reserve M for non empty MetrSpace,
        F,G for open Subset-Family of TopSpaceMetr M;
reserve L for Lebesgue_number of F;
reserve n,k for Nat,
        r for Real,
        X for set,
        M for Reflexive non empty MetrStruct,
        A for Subset of M,
        K for SimplicialComplexStr;
reserve V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve A for Subset of TOP-REAL n;
reserve A for affinely-independent Subset of TOP-REAL n;

theorem Th25:
  card A = n+1 implies ind conv A = n
  proof
  set TR=TOP-REAL n;
  assume A1: card A=n+1;
  set C=conv A;
  A2: ind C>=n
  proof
   set E=the Enumeration of A;
   assume A3: ind C<n;
   A is non empty by A1;
   then reconsider C as non empty Subset of TR;
   ind C is natural;
   then reconsider n1=n-1 as Nat by A3;
   deffunc F(object)=C\conv(A\{E.$1});
   reconsider c =C as Subset of the TopStruct of TR;
   set TRC=TR|C;
   set carr=the carrier of TRC;
   A4: the TopStruct of TR=TopSpaceMetr Euclid n & the TopStruct of TRC=(the
TopStruct of TR)|c by EUCLID:def 8,PRE_TOPC:36;
   consider f be FinSequence such that
    A5: len f=len E & for k st k in dom f holds f.k=F(k) from FINSEQ_1:sch 2;
   A6: [#]TRC=C by PRE_TOPC:def 5;
   rng f c=bool carr
   proof
    let y be object;
    assume y in rng f;
    then consider x be object such that
     A7: x in dom f and
     A8: f.x=y by FUNCT_1:def 3;
    f.x=F(x) by A5,A7;
    then f.x c=C by XBOOLE_1:36;
    hence thesis by A6,A8;
   end;
   then reconsider R=rng f as finite Subset-Family of TRC;
   A9: rng E=A by RLAFFIN3:def 1;
   then A10: len E=card A by FINSEQ_4:62;
   A11: dom f=dom E by A5,FINSEQ_3:29;
   the carrier of TRC c=union R
   proof
    let x be object;
    assume A12: x in the carrier of TRC;
    assume A13: not x in union R;
    now let y be set;
     assume A14: y in A;
     then consider n be object such that
      A15: n in dom E and
      A16: E.n=y by A9,FUNCT_1:def 3;
     reconsider n as Nat by A15;
     f.n=F(n) by A5,A11,A15;
     then F(n) in R by A11,A15,FUNCT_1:def 3;
     then not x in F(n) by A13,TARSKI:def 4;
     then conv(A\{E.n})c=Affin(A\{E.n}) & x in conv(A\{E.n}) by A6,A12,
RLAFFIN1:65,XBOOLE_0:def 5;
     then A17: x|--A=x|--(A\{E.n}) by RLAFFIN1:77,XBOOLE_1:36;
     Carrier(x|--(A\{E.n}))c=A\{E.n} & E.n in {E.n} by RLVECT_2:def 6
,TARSKI:def 1;
     then not E.n in Carrier(x|--(A\{E.n})) by XBOOLE_0:def 5;
     hence (x|--A).y=0 by A14,A16,A17,RLVECT_2:19;
    end;
    then A18: x in conv(A\A) by A6,A12,RLAFFIN1:76;
    A\A={} by XBOOLE_1:37;
    hence contradiction by A18;
   end;
   then A19: R is Cover of TRC by SETFAM_1:def 11;
   now let U be Subset of TRC;
    assume U in R;
    then consider x be object such that
     A20: x in dom f & f.x=U by FUNCT_1:def 3;
    reconsider cAE=conv(A\{E.x}) as Subset of TRC by A6,RLAFFIN1:3,XBOOLE_1:36;
    A\{E.x} is affinely-independent by RLAFFIN1:43,XBOOLE_1:36;
    then A21: cAE is closed by TSEP_1:8;
    U=cAE` by A6,A5,A20;
    hence U is open by A21;
   end;
   then A22: ind TRC=ind C & R is open by TOPDIM_1:17,TOPS_2:def 1;
   ind C<n1+1 by A3;
   then ind C<=n1 by NAT_1:13;
   then consider G be finite Subset-Family of TRC such that
    A23: G is open and
    A24: G is Cover of TRC and
    A25: G is_finer_than R and
    card G<=card R*(n1+1) and
    A26: order G<=n1 by A4,A22,A19,TOPDIM_2:23;
   defpred P[Nat,set,set] means
    $3={g where g is Subset of TRC:g in G & (g c=f.($1+1) or g in $2)};
   defpred P[set] means
    $1 in G & $1c=f.1;
   consider G0 be Subset-Family of TRC such that
    A27: for x be set holds x in G0 iff x in bool carr & P[x] from SUBSET_1:sch
1;
   A28: for k be Nat st 1<=k & k<len E for x be Element of bool bool carr ex y
be Element of bool bool carr st P[k,x,y]
   proof
    let k be Nat such that
     1<=k and
     k<len E;
    let x be Element of bool bool carr;
    set y={g where g is Subset of TRC:g in G & (g c=f.(k+1) or g in x)};
    y c=bool carr
    proof
     let z be object;
     assume z in y;
     then ex g be Subset of TRC st g=z & g in G & (g c=f.(k+1) or g in x);
     hence thesis;
    end;
    hence thesis;
   end;
   consider p be FinSequence of bool bool carr such that
    A29: len p=len E and
    A30: p/.1=G0 or len E=0 and
    A31: for k be Nat st 1<=k & k<len E holds P[k,p/.k,p/.(k+1)] from NAT_2:sch
1(A28);
   defpred H[Nat,object] means
    ($1=1 implies $2=union G0) & ($1<>1 implies $2=union(p.$1\p.($1-1)));
   A32: for k be Nat st k in Seg len E ex x be object st H[k,x]
   proof
    let k be Nat;
    k=1 or k<>1;
    hence thesis;
   end;
   consider h be FinSequence such that
    A33: dom h=Seg len E and
    A34: for k be Nat st k in Seg len E holds H[k,h.k] from FINSEQ_1:sch 1(A32
);
   A35: dom p=Seg len E by A29,FINSEQ_1:def 3;
   rng h c=bool carr
   proof
    let y be object;
    assume y in rng h;
    then consider x be object such that
     A36: x in dom h and
     A37: h.x=y by FUNCT_1:def 3;
    reconsider x as Nat by A36;
    p.x in rng p by A35,A33,A36,FUNCT_1:def 3;
    then reconsider px=p.x as Subset-Family of TRC;
    y=union G0 or y=union(px\p.(x-1)) by A33,A34,A36,A37;
    hence thesis;
   end;
   then reconsider h as FinSequence of bool carr by FINSEQ_1:def 4;
   A38: A is non empty affinely-independent Subset of TOP-REAL n by A1;
   A39: 1<=n+1 by NAT_1:11;
   the carrier of TRC c=union rng h
   proof
    let x be object;
    assume x in the carrier of TRC;
    then x in union G by A24,SETFAM_1:45;
    then consider y be set such that
     A40: x in y and
     A41: y in G by TARSKI:def 4;
    consider z be set such that
     A42: z in R and
     A43: y c=z by A25,A41,SETFAM_1:def 2;
    consider k be object such that
     A44: k in dom f and
     A45: f.k=z by A42,FUNCT_1:def 3;
    reconsider k as Nat by A44;
    A46: k>=1 by A44,FINSEQ_3:25;
    A47: len E>=k by A5,A44,FINSEQ_3:25;
    per cases by A46,XXREAL_0:1;
    suppose A48: k=1 or y in G0;
     A49: 1 in Seg len E by A1,A10,A39,FINSEQ_1:1;
     then A50: h.1=union G0 by A34;
     y in G0 by A27,A41,A43,A45,A48;
     then A51: x in h.1 by A40,A50,TARSKI:def 4;
     h.1 in rng h by A33,A49,FUNCT_1:def 3;
     hence thesis by A51,TARSKI:def 4;
    end;
    suppose A52: k>1 & not y in G0;
     defpred Q[Nat] means
      $1>1 & $1<=len E & y c=f.$1;
     A53: ex k be Nat st Q[k] by A43,A45,A47,A52;
     consider m be Nat such that
      A54: Q[m] & for n be Nat st Q[n] holds m<=n from NAT_1:sch 5(A53);
     reconsider m1=m-1 as Element of NAT by A54,NAT_1:20;
     defpred R[Nat] means
      1<=$1 & $1<=m1 implies not y in p/.$1;
     m1+1<=len E by A54;
     then A55: m1<len E by NAT_1:13;
     A56: for n be Nat st R[n] holds R[n+1]
     proof
      let n be Nat such that
       A57: R[n];
      set n1=n+1;
      assume that
       1<=n1 and
       A58: n1<=m1;
      n<m1 by A58,NAT_1:13;
      then A59: n<len E by A55,XXREAL_0:2;
      per cases by NAT_1:14;
      suppose n=0;
       hence thesis by A1,A9,A30,A52,FINSEQ_4:62;
      end;
      suppose A60: n>=1;
       assume A61: y in p/.n1;
       p/.n1={g where g is Subset of TRC:g in G & (g c=f.n1 or g in p/.n)} by
A31,A59,A60;
       then ex g be Subset of TRC st y=g & g in G & (g c=f.n1 or g in p/.n) by
A61;
       then Q[n1] by A55,A57,A58,A60,NAT_1:13,XXREAL_0:2;
       then m<=n1 by A54;
       then m1+1<=m1 by A58,XXREAL_0:2;
       hence contradiction by NAT_1:13;
      end;
     end;
     A62: R[0];
     A63: for n holds R[n] from NAT_1:sch 2(A62,A56);
     A64: m in dom p by A29,A54,FINSEQ_3:25;
     then A65: h.m in rng h by A35,A33,FUNCT_1:def 3;
     m1+1>1 by A54;
     then A66: m1>=1 by NAT_1:13;
     then A67: p/.m={g where g is Subset of TRC:g in G & (g c=f.(m1+1) or g in
p/.m1)} by A31,A55;
     m1 in dom p by A29,A66,A55,FINSEQ_3:25;
     then p.m1=p/.m1 by PARTFUN1:def 6;
     then A68: not y in p.m1 by A66,A63;
     p.m=p/.m by A64,PARTFUN1:def 6;
     then y in p.m by A41,A54,A67;
     then y in p.m\p.m1 by A68,XBOOLE_0:def 5;
     then A69: x in union(p.m\p.m1) by A40,TARSKI:def 4;
     h.m=union(p.m\p.m1) by A35,A34,A54,A64;
     hence thesis by A69,A65,TARSKI:def 4;
    end;
   end;
   then A70: rng h is Cover of TRC by SETFAM_1:def 11;
   now let A be Subset of TRC;
    assume A in rng h;
    then consider k be object such that
     A71: k in dom h and
     A72: h.k=A by FUNCT_1:def 3;
    reconsider k as Nat by A71;
    A73: k>=1 by A33,A71,FINSEQ_1:1;
    per cases by A73,XXREAL_0:1;
    suppose A74: k=1;
     A75: G0 c=G
     by A27;
     h.k=union G0 by A33,A34,A71,A74;
     hence A is open by A23,A72,A75,TOPS_2:11,19;
    end;
    suppose A76: k>1;
     then reconsider k1=k-1 as Element of NAT by NAT_1:20;
     k1+1>1 by A76;
     then A77: k1>=1 by NAT_1:13;
     k1+1<=len E by A33,A71,FINSEQ_1:1;
     then A78: k1<len E by NAT_1:13;
     then k1 in dom p by A29,A77,FINSEQ_3:25;
     then A79: p.k1=p/.k1 by PARTFUN1:def 6;
     A80: P[k1,p/.k1,p/.(k1+1)] by A31,A77,A78;
     p/.k c=G
     proof
      let x be object;
      assume x in p/.k;
      then ex g be Subset of TRC st x=g & g in G & (g c=f.k or g in p/.k1) by
A80;
      hence thesis;
     end;
     then p/.k is open by A23,TOPS_2:11;
     then A81: p/.k\p/.(k-1) is open by TOPS_2:15;
     A82: p.k=p/.k by A35,A33,A71,PARTFUN1:def 6;
     A=union(p.k\p.(k-1)) by A33,A34,A71,A72,A76;
     hence A is open by A82,A81,A79,TOPS_2:19;
    end;
   end;
   then A83: rng h is open by TOPS_2:def 1;
   TRC is compact by COMPTS_1:3;
   then consider gx be Subset-Family of TRC such that
    gx is open and
    A84: gx is Cover of TRC and
    A85: clf gx is_finer_than rng h and
    A86: gx is locally_finite by A4,A70,A83,PCOMPS_1:22,PCOMPS_2:3;
   set cgx=clf gx;
   defpred G[object,object] means
    $2=union{u where u is Subset of TRC:u in cgx & u c=h.$1};
   A87: for k be Nat st k in Seg len E ex x be Element of bool carr st G[k,x]
   proof
    let k be Nat;
    set U={u where u is Subset of TRC:u in cgx & u c=h.k};
    U c=bool carr
    proof
     let x be object;
     assume x in U;
     then ex u be Subset of TRC st u=x & u in cgx & u c=h.k;
     hence thesis;
    end;
    then reconsider U as Subset-Family of TRC;
    union U is Subset of TRC;
    hence thesis;
   end;
   consider GX be FinSequence of bool carr such that
    A88: dom GX=Seg len E & for k be Nat st k in Seg len E holds G[k,GX.k]
from FINSEQ_1:sch 5(A87);
   A89: for i be Nat st i in dom GX holds GX.i c=h.i
   proof
    let i be Nat;
    set U={u where u is Subset of TRC:u in cgx & u c=h.i};
    now let x be set;
     assume x in U;
     then ex u be Subset of TRC st x=u & u in cgx & u c=h.i;
     hence x c=h.i;
    end;
    then A90: union U c=h.i by ZFMISC_1:76;
    assume i in dom GX;
    hence thesis by A88,A90;
   end;
   A91: dom E=Seg len E by FINSEQ_1:def 3;
   A92: for k be Nat st k in Seg len E holds h.k misses conv(A\{E.k})
   proof
    let k be Nat;
    assume A93: k in Seg len E;
    then A94: k>=1 by FINSEQ_1:1;
    A95: H[k,h.k] by A34,A93;
    assume h.k meets conv(A\{E.k});
    then consider x be object such that
     A96: x in h.k and
     A97: x in conv(A\{E.k}) by XBOOLE_0:3;
    per cases by A94,XXREAL_0:1;
    suppose A98: k=1;
     then consider y be set such that
      A99: x in y and
      A100: y in G0 by A95,A96,TARSKI:def 4;
     P[y] by A27,A100;
     then y c=F(k) by A5,A11,A91,A93,A98;
     hence contradiction by A97,A99,XBOOLE_0:def 5;
    end;
    suppose A101: k>1;
     then reconsider k1=k-1 as Element of NAT by NAT_1:20;
     len E>=k1+1 by A93,FINSEQ_1:1;
     then A102: len E>k1 by NAT_1:13;
     k1+1>1 by A101;
     then A103: k1>=1 by NAT_1:13;
     then A104: P[k1,p/.k1,p/.(k1+1)] by A31,A102;
     k1 in dom p by A29,A103,A102,FINSEQ_3:25;
     then A105: p/.k1=p.k1 by PARTFUN1:def 6;
     A106: p/.k=p.k by A35,A93,PARTFUN1:def 6;
     consider y be set such that
      A107: x in y and
      A108: y in p.k\p.(k-1) by A95,A96,A101,TARSKI:def 4;
     y in p.k by A108;
     then consider g be Subset of TRC such that
      A109: y=g and
      g in G and
      A110: g c=f.k or g in p.k1 by A104,A106,A105;
     f.k=F(k) by A5,A11,A91,A93;
     hence contradiction by A97,A107,A108,A109,A110,XBOOLE_0:def 5;
    end;
   end;
   carr c=union rng GX
   proof
    let x be object;
    assume A111: x in carr;
    union gx=carr & union gx c=union cgx by A84,PCOMPS_1:19,SETFAM_1:45;
    then consider y be set such that
     A112: x in y and
     A113: y in cgx by A111,TARSKI:def 4;
    consider r be set such that
     A114: r in rng h and
     A115: y c=r by A85,A113,SETFAM_1:def 2;
    consider k be object such that
     A116: k in dom h and
     A117: h.k=r by A114,FUNCT_1:def 3;
    A118: G[k,GX.k] by A33,A88,A116;
    A119: GX.k in rng GX by A33,A88,A116,FUNCT_1:def 3;
    y in {u where u is Subset of TRC:u in cgx & u c=h.k} by A113,A115,A117;
    then x in GX.k by A112,A118,TARSKI:def 4;
    hence thesis by A119,TARSKI:def 4;
   end;
   then A120: rng GX is Cover of TRC by SETFAM_1:def 11;
   A121: for S be Subset of dom GX holds conv(E.:S)c=union(GX.:S)
   proof
    let S be Subset of dom GX;
    A122: rng GX=GX.:dom GX by RELAT_1:113;
    A123: union rng GX=carr by A120,SETFAM_1:45;
    per cases by XBOOLE_0:def 10;
    suppose S=dom GX;
     hence thesis by A6,A9,A91,A88,A122,A123,RELAT_1:113;
    end;
    suppose A124: not dom GX c=S;
     set U={conv(A\{E.i}) where i is Element of NAT:i in dom E\S};
     dom GX\S is non empty by A124,XBOOLE_1:37;
     then A125: conv(E.:S)=meet U by A91,A88,Th12;
     A126: meet U misses union(GX.:(dom E\S))
     proof
      assume meet U meets union(GX.:(dom E\S));
      then consider x be object such that
       A127: x in meet U and
       A128: x in union(GX.:(dom E\S)) by XBOOLE_0:3;
      consider y be set such that
       A129: x in y and
       A130: y in GX.:(dom E\S) by A128,TARSKI:def 4;
      consider i be object such that
       A131: i in dom GX and
       A132: i in dom E\S and
       A133: GX.i=y by A130,FUNCT_1:def 6;
      reconsider i as Element of NAT by A131;
      conv(A\{E.i}) in U by A132;
      then A134: meet U c=conv(A\{E.i}) by SETFAM_1:7;
      y c=h.i by A89,A131,A133;
      then h.i meets conv(A\{E.i}) by A127,A129,A134,XBOOLE_0:3;
      hence contradiction by A92,A88,A131;
     end;
     dom GX=dom E by A88,FINSEQ_1:def 3;
     then rng GX=GX.:(S\/(dom E\S)) by A122,XBOOLE_1:45
      .=GX.:S\/GX.:(dom E\S) by RELAT_1:120;
     then A135: union(GX.:S)\/union(GX.:(dom E\S)) =C by A6,A123,ZFMISC_1:78;
     conv(E.:S)c=C by A9,RELAT_1:111,RLAFFIN1:3;
     hence thesis by A125,A135,A126,XBOOLE_1:73;
    end;
   end;
   A136: cgx is locally_finite by A86,PCOMPS_1:18;
   now let A be Subset of TRC;
    assume A in rng GX;
    then consider k be object such that
     A137: k in dom GX & GX.k=A by FUNCT_1:def 3;
    set U={u where u is Subset of TRC:u in cgx & u c=h.k};
    A138: U c=cgx
    proof
     let x be object;
     assume x in U;
     then ex u be Subset of TRC st x=u & u in cgx & u c=h.k;
     hence thesis;
    end;
    then reconsider U as Subset-Family of TRC by XBOOLE_1:1;
    U is closed by A138,PCOMPS_1:11,TOPS_2:12;
    then union U is closed by A136,A138,PCOMPS_1:9,21;
    hence A is closed by A88,A137;
   end;
   then A139: rng GX is closed by TOPS_2:def 2;
   len GX=card A by A10,A88,FINSEQ_1:def 3;
   then meet rng GX is non empty by A139,A38,A121,Th22;
   then consider I be object such that
    A140: I in meet rng GX by XBOOLE_0:def 1;
   defpred T[Nat,set] means
    $2 in G & I in $2 & $2 in p.$1 & ($1<>1 implies not$2 in p.($1-1));
   A141: for k be Nat st k in Seg len E ex x be Element of bool carr st T[k,x]
   proof
    let k be Nat;
    assume A142: k in Seg len E;
    then A143: k>=1 by FINSEQ_1:1;
    A144: k<=len E by A142,FINSEQ_1:1;
    A145: GX.k c=h.k & H[k,h.k] by A34,A88,A89,A142;
    GX.k in rng GX by A88,A142,FUNCT_1:def 3;
    then meet rng GX c=GX.k by SETFAM_1:7;
    then A146: I in GX.k by A140;
    per cases by A143,XXREAL_0:1;
    suppose A147: k=1;
     1 in dom p by A1,A10,A35,A39,FINSEQ_1:1;
     then A148: p.1=G0 by A1,A9,A30,FINSEQ_4:62,PARTFUN1:def 6;
     consider g be set such that
      A149: I in g and
      A150: g in G0 by A146,A145,A147,TARSKI:def 4;
     g in G by A27,A150;
     hence thesis by A147,A149,A150,A148;
    end;
    suppose A151: k>1;
     then reconsider k1=k-1 as Nat;
     A152: k1+1=k;
     then A153: k1<len E by A144,NAT_1:13;
     k1>=1 by A151,A152,NAT_1:13;
     then A154: P[k1,p/.k1,p/.k] by A31,A153;
     A155: p.k=p/.k by A35,A142,PARTFUN1:def 6;
     consider g be set such that
      A156: I in g and
      A157: g in p.k\p.(k-1) by A146,A145,A151,TARSKI:def 4;
     A158: not g in p.(k-1) by A157,XBOOLE_0:def 5;
     g in p.k by A157;
     then ex gg be Subset of TRC st g=gg & gg in G & (gg c=f.(k1+1) or gg in p
/.k1) by A154,A155;
     hence thesis by A156,A157,A158;
    end;
   end;
   consider t be FinSequence of bool carr such that
    A159: dom t=Seg len E & for k be Nat st k in Seg len E holds T[k,t.k] from
FINSEQ_1:sch 5(A141);
   A160: now let i,j be Nat;
    assume that
     A161: i in dom t and
     A162: j in dom t and
     A163: i<j;
    A164: j<=len E by A159,A162,FINSEQ_1:1;
    defpred P[Nat] means
     i<=$1 & $1<j implies t.i in p.$1;
    A165: T[i,t.i] by A159,A161;
    A166: 1<=i by A159,A161,FINSEQ_1:1;
    A167: for k st P[k] holds P[k+1]
    proof
     let k;
     assume A168: P[k];
     set k1=k+1;
     assume that
      A169: i<=k1 and
      A170: k1<j;
     A171: k<j by A170,NAT_1:13;
     per cases by A169,NAT_1:8;
     suppose i=k1;
      hence thesis by A159,A161;
     end;
     suppose A172: i<=k;
      1<=k1 & k1<=len E by A166,A164,A169,A170,XXREAL_0:2;
      then A173: k1 in dom p by A35,FINSEQ_1:1;
      A174: k<len E by A164,A171,XXREAL_0:2;
      A175: 1<=k by A166,A172,XXREAL_0:2;
      then k in dom p by A35,A174,FINSEQ_1:1;
      then A176: p.k=p/.k by PARTFUN1:def 6;
      P[k,p/.k,p/.k1] by A31,A175,A174;
      then p.k1={g where g is Subset of TRC:g in G & (g c=f.k1 or g in p.k)}
by A173,A176,PARTFUN1:def 6;
      hence thesis by A165,A168,A170,A172,NAT_1:13;
     end;
    end;
    A177: P[0] by A165;
    A178: for k holds P[k] from NAT_1:sch 2(A177,A167);
    reconsider j1=j-1 as Nat by A163;
    assume A179: t.i=t.j;
    A180: j1+1=j;
    then A181: j1<j by NAT_1:13;
    A182: j<>1 by A159,A161,A163,FINSEQ_1:1;
    i<=j1 by A163,A180,NAT_1:13;
    then t.i in p.j1 by A181,A178;
    hence contradiction by A159,A162,A179,A182;
   end;
   now let x1,x2 be object;
    assume A183: x1 in dom t & x2 in dom t;
    then reconsider i1=x1,i2=x2 as Nat;
    assume A184: t.x1=t.x2;
    assume x1<>x2;
    then i1>i2 or i1<i2 by XXREAL_0:1;
    hence contradiction by A160,A183,A184;
   end;
   then t is one-to-one by FUNCT_1:def 4;
   then A185: card rng t=len t by FINSEQ_4:62
    .=len E by A159,FINSEQ_1:def 3
    .=n+1 by A1,A9,FINSEQ_4:62;
   then A186: rng t is non empty;
   A187: now let x be set;
    assume x in rng t;
    then consider i be object such that
     A188: i in dom t & t.i=x by FUNCT_1:def 3;
    thus I in x by A159,A188;
   end;
   A189: rng t c=G
   proof
    let y be object;
    assume y in rng t;
    then consider x be object such that
     A190: x in dom t & t.x=y by FUNCT_1:def 3;
    thus thesis by A159,A190;
   end;
   n<card rng t by A185,NAT_1:13;
   then card Segm n in card Segm card rng t by NAT_1:41;
   then n1+1 in card rng t;
   then meet rng t is empty by A26,A189,TOPDIM_2:2;
   hence contradiction by A186,A187,SETFAM_1:def 1;
  end;
  ind C<=ind TR & ind TR<=n by TOPDIM_1:20,TOPDIM_2:21;
  then ind C<=n by XXREAL_0:2;
  hence thesis by A2,XXREAL_0:1;
 end;
