11;
          set h=(x,y)-->({}TM,{}TM);
A31:      dom h={x,y} by FUNCT_4:62;
          not x in F\X by ZFMISC_1:56;
          then
A32:      not x in Fxy by ZFMISC_1:56;
          not y in Fxy by ZFMISC_1:56;
          then
A33:      Fxy misses{x,y} by A32,ZFMISC_1:51;
A34:      x<>y by A18,ZFMISC_1:56;
          then
A35:      rng h={{}TM,{}TM} by FUNCT_4:64;
A36:      Fxy\/{x,y}=Fxy\/(Y\/X) by ENUMSET1:1
            .=Fxy\/Y\/X by XBOOLE_1:4
            .=F by A18,A19,ZFMISC_1:116;
A37:      dom g=Fxy by FUNCT_2:def 1;
          then
A38:      rng(g+*h)=rng g\/rng h by A31,A33,NECKLACE:6;
          dom(g+*h)=Fxy\/{x,y} by A31,A37,FUNCT_4:def 1;
          then reconsider gh=g+*h as Function of F,bool the carrier of TM
            by A36,A38,FUNCT_2:2;
          take gh;
          h.y={}TM & y in {x,y} by FUNCT_4:63,TARSKI:def 2;
          then
A39:      gh.y={}TM by A31,FUNCT_4:13;
          for A be Subset of TM st A in {{}TM,{}TM} holds A is open
            by TARSKI:def 2;
          then {{}TM,{}TM} is open;
          hence rng gh is open by A26,A35,A38,TOPS_2:13;
          union rng gh=union rng g\/union rng h by A38,ZFMISC_1:78
            .=union rng g\/union{{}TM} by A35,ENUMSET1:29
            .=union rng g\/{}TM by ZFMISC_1:25
            .=union rng g;
          hence rng gh is Cover of A by A30,SETFAM_1:def 11;
          x in {x,y} & h.x={}TM by A34,FUNCT_4:63,TARSKI:def 2;
          then
A40:      gh.x={}TM by A31,FUNCT_4:13;
          hereby
            let a be set;
            assume
A41:        a in F;
            per cases by A36,A41,XBOOLE_0:def 3;
            suppose a in Fxy;
              then (not a in dom h) & g.a c=a by A28,A33,XBOOLE_0:3;
              hence gh.a c=a by FUNCT_4:11;
            end;
            suppose a in {x,y};
              then a=x or a=y by TARSKI:def 2;
              hence gh.a c=a by A39,A40;
            end;
          end;
          let a,b be set such that
A42:      a in F & b in F and
A43:      a<>b;
          per cases by A36,A42,XBOOLE_0:def 3;
          suppose
A44:        a in Fxy & b in Fxy;
            then not a in dom h by A33,XBOOLE_0:3;
            then
A45:        gh.a=g.a by FUNCT_4:11;
            (not b in dom h) & g.a misses g.b by A29,A33,A43,A44,XBOOLE_0:3;
            hence thesis by A45,FUNCT_4:11;
          end;
          suppose a in {x,y} or b in {x,y};
            then gh.a={}TM or gh.b={}TM by A39,A40,TARSKI:def 2;
            hence thesis;
          end;
        end;
        suppose
A46:      not Fxy is Cover of A;
A47:      union Fxy\/xy=union Fxy\/union XY by ZFMISC_1:25
            .=union FXY by ZFMISC_1:78;
A48:      FXY is open by A22,A24,TOPS_2:13;
A49:      union F=union(F\X)\/union X by A19,ZFMISC_1:78
            .=union(F\X)\/x by ZFMISC_1:25
            .=union Fxy\/union Y\/x by A23,ZFMISC_1:78
            .=union Fxy\/y\/x by ZFMISC_1:25
            .=union Fxy\/(y\/x) by XBOOLE_1:4;
          A c=union F by A9,SETFAM_1:def 11;
          then FXY is Cover of A by A47,A49,SETFAM_1:def 11;
          then consider g be Function of FXY,bool the carrier of TM such that
A50:      rng g is open and
A51:      rng g is Cover of A and
A52:      for a be set st a in FXY holds g.a c=a and
A53:      for a,b be set st a in FXY & b in FXY & a<>b holds g.a misses g.b
          by A4,A5,A6,A7,A21,A48;
A54:      rng(g|Fxy) is open by A50,RELAT_1:70,TOPS_2:11;
          xy in XY by TARSKI:def 1;
          then
A55:      xy in FXY by XBOOLE_0:def 3;
          then
A56:      g.xy c=xy by A52;
A57:      dom g=FXY by FUNCT_2:def 1;
          then
A58:      dom(g|Fxy)=FXY/\Fxy by RELAT_1:61
            .=Fxy by XBOOLE_1:21;
          g.xy in rng g by A55,A57,FUNCT_1:def 3;
          then reconsider gxy=g.xy as open Subset of TM by A50;
          set gxyA=gxy/\A;
          gxyA c=gxy by XBOOLE_1:17;
          then
A59:      gxyA c=xy by A56;
          [#](TM|A)=A by PRE_TOPC:def 5;
          then reconsider GxyA=gxyA as Subset of TM|A by XBOOLE_1:17;
A60:      TM|A|GxyA=TM|gxyA by METRIZTS:9;
          TM|A is finite-ind by A6;
          then
A61:      gxyA is finite-ind by A60,TOPDIM_1:18;
          then
A62:      ind gxyA=ind TM|gxyA by TOPDIM_1:17;
          ind GxyA<=ind TM|A by A6,TOPDIM_1:19;
          then ind GxyA<=0 by A6,A7,TOPDIM_1:17;
          then ind gxyA<=0 by A61,TOPDIM_1:21;
          then consider V1,V2 be open Subset of TM such that
A63:      V1 c=x & V2 c=y and
A64:      V1 misses V2 and
A65:      gxyA c=V1\/V2 by A5,A59,A60,A61,A62,Lm10;
          reconsider gV1=gxy/\V1,gV2=gxy/\V2 as open Subset of TM;
          set h=(x,y)-->(gV1,gV2);
A66:      dom h={x,y} by FUNCT_4:62;
          then
A67:      dom((g|Fxy)+*h)=Fxy\/{x,y} by A58,FUNCT_4:def 1;
          not x in F\X by ZFMISC_1:56;
          then
A68:      not x in Fxy by ZFMISC_1:56;
A69:      x in {x,y} by TARSKI:def 2;
A70:      Fxy\/{x,y}=Fxy\/(Y\/X) by ENUMSET1:1
            .=Fxy\/Y\/X by XBOOLE_1:4
            .=F by A18,A19,ZFMISC_1:116;
          for A be Subset of TM st A in {gV1,gV2} holds A is open
            by TARSKI:def 2;
          then
A71:      {gV1,gV2} is open;
A72:      y in {x,y} by TARSKI:def 2;
          not y in Fxy by ZFMISC_1:56;
          then
A73:      Fxy misses{x,y} by A68,ZFMISC_1:51;
          then
A74:      rng((g|Fxy)+*h)=rng(g|Fxy)\/rng h by A58,A66,NECKLACE:6;
          then reconsider gh=(g|Fxy)+*h as
            Function of F,bool the carrier of TM by A67,A70,FUNCT_2:2;
A75:      Fxy c=dom gh by A67,XBOOLE_1:7;
          take gh;
A76:      x<>y by A18,ZFMISC_1:56;
          then rng h={gV1,gV2} by FUNCT_4:64;
          hence rng gh is open by A54,A71,A74,TOPS_2:13;
          h.x=gV1 by A76,FUNCT_4:63;
          then
A77:      gh.x=gV1 by A66,A69,FUNCT_4:13;
          h.y=gV2 by FUNCT_4:63;
          then
A78:      gh.y=gV2 by A66,A72,FUNCT_4:13;
A79:      for a,b be set st a in Fxy & b in {x,y} & a<>b holds gh.a misses gh
          .b
          proof
            xy in XY by TARSKI:def 1; then
A80:        xy in FXY by XBOOLE_0:def 3;
            let a,b be set such that
A81:        a in Fxy and
A82:        b in {x,y} and
            a<>b;
            (g|Fxy).a=g.a & not a in dom h
            by A58,A73,A81,FUNCT_1:47,XBOOLE_0:3;
            then
A83:        gh.a=g.a by FUNCT_4:11;
A84:        a<>xy
            proof
              assume a=xy; then
A85:          union F=union Fxy by A49,A81,XBOOLE_1:12,ZFMISC_1:74;
              not A c=union Fxy by A46,SETFAM_1:def 11;
              hence thesis by A9,A85,SETFAM_1:def 11;
            end;
            assume gh.a meets gh.b;
            then consider z be object such that
A86:        z in gh.a and
A87:        z in gh.b by XBOOLE_0:3;
            z in gV1 or z in gV2 by A78,A77,A82,A87,TARSKI:def 2;
            then z in gxy by XBOOLE_0:def 4;
            then g.xy meets g.a by A83,A86,XBOOLE_0:3;
            hence thesis by A53,A58,A80,A81,A84;
          end;
A88:      {x,y}c=dom gh by A67,XBOOLE_1:7;
          then
A89:      gh.y in rng gh by A72,FUNCT_1:def 3;
A90:      gh.x in rng gh by A69,A88,FUNCT_1:def 3;
          A c=union rng gh
          proof
            let z be object such that
A91:        z in A;
            A c=union rng g by A51,SETFAM_1:def 11;
            then consider G be set such that
A92:        z in G and
A93:        G in rng g by A91,TARSKI:def 4;
            consider Gx be object such that
A94:        Gx in FXY and
A95:        g.Gx=G by A57,A93,FUNCT_1:def 3;
            per cases by A94,XBOOLE_0:def 3;
            suppose
A96:          Gx in Fxy;
              then (g|Fxy).Gx=G & not Gx in dom h by A58,A73,A95,FUNCT_1:47
,XBOOLE_0:3;
              then
A97:          gh.Gx=G by FUNCT_4:11;
              gh.Gx in rng gh by A75,A96,FUNCT_1:def 3;
              hence thesis by A92,A97,TARSKI:def 4;
            end;
            suppose Gx in XY;
              then
A98:          z in gxy by A92,A95,TARSKI:def 1;
              then z in gxyA by A91,XBOOLE_0:def 4;
              then z in V1 or z in V2 by A65,XBOOLE_0:def 3;
              then z in gV1 or z in gV2 by A98,XBOOLE_0:def 4;
              hence thesis by A78,A77,A90,A89,TARSKI:def 4;
            end;
          end;
          hence rng gh is Cover of A by SETFAM_1:def 11;
A99:      gV1 c=V1 & gV2 c=V2 by XBOOLE_1:17;
          hereby
            let a be set;
            assume
A100:       a in F;
            per cases by A70,A100,XBOOLE_0:def 3;
            suppose
A101:         a in Fxy; then
A102:         (g|Fxy).a=g.a by A58,FUNCT_1:47;
              (not a in dom h) & g.a c=a by A52,A58,A73,A101,XBOOLE_0:3;
              hence gh.a c=a by A102,FUNCT_4:11;
            end;
            suppose a in {x,y};
              then a=x or a=y by TARSKI:def 2;
              hence gh.a c=a by A63,A78,A77,A99;
            end;
          end;
          let a,b be set such that
A103:     a in F & b in F and
A104:     a<>b;
          per cases by A70,A103,XBOOLE_0:def 3;
          suppose
A105:       a in Fxy & b in Fxy;
            then (g|Fxy).a=g.a & not a in dom h
              by A58,A73,FUNCT_1:47,XBOOLE_0:3;
            then
A106:       gh.a=g.a by FUNCT_4:11;
A107:       (g|Fxy).b=g.b & not b in dom h by A58,A73,A105,FUNCT_1:47
,XBOOLE_0:3;
            g.a misses g.b by A53,A58,A104,A105;
            hence thesis by A106,A107,FUNCT_4:11;
          end;
          suppose a in Fxy & b in {x,y} or a in {x,y} & b in Fxy;
            hence thesis by A79,A104;
          end;
          suppose
A108:       a in {x,y} & b in {x,y};
            then a=x or a=y by TARSKI:def 2;
            then a=x & b=y or a=y & b=x by A104,A108,TARSKI:def 2;
            hence thesis by A64,A78,A77,A99,XBOOLE_1:64;
          end;
        end;
      end;
    end;
  end;
A109: P[0]
  proof
    let A be Subset of TM such that
    TM|A is second-countable and
    A is finite-ind and
    ind A<=0;
    let F be finite Subset-Family of TM such that
    F is open and
A110: F is Cover of A and
A111: card F<=0;
    rng{}c=bool the carrier of TM & dom{}=F by A111;
    then reconsider g={} as Function of F,bool the carrier of TM by FUNCT_2:2;
    take g;
    F={} by A111;
    hence thesis by A110;
  end;
  for n holds P[n] from NAT_1:sch 2(A109,A3);
  then P[card F];
  hence thesis by A1,A2,TOPDIM_1:18;
end;
