reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P1,P2 for Subset of TOP-REAL 2,
  f,f1,f2,g1,g2 for FinSequence of TOP-REAL 2,
  n,m,i,j,k for Nat,
  G,G1 for Go-board,
  x,y for set;

theorem Th1:
  for G,f1,f2 st 1<=len f1 & 1<=len f2 &
    f1 is_sequence_on G & f2 is_sequence_on G &
    f1/.1 in rng Line(G,1) & f1/.len f1 in rng Line(G,len G) &
    f2/.1 in rng Col(G,1) & f2/.len f2 in rng Col(G,width G)
  holds rng f1 meets rng f2
proof
  defpred P[Nat] means
    for G1,g1,g2 st $1=width G1 & $1>0 & 1<=len  g1 & 1<=len g2 &
      g1 is_sequence_on G1 & g2 is_sequence_on G1 &
      g1/.1 in rng Line(G1,1) & g1/.len g1 in rng Line(G1,len G1) &
      g2/.1 in rng Col(G1,1) & g2/.len g2 in rng Col(G1,width G1)
     holds rng g1 /\ rng g2 <> {};
  let G,f1,f2;
A1: for k st P[k] holds P[k+1]
  proof
    let k such that
A2: P[k];
    let G1,g1,g2 such that
A3: k+1=width G1 and
    k+1>0 and
A4: 1<=len g1 and
A5: 1<=len g2 and
A6: g1 is_sequence_on G1 and
A7: g2 is_sequence_on G1 and
A8: g1/.1 in rng Line(G1,1) and
A9: g1/.len g1 in rng Line(G1,len G1) and
A10: g2/.1 in rng Col(G1,1) and
A11: g2/.len g2 in rng Col(G1,width G1);
    defpred P[Nat] means $1 in dom g2 & g2/.$1 in rng Col(G1,width G1);
A12: ex m be Nat st P[m] by A5,A11,FINSEQ_3:25;
    consider m be Nat such that
A13: P[m] & for i be Nat st P[i] holds m<=i from NAT_1:sch 5(A12);
    reconsider m as Nat;
    set g = g2|m;
A14: g/.1 in rng Col(G1,1) by A10,A13,FINSEQ_4:92;
A15: g is_sequence_on G1 by A7,GOBOARD1:22;
A16: m<=len g2 by A13,FINSEQ_3:25;
    then
A17: len g = m by FINSEQ_1:59;
A18: rng g c= rng g2
    proof
      let x be object;
      assume x in rng g;
      then consider n being Element of NAT such that
A19:  n in dom g and
A20:  x=g/.n by PARTFUN2:2;
A21:  n in Seg m by A17,A19,FINSEQ_1:def 3;
      then
A22:  n in dom g2 by A13,FINSEQ_4:71;
      x=g2/.n by A13,A20,A21,FINSEQ_4:71;
      hence thesis by A22,PARTFUN2:2;
    end;
    reconsider L1 = Line(G1,1), Ll = Line(G1,len G1) as FinSequence of
    TOP-REAL 2;
A23: dom g2 = Seg len g2 by FINSEQ_1:def 3;
A24: dom g = Seg len g by FINSEQ_1:def 3;
    0<>len G1 by MATRIX_0:def 10;
    then
A25: 0+1<=len G1 by NAT_1:14;
    then
A26: 1 in dom G1 by FINSEQ_3:25;
A27: len G1 in dom G1 by A25,FINSEQ_3:25;
A28: g/.len g in rng Col(G1,width G1) by A13,FINSEQ_4:93;
    defpred P[Nat] means $1 in dom G1 & rng g /\ rng Line(G1,$1) <> {};
A29: for n be Nat st P[n] holds n<=len G1 by FINSEQ_3:25;
    0<>width G1 by MATRIX_0:def 10;
    then
A30: 0+1<=width G1 by NAT_1:14;
    then
A31: 1 in Seg width G1 by FINSEQ_1:1;
A32: 1<=len g by A13,GOBOARD1:22;
    then
A33: 1 in dom g by FINSEQ_3:25;
A34: ex n be Nat st P[n]
    proof
      m in dom g2 implies g2/.1 = (g2|m)/.1 by FINSEQ_4:92;
      then consider i being Nat such that
A35:  i in dom Col(G1,1) and
A36:  g/.1=Col(G1,1).i by A10,A13,FINSEQ_2:10;
      reconsider i as Nat;
      take i;
      i in Seg len Col(G1,1) by A35,FINSEQ_1:def 3;
      then i in Seg len G1 by MATRIX_0:def 8;
      hence i in dom G1 by FINSEQ_1:def 3;
      then
A37:  g/.1=Line(G1,i).1 by A31,A36,MATRIX_0:42;
A38:  g/.1 in rng g by A33,PARTFUN2:2;
      len Line(G1,i)=width G1 by MATRIX_0:def 7;
      then dom Line(G1,i) = Seg width G1 by FINSEQ_1:def 3;
      then g/.1 in rng Line(G1,i) by A31,A37,FUNCT_1:def 3;
      hence thesis by A38,XBOOLE_0:def 4;
    end;
    consider mi be Nat such that
A39: P[mi] & for n be Nat st P[n] holds mi<=n from NAT_1:sch 5(A34);
A40: ex n be Nat st P[n] by A34;
    consider ma be Nat such that
A41: P[ma] & for n be Nat st P[n] holds n<=ma from NAT_1:sch 6(A29,A40
    );
    reconsider mi,ma as Nat;
A42: 1<=mi by A39,FINSEQ_3:25;
    reconsider Lmi = Line(G1,mi) as FinSequence of TOP-REAL 2;
    defpred P[Nat] means $1 in dom g1 & g1/.$1 in rng Line(G1,mi);
A43: for n be Nat st P[n] holds n<=len g1 by FINSEQ_3:25;
A44: mi<=len G1 by A39,FINSEQ_3:25;
    then ex n st P[n] by A4,A6,A8,A9,A42,GOBOARD1:29;
    then
A45: ex n be Nat st P[n];
    consider ma1 be Nat such that
A46: P[ma1] & for n be Nat st P[n] holds n<= ma1 from NAT_1:sch 6(A43,
    A45);
A47: ma<=len G1 by A41,FINSEQ_3:25;
    1<=mi by A39,FINSEQ_3:25;
    then reconsider l1=mi-1, l2=len G1-ma as Element of NAT by A47,INT_1:5;
A48: ma<=len G1 by A41,FINSEQ_3:25;
    reconsider ma1 as Nat;
    consider k1 be Element of NAT such that
A49: k1 in dom Lmi and
A50: g1/.ma1 = Lmi/.k1 by A46,PARTFUN2:2;
    Seg len Lmi = dom Lmi by FINSEQ_1:def 3;
    then
A51: k1 in Seg width G1 by A49,MATRIX_0:def 7;
A52: dom G1 = Seg len G1 by FINSEQ_1:def 3;
A53: mi=ma implies rng g1 /\ rng g2 <> {}
    proof
      consider n such that
A54:  n in dom g1 and
A55:  g1/.n in rng Line(G1,mi) by A4,A6,A8,A9,A42,A44,GOBOARD1:29;
      consider i being Element of NAT such that
A56:  i in dom Line(G1,mi) and
A57:  g1/.n=Lmi/.i by A55,PARTFUN2:2;
A58:  1<=i by A56,FINSEQ_3:25;
A59:  len Line(G1,mi)=width G1 by MATRIX_0:def 7;
      then i<=width G1 by A56,FINSEQ_3:25;
      then consider m such that
A60:  m in dom g and
A61:  g/.m in rng Col(G1,i) by A32,A15,A14,A28,A58,GOBOARD1:33;
A62:  g/.m in rng g by A60,PARTFUN2:2;
      reconsider Ci = Col(G1,i) as FinSequence of TOP-REAL 2;
A63:  len Col(G1,i)= len G1 by MATRIX_0:def 8;
      then
A64:  dom Col(G1,i) = Seg len G1 by FINSEQ_1:def 3
        .= dom G1 by FINSEQ_1:def 3;
      assume
A65:  mi=ma;
A66:  dom Line(G1,mi) = Seg len Line(G1,mi) by FINSEQ_1:def 3;
      consider j being Element of NAT such that
A67:  j in dom Ci and
A68:  g/.m=Ci/.j by A61,PARTFUN2:2;
      reconsider Lj = Line(G1,j) as FinSequence of TOP-REAL 2;
      len Line(G1,mi) = width G1 by MATRIX_0:def 7
        .= len Lj by MATRIX_0:def 7;
      then
A69:  dom Line(G1,mi) = dom Lj by A66,FINSEQ_1:def 3;
A70:  g/.m = Ci.j by A67,A68,PARTFUN1:def 6
        .= Lj.i by A56,A66,A59,A64,A67,MATRIX_0:42
        .= Lj/.i by A56,A69,PARTFUN1:def 6;
      len Lj=width G1 by MATRIX_0:def 7;
      then i in dom Lj by A56,A66,A59,FINSEQ_1:def 3;
      then g/.m in rng Lj by A70,PARTFUN2:2;
      then
A71:  dom Ci = Seg len Ci & rng g /\ rng Line(G1,j) <> {}
        by A62,FINSEQ_1:def 3,XBOOLE_0:def 4;
A72:  now
        assume j<>mi;
        then j<mi or j>mi by XXREAL_0:1;
        hence contradiction by A39,A52,A41,A65,A63,A67,A71;
      end;
      g1/.n in rng g1 by A54,PARTFUN2:2;
      hence thesis by A18,A57,A62,A70,A72,XBOOLE_0:def 4;
    end;
A73: width G1 in Seg width G1 by A30,FINSEQ_1:1;
    deffunc F(Nat) = G1*($1,k1);
    reconsider Ck1 = Col(G1,k1) as FinSequence of TOP-REAL 2;
    consider h1 be FinSequence of TOP-REAL 2 such that
A74: len h1 = l1 & for n being Nat st n in dom h1 holds h1/.n=F(n)
    from FINSEQ_4:sch 2;
A75: dom g1 = Seg len g1 by FINSEQ_1:def 3;
    now
      per cases;
      suppose
A76:    k=0;
        reconsider c1 = Col(G1,1) as FinSequence of TOP-REAL 2;
        consider x being Element of NAT such that
A77:    x in dom c1 and
A78:    g2/.1=c1/.x by A10,PARTFUN2:2;
        reconsider lx = Line(G1,x) as FinSequence of TOP-REAL 2;
A79:    1<=x by A77,FINSEQ_3:25;
A80:    len c1 = len G1 by MATRIX_0:def 8;
        then x<=len G1 by A77,FINSEQ_3:25;
        then consider m such that
A81:    m in dom g1 and
A82:    g1/.m in rng lx by A4,A6,A8,A9,A79,GOBOARD1:29;
A83:    g1/.m in rng g1 by A81,PARTFUN2:2;
        Seg len c1 = dom c1 by FINSEQ_1:def 3;
        then
A84:    x in dom G1 by A77,A80,FINSEQ_1:def 3;
A85:    c1/.x = c1.x by A77,PARTFUN1:def 6;
        consider y being Element of NAT such that
A86:    y in dom lx and
A87:    lx/.y=g1/.m by A82,PARTFUN2:2;
        Seg len lx=dom lx & len lx=width G1 by FINSEQ_1:def 3,MATRIX_0:def 7;
        then
A88:    y =1 by A3,A76,A86,FINSEQ_1:2,TARSKI:def 1;
        0<>width G1 by MATRIX_0:def 10;
        then 0+1<=width G1 by NAT_1:14;
        then
A89:    1 in Seg width G1 by FINSEQ_1:1;
        1 in dom g2 by A5,FINSEQ_3:25;
        then
A90:    g2/.1 in rng g2 by PARTFUN2:2;
        lx/.y = lx.y by A86,PARTFUN1:def 6;
        then g1/.m=g2/.1 by A78,A89,A84,A87,A85,A88,MATRIX_0:42;
        hence thesis by A83,A90,XBOOLE_0:def 4;
      end;
      suppose
A91:    k<>0;
        then
A92:    0<k;
        then
A93: width G1 > 1+0 by A3,XREAL_1:8;
    then
A94: width G1 in Seg width G1 by FINSEQ_1:1;
        now
          per cases;
          suppose
            mi=ma;
            hence thesis by A53;
          end;
          suppose
A95:        mi<>ma;
            1<=ma1 by A46,FINSEQ_3:25;
            then reconsider w=ma1-1 as Element of NAT by INT_1:5;
            reconsider Lma = Line(G1,ma) as FinSequence of TOP-REAL 2;
A96:        Indices G1=[:dom G1,Seg width G1:] by MATRIX_0:def 4;
A97:        now
              let n;
A98:          l1<=mi by XREAL_1:43;
              assume
A99:          n in dom h1;
              then
A100:          1<=n by FINSEQ_3:25;
              n<=l1 by A74,A99,FINSEQ_3:25;
              then
A101:         n<=mi by A98,XXREAL_0:2;
              mi<=len G1 by A39,FINSEQ_3:25;
              then n<=len G1 by A101,XXREAL_0:2;
              hence n in dom G1 by A100,FINSEQ_3:25;
            end;
A102:       now
              let n;
              assume that
A103:         n in dom h1 and
A104:         n+1 in dom h1;
              n+1 in dom G1 by A97,A104;
              then
A105:         [n+1,k1] in Indices G1 by A51,A96,ZFMISC_1:87;
              let i1,i2,j1,j2 be Nat;
              assume that
A106:         [i1,i2] in Indices G1 and
A107:         [j1,j2] in Indices G1 and
A108:         h1/.n=G1*(i1,i2) and
A109:         h1/.(n+1)=G1*(j1,j2);
              h1/.(n+1)=G1*(n+1,k1) by A74,A104;
              then
A110:         j1=n+1 & j2=k1 by A105,A107,A109,GOBOARD1:5;
              n in dom G1 by A97,A103;
              then
A111:         [n,k1] in Indices G1 by A51,A96,ZFMISC_1:87;
              h1/.n=G1*(n,k1) by A74,A103;
              then i1=n & i2=k1 by A111,A106,A108,GOBOARD1:5;
              hence |.i1-j1.|+|.i2-j2.|= |.n-n+-1.|+0 by A110,ABSVALUE:2
                .= |.1.| by COMPLEX1:52
                .= 1 by ABSVALUE:def 1;
            end;
A112:       rng h1 /\ rng g = {}
            proof
              set x = the Element of rng h1 /\ rng g;
              assume
A113:         not thesis;
              then x in rng h1 by XBOOLE_0:def 4;
              then consider n1 be Element of NAT such that
A114:         n1 in dom h1 and
A115:         x=h1/.n1 by PARTFUN2:2;
A116:         n1<=l1 by A74,A114,FINSEQ_3:25;
              n1 in dom G1 by A97,A114;
              then
A117:         [n1,k1] in Indices G1 by A51,A96,ZFMISC_1:87;
              x in rng g by A113,XBOOLE_0:def 4;
              then consider n2 be Element of NAT such that
A118:         n2 in dom g and
A119:         x=g/.n2 by PARTFUN2:2;
A120:         g/.n2 in rng g by A118,PARTFUN2:2;
              consider i1,i2 be Nat such that
A121:         [i1,i2] in Indices G1 and
A122:         g/.n2=G1*(i1,i2) by A15,A118,GOBOARD1:def 9;
              reconsider L=Line(G1,i1) as FinSequence of TOP-REAL 2;
A123:         i2 in Seg width G1 by A96,A121,ZFMISC_1:87;
A124:         Seg len L = dom L & len L=width G1
                 by FINSEQ_1:def 3,MATRIX_0:def 7;
              then L/.i2 = L.i2 by A123,PARTFUN1:def 6;
              then g/.n2=L/.i2 by A122,A123,MATRIX_0:def 7;
              then g/.n2 in rng L by A123,A124,PARTFUN2:2;
              then
A125:         rng g /\ rng L <> {} by A120,XBOOLE_0:def 4;
              i1 in dom G1 by A96,A121,ZFMISC_1:87;
              then
A126:         mi<=i1 by A39,A125;
              x=G1*(n1,k1) by A74,A114,A115;
              then i1=n1 by A119,A121,A122,A117,GOBOARD1:5;
              then mi<=mi-1 by A116,A126,XXREAL_0:2;
              hence contradiction by XREAL_1:44;
            end;
            defpred P[Nat] means $1 in dom g1 & ma1<$1 & g1/.$1 in rng Line(G1
            ,ma);
A127:       mi<=ma by A39,A41;
            then
A128:       mi<ma by A95,XXREAL_0:1;
            then ex n st P[n] by A6,A9,A39,A48,A46,GOBOARD1:37;
            then
A129:       ex n be Nat st P[n];
            consider mi1 be Nat such that
A130:       P[mi1] & for n be Nat st P[n] holds mi1<=n from NAT_1:
            sch 5(A129);
            consider k2 be Element of NAT such that
A131:       k2 in dom Lma and
A132:       g1/.mi1=Lma/.k2 by A130,PARTFUN2:2;
            deffunc F(Nat) = G1*(ma+$1,k2);
            consider h2 be FinSequence of TOP-REAL 2 such that
A133:       len h2 = l2 & for n being Nat st n in dom h2 holds h2/.n
            = F(n) from FINSEQ_4:sch 2;
            reconsider Ck2 = Col(G1,k2) as FinSequence of TOP-REAL 2;
            dom Lma = Seg len Lma by FINSEQ_1:def 3;
            then
A134:       k2 in Seg width G1 by A131,MATRIX_0:def 7;
A135:       now
              let n;
A136:         n<=n+ma by NAT_1:11;
              assume
A137:         n in dom h2;
              then n<=l2 by A133,FINSEQ_3:25;
              then
A138:         ma+n<=ma+l2 by XREAL_1:7;
              1<=n by A137,FINSEQ_3:25;
              then 1<=n+ma by A136,XXREAL_0:2;
              hence ma+n in dom G1 by A138,FINSEQ_3:25;
            end;
A139:       now
              let n;
              assume
A140:         n in dom h2;
              reconsider m=ma+n, k2 as Nat;
              take m,k2;
              ma+n in dom G1 by A135,A140;
              hence [m,k2] in Indices G1 & h2/.n=G1*(m,k2) by A134,A133,A96
,A140,ZFMISC_1:87;
            end;
A141:       now
              let n;
              assume that
A142:         n in dom h2 and
A143:         n+1 in dom h2;
              ma+(n+1) in dom G1 by A135,A143;
              then
A144:         [ma+n+1,k2] in Indices G1 by A134,A96,ZFMISC_1:87;
              let i1,i2,j1,j2 be Nat;
              assume that
A145:         [i1,i2] in Indices G1 and
A146:         [j1,j2] in Indices G1 and
A147:         h2/.n=G1* (i1,i2) and
A148:         h2/.(n+1)=G1*(j1,j2);
              h2/.(n+1)=G1*(ma+(n+1),k2) by A133,A143;
              then
A149:         j1=ma+n+1 & j2=k2 by A144,A146,A148,GOBOARD1:5;
              ma+n in dom G1 by A135,A142;
              then
A150:         [ma+n,k2] in Indices G1 by A134,A96,ZFMISC_1:87;
              h2/.n=G1 *(ma+n,k2) by A133,A142;
              then i1=ma+n & i2=k2 by A150,A145,A147,GOBOARD1:5;
              hence |.i1-j1.|+|.i2-j2.|= |.ma+n-(ma+n)+-1.|+0 by A149,
ABSVALUE:2
                .= |.1.| by COMPLEX1:52
                .= 1 by ABSVALUE:def 1;
            end;
A151:       mi1<=mi1+1 by NAT_1:11;
A152:       0+1<width G1 by A3,A92,XREAL_1:6;
A153:       len DelCol(G1,width G1)= len G1 by MATRIX_0:def 13;
            ma1<=mi1 by A130;
            then reconsider l=mi1+1-ma1 as Element of NAT by A151,INT_1:5
,XXREAL_0:2;
            set f1=g1|mi1;
            defpred P[Nat,Element of TOP-REAL 2] means $2=f1/.(w+$1);
A154:       for n being Nat st n in Seg l ex p st P[n,p];
            consider f such that
A155:       len f = l & for n being Nat st n in Seg l holds P[n,f/.n
            ] from FINSEQ_4:sch 1(A154);
A156:       w+l= mi1;
A157:       dom f = Seg l by A155,FINSEQ_1:def 3;
            set ff = h1^f^h2;
            ma1+1<=mi1 by A130,NAT_1:13;
            then ma1+1<=mi1+1 by NAT_1:13;
            then
A158:       1<=l by XREAL_1:19;
A159:       now
              per cases;
              suppose
A160:           mi=1;
                1<=ma1 by A75,A46,FINSEQ_1:1;
                then
A161:           ma1 in Seg mi1 by A130,FINSEQ_1:1;
A162:           w+1=ma1;
A163:           1 in Seg l by A158,FINSEQ_1:1;
                h1 = {} by A74,A160;
                then ff=f^h2 by FINSEQ_1:34;
                then ff/.1=f/.1 by A157,A163,FINSEQ_4:68
                  .= f1/.ma1 by A155,A163,A162
                  .= g1/.ma1 by A130,A161,FINSEQ_4:71;
                hence ff/.1 in rng Line(G1,1) by A46,A160;
              end;
              suppose
A164:           mi<>1;
                1<=mi by A39,FINSEQ_3:25;
                then 1<mi by A164,XXREAL_0:1;
                then 1+1<=mi by NAT_1:13;
                then
A165:           1<=l1 by XREAL_1:19;
                then
A166:           1 in dom h1 by A74,FINSEQ_3:25;
A167:           len Line(G1,1)=width G1 by MATRIX_0:def 7;
                then
A168:           k1 in dom L1 by A51,FINSEQ_1:def 3;
                dom Line(G1,1) = Seg width G1 by A167,FINSEQ_1:def 3;
                then
A169:           L1/.k1 = L1.k1 by A51,PARTFUN1:def 6;
                len(h1^f)=len h1 + len f & 0<=len f by FINSEQ_1:22;
                then 0+1<=len(h1^f) by A74,A165,XREAL_1:7;
                then 1 in dom(h1^f) by FINSEQ_3:25;
                then ff/.1=(h1^f)/.1 by FINSEQ_4:68
                  .=h1/.1 by A166,FINSEQ_4:68
                  .=G1*(1,k1) by A74,A166
                  .=L1/.k1 by A51,A169,MATRIX_0:def 7;
                hence ff/.1 in rng Line(G1,1) by A168,PARTFUN2:2;
              end;
            end;
A170:       for n st n in Seg l holds f1/.(w+n)=g1/.(w+n) & w+n in dom g1
            proof
              0+1<=ma1 by A46,FINSEQ_3:25;
              then
A171:         0<=ma1-1 by XREAL_1:19;
              let n such that
A172:         n in Seg l;
              n<=l by A172,FINSEQ_1:1;
              then n+ma1<=l+ma1 by XREAL_1:7;
              then
A173:         n+ma1-1<=mi1 by XREAL_1:20;
              1<=n by A172,FINSEQ_1:1;
              then 0+1<=ma1-1+n by A171,XREAL_1:7;
              then w+n in Seg mi1 by A173,FINSEQ_1:1;
              hence thesis by A130,FINSEQ_4:71;
            end;
A174:       now
              let n;
              assume
A175:         n in dom f;
              then w+n in dom g1 by A170,A157;
              then consider i,j such that
A176:         [i,j] in Indices G1 & g1/.(w+n)=G1*(i,j) by A6,GOBOARD1:def 9;
              take i,j;
              f/.n=f1/.(w+n) by A155,A157,A175;
              hence [i,j] in Indices G1 & f/.n=G1*(i,j) by A170,A157,A175,A176;
            end;
A177:       Seg len f=dom f by FINSEQ_1:def 3;
A178:       rng f c= rng g1
            proof
              let x be object;
              assume x in rng f;
              then consider n being Element of NAT such that
A179:         n in dom f and
A180:         x=f/.n by PARTFUN2:2;
              f/.n=f1/.(w+n) by A155,A177,A179;
              then
A181:         f/.n=g1/.(w+n) by A170,A155,A177,A179;
              w+n in dom g1 by A170,A155,A177,A179;
              hence thesis by A180,A181,PARTFUN2:2;
            end;
A182:       dom h2 = Seg len h2 by FINSEQ_1:def 3;
A183:       now
              per cases;
              suppose
A184:           ma=len G1;
                1<=mi1 by A130,FINSEQ_3:25;
                then
A185:           mi1 in Seg mi1 by FINSEQ_1:1;
A186:           len f in dom f by A155,A177,A158,FINSEQ_1:1;
                h2 = {} by A133,A184;
                then
A187:           ff=h1^f by FINSEQ_1:34;
                then ff/.len ff=ff/.(len h1+len f) by FINSEQ_1:22
                  .= f/.l by A155,A187,A186,FINSEQ_4:69
                  .= f1/.mi1 by A155,A157,A156,A186
                  .= g1/.mi1 by A130,A185,FINSEQ_4:71;
                hence ff/.len ff in rng Line(G1,len G1) by A130,A184;
              end;
              suppose
A188:           ma<>len G1;
                ma<=len G1 by A41,FINSEQ_3:25;
                then ma<len G1 by A188,XXREAL_0:1;
                then ma+1<=len G1 by NAT_1:13;
                then
A189:           1<=l2 by XREAL_1:19;
                then
A190:           l2 in Seg l2 by FINSEQ_1:1;
A191:           len h2 in dom h2 by A133,A189,FINSEQ_3:25;
A192:           len Line(G1,len G1)=width G1 by MATRIX_0:def 7;
                then
A193:           k2 in dom Ll by A134,FINSEQ_1:def 3;
                k2 in dom Ll by A134,A192,FINSEQ_1:def 3;
                then
A194:           Ll/.k2 = Ll.k2 by PARTFUN1:def 6;
                ff/.len ff=ff/.(len(h1^f)+len h2) by FINSEQ_1:22
                  .=h2/.l2 by A133,A191,FINSEQ_4:69
                  .=G1*(ma+l2,k2) by A133,A182,A190
                  .=Ll/.k2 by A134,A194,MATRIX_0:def 7;
                hence ff/.len ff in rng Line(G1,len G1) by A193,PARTFUN2:2;
              end;
            end;
A195:       0+1<=len f+(len h1+len h2) by A155,A158,XREAL_1:7;
A196:       rng ff = rng(h1^f) \/ rng h2 by FINSEQ_1:31
              .= rng h1 \/ rng f \/ rng h2 by FINSEQ_1:31;
A197:       for k st k in Seg width G1 & rng f /\ rng Col(G1,k)={} holds
            rng ff /\ rng Col(G1,k)={}
            proof
A198:         len Col(G1,k2)=len G1 by MATRIX_0:def 8;
A199:         len Col(G1,k1)=len G1 by MATRIX_0:def 8;
A200:         w+1=ma1;
              let k;
              assume that
A201:         k in Seg width G1 and
A202:         rng f /\ rng Col(G1,k)={};
              set gg=Col(G1,k);
              assume
A203:         rng ff /\ rng gg <> {};
              set x = the Element of rng ff /\ rng gg;
              rng ff = rng f \/ (rng h1 \/ rng h2) by A196,XBOOLE_1:4;
              then
A204:         rng ff /\ rng gg = {} \/ (rng h1 \/ rng h2) /\ rng gg by A202,
XBOOLE_1:23
                .= rng h1 /\ rng gg \/ rng h2 /\ rng gg by XBOOLE_1:23;
              now
                per cases by A203,A204,XBOOLE_0:def 3;
                suppose
A205:             x in rng h1 /\ rng gg;
                  then x in rng h1 by XBOOLE_0:def 4;
                  then consider i being Element of NAT such that
A206:             i in dom h1 and
A207:             x=h1/.i by PARTFUN2:2;
A208:             x=G1*(i,k1) by A74,A206,A207;
                  reconsider y=x as Point of TOP-REAL 2 by A207;
A209:             Lmi/.k1 = Lmi.k1 by A49,PARTFUN1:def 6;
A210:             x in rng gg by A205,XBOOLE_0:def 4;
A211:             dom Col(G1,k1) = Seg len G1 by A199,FINSEQ_1:def 3
                    .= dom G1 by FINSEQ_1:def 3;
                  then
A212:             Ck1/.mi = Ck1.mi by A39,PARTFUN1:def 6;
A213:             i in dom Ck1 by A97,A206,A211;
                  Ck1/.i = Ck1.i by A97,A206,A211,PARTFUN1:def 6;
                  then y=Ck1/.i by A208,A211,A213,MATRIX_0:def 8;
                  then y in rng Ck1 by A213,PARTFUN2:2;
                  then
A214:             k1 =k by A51,A201,A210,GOBOARD1:4;
A215:             1 in Seg l by A158,FINSEQ_1:1;
                  then f/.1=f1/.ma1 & f1/.ma1=g1/.(w+1) by A170,A155,A200;
                  then f/.1= Ck1/.mi by A39,A50,A51,A209,A212,MATRIX_0:42;
                  then
A216:             f/.1 in rng Col(G1,k) by A39,A211,A214,PARTFUN2:2;
                  f/.1 in rng f by A157,A215,PARTFUN2:2;
                  hence contradiction by A202,A216,XBOOLE_0:def 4;
                end;
                suppose
A217:             x in rng h2 /\ rng gg;
                  then x in rng h2 by XBOOLE_0:def 4;
                  then consider i being Element of NAT such that
A218:             i in dom h2 and
A219:             x=h2/.i by PARTFUN2:2;
A220:             x=G1*(ma+i,k2) by A133,A218,A219;
                  reconsider y=x as Point of TOP-REAL 2 by A219;
A221:             Lma/.k2 = Lma.k2 by A131,PARTFUN1:def 6;
A222:             x in rng gg by A217,XBOOLE_0:def 4;
A223:             dom Col(G1,k2) = Seg len G1 by A198,FINSEQ_1:def 3
                    .= dom G1 by FINSEQ_1:def 3;
                  then
A224:             Ck2/.ma = Ck2.ma by A41,PARTFUN1:def 6;
A225:             ma+i in dom Ck2 by A135,A218,A223;
                  Ck2/.(ma+i) = Ck2.(ma+i) by A135,A218,A223,PARTFUN1:def 6;
                  then y=Ck2/.(ma+i) by A220,A223,A225,MATRIX_0:def 8;
                  then y in rng Ck2 by A225,PARTFUN2:2;
                  then
A226:             k2=k by A134,A201,A222,GOBOARD1:4;
A227:             l in Seg l by A158,FINSEQ_1:1;
                  then f/.l=f1/.(w+l) & f1/.(w+l)=g1/.(w+l) by A170,A155;
                  then f/.l=Ck2/.ma by A41,A132,A134,A221,A224,MATRIX_0:42;
                  then
A228:             f/.l in rng Col(G1,k) by A41,A223,A226,PARTFUN2:2;
                  f/.l in rng f by A157,A227,PARTFUN2:2;
                  hence contradiction by A202,A228,XBOOLE_0:def 4;
                end;
              end;
              hence contradiction;
            end;
            rng h2 /\ rng g = {}
            proof
              set x = the Element of rng h2 /\ rng g;
              assume
A229:         not thesis;
              then x in rng h2 by XBOOLE_0:def 4;
              then consider n1 be Element of NAT such that
A230:         n1 in dom h2 and
A231:         x=h2/.n1 by PARTFUN2:2;
A232:         1<=n1 by A230,FINSEQ_3:25;
              ma+n1 in dom G1 by A135,A230;
              then
A233:         [ma+n1,k2] in Indices G1 by A134,A96,ZFMISC_1:87;
              x in rng g by A229,XBOOLE_0:def 4;
              then consider n2 be Element of NAT such that
A234:         n2 in dom g and
A235:         x=g/.n2 by PARTFUN2:2;
              consider i1,i2 be Nat such that
A236:         [i1,i2] in Indices G1 and
A237:         g/.n2=G1*(i1,i2) by A15,A234,GOBOARD1:def 9;
              reconsider L=Line(G1,i1) as FinSequence of TOP-REAL 2;
A238:         i2 in Seg width G1 by A96,A236,ZFMISC_1:87;
A239:         Seg len L = dom L & len L=width G1 by FINSEQ_1:def 3
,MATRIX_0:def 7;
              then L/.i2 = L.i2 by A238,PARTFUN1:def 6;
              then g/.n2=L/.i2 by A237,A238,MATRIX_0:def 7;
              then
A240:         g/.n2 in rng L by A238,A239,PARTFUN2:2;
              g/.n2 in rng g by A234,PARTFUN2:2;
              then
A241:         rng g /\ rng L <> {} by A240,XBOOLE_0:def 4;
A242:         i1 in dom G1 by A96,A236,ZFMISC_1:87;
              x=G1*(ma+n1,k2) by A133,A230,A231;
              then i1=ma+n1 by A235,A236,A237,A233,GOBOARD1:5;
              then n1<=0 by A41,A242,A241,XREAL_1:29;
              hence contradiction by A232;
            end;
            then rng ff /\ rng g = ((rng h1 \/ rng f) /\ rng g) \/ {} by A196,
XBOOLE_1:23
              .= {} \/ rng f /\ rng g by A112,XBOOLE_1:23
              .= rng f /\ rng g;
            then
A243:       rng ff /\ rng g c= rng g1 /\ rng g2 by A18,A178,XBOOLE_1:27;
A244:       len DelCol(G1,1)=len G1 by MATRIX_0:def 13;
            then
A245:       dom DelCol(G1,1) = Seg len G1 by FINSEQ_1:def 3
              .= dom G1 by FINSEQ_1:def 3;
A246:       now
              let n;
              assume that
A247:         n in dom f and
A248:         n+1 in dom f;
              f/.n=f1/.(w+n) by A155,A157,A247;
              then
A249:         f/.n=g1 /.(w+n) by A170,A157,A247;
              f/.(n+1)=f1/.(w+(n+1)) by A155,A157,A248;
              then
A250:         w+n+1 in dom g1 & f/.(n+1)=g1/.(w+n+1) by A170,A157,A248;
              let i1,i2,j1,j2 be Nat;
              assume
A251:         [i1,i2] in Indices G1 & [j1,j2] in Indices G1 & f/.n=
              G1*(i1,i2) & f/. (n+1)=G1*(j1,j2);
              w+n in dom g1 by A170,A157,A247;
              hence |.i1-j1.|+|.i2-j2.|=1 by A6,A249,A250,A251,GOBOARD1:def 9;
            end;
            set hf=h1^f;
A252:       len ff=len(h1^f)+len h2 by FINSEQ_1:22
              .=len h1+len f+len h2 by FINSEQ_1:22;
A253:       now
              let i1,i2,j1,j2 be Nat;
              assume that
A254:         [ i1,i2] in Indices G1 and
A255:         [j1,j2] in Indices G1 and
A256:         hf/.len hf=G1*(i1,i2) and
A257:         h2/.1=G1*(j1,j2) and
              len hf in dom hf and
A258:         1 in dom h2;
              ma+1 in dom G1 by A135,A258;
              then
A259:         [ma+1,k2] in Indices G1 by A134,A96,ZFMISC_1:87;
A260:         [ma,k2] in Indices G1 by A41,A134,A96,ZFMISC_1:87;
A261:         Lma/.k2 = Lma.k2 by A131,PARTFUN1:def 6;
A262:         len f in dom f by A155,A157,A158,FINSEQ_1:1;
              hf/.len hf=hf/.(len h1+len f) by FINSEQ_1:22
                .= f/.len f by A262,FINSEQ_4:69
                .= f1/.(w+l) by A155,A157,A262
                .= g1/.mi1 by A170,A155,A157,A262
                .= G1*(ma,k2) by A132,A134,A261,MATRIX_0:def 7;
              then
A263:         i1=ma & i2=k2 by A254,A256,A260,GOBOARD1:5;
              h2/.1=G1*(ma+1,k2) by A133,A258;
              then j1=ma+1 & j2=k2 by A255,A257,A259,GOBOARD1:5;
              hence |.i1-j1.|+|.i2-j2.|=|.ma-(ma+1).|+0 by A263,ABSVALUE:2
                .=|.-(ma+1 -ma).|
                .=|.1.| by COMPLEX1:52
                .=1 by ABSVALUE:def 1;
            end;
            now
A264:         [mi,k1] in Indices G1 by A39,A51,A96,ZFMISC_1:87;
A265:         Lmi/.k1 = Lmi.k1 by A49,PARTFUN1:def 6;
              let i1,i2,j1,j2 be Nat;
              assume that
A266:         [i1,i2] in Indices G1 and
A267:         [j1,j2] in Indices G1 and
A268:         h1/.len h1=G1*(i1,i2) and
A269:         f/.1=G1*(j1,j2) and
A270:         len h1 in dom h1 and
A271:         1 in dom f;
              l1 in dom G1 by A74,A97,A270;
              then
A272:         [l1,k1] in Indices G1 by A51,A96,ZFMISC_1:87;
A273:         w+1=ma1;
              then f/.1=f1/.ma1 by A155,A157,A271;
              then f/.1=g1/.ma1 by A170,A157,A271,A273
                .= G1*(mi,k1) by A50,A51,A265,MATRIX_0:def 7;
              then
A274:         j1=mi & j2=k1 by A267,A269,A264,GOBOARD1:5;
              h1/.len h1=G1*(l1,k1) by A74,A270;
              then i1=l1 & i2=k1 by A266,A268,A272,GOBOARD1:5;
              hence |.i1-j1.|+|.i2-j2.|= |.mi-1-mi.|+0 by A274,ABSVALUE:2
                .= |.-1.|
                .= |.1.| by COMPLEX1:52
                .= 1 by ABSVALUE:def 1;
            end;
            then for n st n in dom(h1^f) & n+1 in dom(h1^f) for i1,i2,j1,j2
be Nat st [i1,i2] in Indices G1 & [j1,j2] in Indices G1 & (h1^f)/.n=
G1*(i1,i2) & (h1^f)/.(n+1)=G1*(j1,j2) holds |.i1-j1.|+|.i2-j2.|=1 by A102
,A246,GOBOARD1:24;
            then
A275:       for n st n in dom ff & n+1 in dom ff for m,k,i,j st [m,k] in
Indices G1 & [i,j] in Indices G1 & ff/.n=G1*(m,k) & ff/.(n+1)=G1*(i,j) holds
            |.m-i.|+|.k-j.|=1 by A141,A253,GOBOARD1:24;
            now
              let n;
              assume
A276:         n in dom h1;
               reconsider k1 as Nat;
              take i=n,k1;
              n in dom G1 by A97,A276;
              hence
              [i,k1] in Indices G1 & h1/.n=G1*(i,k1) by A74,A51,A96,A276,
ZFMISC_1:87;
            end;
            then for n st n in dom(h1^f) ex i,j st [i,j] in Indices G1 & (h1^
            f)/.n=G1*(i,j) by A174,GOBOARD1:23;
            then for n st n in dom ff ex i,j st [i,j] in Indices G1 & ff/.n=
            G1*(i,j) by A139,GOBOARD1:23;
            then
A277:       ff is_sequence_on G1 by A275,GOBOARD1:def 9;
A278:       Seg len ff=dom ff by FINSEQ_1:def 3;
            then
A279:       len ff in dom ff by A252,A195,FINSEQ_1:1;
A280:       1 in dom ff by A252,A195,A278,FINSEQ_1:1;
            thus thesis
            proof
              per cases;
              suppose
A281:           rng f /\ rng Col(G1,1)={};
                set D = DelCol(G1,1);
A282: 1 in Seg width G1 by A152,FINSEQ_1:1;
A283:          D is not empty-yielding by A152,A282,MATRIX_0:65;
                rng ff /\ rng Col(G1,1)={} by A31,A197,A281;
                then
A284:           rng ff misses rng Col(G1,1) by XBOOLE_0:def 7;
                then
A285:           ff is_sequence_on D & ff/.1 in rng Line(D,1) by A31,A26,A277
,A159,A152,A280,GOBOARD1:25,MATRIX_0:75;
A286:           ff/.len ff in rng Line(D,len D) by A31,A27,A183,A152,A244,A279
,A284,MATRIX_0:75;
                defpred P[Nat] means $1 in dom g & g/.$1 in rng Col(G1,1);
A287:           ex m be Nat st P[m]
                proof
                  take 1;
                  thus thesis by A10,A13,A32,FINSEQ_3:25,FINSEQ_4:92;
                end;
A288:           for m be Nat st P[m] holds m<=len g by FINSEQ_3:25;
                consider m be Nat such that
A289:           P [m] & for k be Nat st P[k] holds k<=m from NAT_1:
                sch 6(A288,A287);
                reconsider m as Nat;
                reconsider p=g/.m as Point of TOP-REAL 2;
A290:           now
                  assume
A291:             m>=len g;
                  m<=len g by A289,FINSEQ_3:25;
                  then p in rng Col(G1,width G1) by A28,A291,XXREAL_0:1;
                  then 1=k+1 by A3,A31,A73,A289,GOBOARD1:4;
                  hence contradiction by A91;
                end;
                then reconsider ll = len g-m as Element of NAT by INT_1:5;
                deffunc F(Nat) = g/.(m+$1);
                consider t be FinSequence of TOP-REAL 2 such that
A292:           len t = ll & for n being Nat st n in dom t holds t/.
                n = F(n) from FINSEQ_4:sch 2;
A293:           dom t = Seg ll by A292,FINSEQ_1:def 3;
A294:           rng t c= rng g
                proof
                  let y be object;
                  assume y in rng t;
                  then consider x being Element of NAT such that
A295:             x in dom t and
A296:             t/.x=y by PARTFUN2:2;
                  x<=ll by A293,A295,FINSEQ_1:1;
                  then
A297:             m+x<=m+ll by XREAL_1:7;
A298:             x<=x+m by NAT_1:11;
                  1<=x by A293,A295,FINSEQ_1:1;
                  then 1<=x+m by A298,XXREAL_0:2;
                  then m+x in dom g by A297,FINSEQ_3:25;
                  then g/.(m+x) in rng g by PARTFUN2:2;
                  hence thesis by A292,A295,A296;
                end;
A299:           for n st n in dom t holds m+n in dom g
                proof
                  let n;
A300:             n<=n+m by NAT_1:11;
                  assume
A301:             n in dom t;
                  then n<=ll by A293,FINSEQ_1:1;
                  then
A302:             m+n<=m+ll by XREAL_1:7;
                  1<=n by A293,A301,FINSEQ_1:1;
                  then 1<=n+m by A300,XXREAL_0:2;
                  hence thesis by A302,FINSEQ_3:25;
                end;
A303:           Seg len t = dom t by FINSEQ_1:def 3;
                reconsider t as FinSequence of TOP-REAL 2;
A304:           width D = k by A3,A31,MATRIX_0:63;
A305:           Indices D = [:dom D,Seg width D:] by MATRIX_0:def 4;
A306:           now
                  let n;
                  assume
A307:             n in dom t;
                  then m+n in dom g by A299;
                  then consider i1,i2 be Nat such that
A308:             [i1,i2] in Indices G1 and
A309:             g/.(m+n)=G1*(i1,i2) by A15,GOBOARD1:def 9;
A310:             i2 in Seg width G1 by A96,A308,ZFMISC_1:87;
                  then
A311:             1<=i2 by FINSEQ_1:1;
                  then reconsider l=i2-1 as Element of NAT by INT_1:5;
                  reconsider Ci2 = Col(G1,i2) as FinSequence of TOP-REAL 2;
A312:             i1 in dom G1 by A96,A308,ZFMISC_1:87;
                  len Ci2 = len G1 by MATRIX_0:def 8;
                  then
A313:             dom Ci2 = Seg len G1 by FINSEQ_1:def 3
                    .= dom G1 by FINSEQ_1:def 3;
                  then Ci2/.i1 = Ci2.i1 by A312,PARTFUN1:def 6;
                  then Ci2/.i1 = G1*(i1,i2) by A312,MATRIX_0:def 8;
                  then
A314:             g/.(m+n) in rng Col(G1,i2) by A309,A312,A313,PARTFUN2:2;
                  now
                    1<=n by A293,A307,FINSEQ_1:1;
                    then
A315:               m+1<=m+n by XREAL_1:7;
                    assume i2=1;
                    then m+n<=m by A289,A299,A307,A314;
                    then m+1<=m by A315,XXREAL_0:2;
                    then 1<=m-m by XREAL_1:19;
                    then 1<=0;
                    hence contradiction;
                  end;
                  then 1<i2 by A311,XXREAL_0:1;
                  then 1+1<=i2 by NAT_1:13;
                  then
A316:             1<=l by XREAL_1:19;
                   reconsider i1,l as Nat;
                  take i1;
                  take l;
A317:             t/.n=g/.(m+n) by A292,A307;
                  i2<=width G1 by A310,FINSEQ_1:1;
                  then
A318:             l<=width D by A3,A304,XREAL_1:20;
                  then l in Seg width D by A316,FINSEQ_1:1;
                  hence [i1,l] in Indices D by A245,A305,A312,ZFMISC_1:87;
                  l+1=i2;
                  hence
                  t/.n = D*(i1,l) by A3,A31,A304,A309,A312,A317,A316,A318,
MATRIX_0:70;
                end;
                0<width D by A283,MATRIX_0:def 10,NAT_1:3;
                then
A319:           0+1<=width D by NAT_1:13;
                then
A320:           Col(D,1)=Col(G1,1+1) & 1+1 in Seg width G1 by A3,A31,A304,
MATRIX_0:68;
                m+1<=len g by A290,NAT_1:13;
                then
A321:           1<=len t by A292,XREAL_1:19;
                then t/.1 = g/.(m+1) by A292,A303,FINSEQ_1:1;
                then
A322:           t/.1 in rng Col(D,1) by A32,A15,A28,A31,A289,A320,
GOBOARD1:32;
                now
                  let n;
                  assume that
A323:             n in dom t and
A324:             n+1 in dom t;
                  let i1,i2,j1,j2 be Nat;
                  assume that
A325:             [ i1,i2] in Indices D and
A326:             [j1,j2] in Indices D and
A327:             t/.n=D*(i1,i2) and
A328:             t/.(n+1)=D*(j1,j2);
A329:             i1 in dom D by A305,A325,ZFMISC_1:87;
                  i2 in Seg k by A304,A305,A325,ZFMISC_1:87;
                  then
A330:             1<=i2 & i2<=k by FINSEQ_1:1;
                  then i2+1 in Seg width G1 by A3,A31,A245,A329,MATRIX_0:70;
                  then
A331:             [i1,i2+1] in Indices G1 by A96,A245,A329,ZFMISC_1:87;
                  t/.n=g/.(m+n) by A292,A323;
                  then
A332:             g/.(m+n)=G1*(i1,i2+1) by A3,A31,A245,A327,A329,A330,
MATRIX_0:70;
                  m+(n+1)=m+n+1;
                  then
A333:             m+n+1 in dom g by A299,A324;
A334:             j1 in dom D by A305,A326,ZFMISC_1:87;
                  j2 in Seg k by A304,A305,A326,ZFMISC_1:87;
                  then
A335:             1<=j2 & j2<=k by FINSEQ_1:1;
                  then j2+1 in Seg width G1 by A3,A31,A245,A329,MATRIX_0:70;
                  then
A336:             [j1,j2+1] in Indices G1 by A96,A245,A334,ZFMISC_1:87;
                  t/.(n+1)=g/.(m+(n+1)) by A292,A324;
                  then
A337:             g/.(m+n+1)=G1* (j1,j2+1) by A3,A31,A245,A328,A334,A335,
MATRIX_0:70;
                  m+n in dom g by A299,A323;
                  hence 1= |.i1-j1.|+|.i2+1 -(j2+1).| by A15,A332,A337,A331
,A336,A333,GOBOARD1:def 9
                    .=|.i1-j1.|+|.i2-j2.|;
                end;
                then
A338:           t is_sequence_on D by A306,GOBOARD1:def 9;
                set x = the Element of rng ff /\ rng t;
A339:           rng ff /\ rng t c= rng ff /\ rng g by A294,XBOOLE_1:26;
                len t in Seg ll by A292,A321,FINSEQ_1:1;
                then t/.len t = g/.(m+ll) by A292,A303
                  .= g/.len g;
                then t/.len t in rng Col(D,width D) by A3,A28,A31,A304,A319
,MATRIX_0:68;
                then rng ff /\ rng t <> {} by A2,A92,A252,A195,A304,A321,A322
,A338,A285,A286,A283;
                then x in rng ff /\ rng g by A339;
                hence thesis by A243;
              end;
              suppose
A340:           rng f /\ rng Col(G1,1) <> {};
                set D = DelCol(G1,width G1);
A341:            D is not empty-yielding by A93,A94,MATRIX_0:65;
A342:           0+1<k+1 by A92,XREAL_1:6;
                now
                  per cases;
                  suppose
                    rng f /\ rng Col(G1,width G1) = {};
                    then rng ff /\ rng Col(G1,width G1) = {} by A73,A197;
                    then
A343:               rng ff misses rng Col(G1,width G1) by XBOOLE_0:def 7;
                    then
A344:               ff is_sequence_on D by A73,A277,A152,GOBOARD1:25;
                    consider t be FinSequence of TOP-REAL 2 such that
A345:               t/.1 in rng Col(D,1) & t/.len t in rng Col(D,
                    width D) & 1<=len t & t is_sequence_on D and
A346:               rng t c= rng g by A3,A32,A15,A14,A28,A342,GOBOARD1:35;
                    set x = the Element of rng ff /\ rng t;
A347:               rng ff /\ rng t c= rng ff /\ rng g by A346,XBOOLE_1:26;
A348:               width D = k by A3,A73,MATRIX_0:63;
                    ff/.1 in rng Line(D,1) & ff/.len ff in rng Line(D,
len D) by A73,A26,A27,A159,A183,A152,A153,A280,A279,A343,MATRIX_0:75;
                    then rng ff /\ rng t <> {} by A2,A92,A252,A195,A345,A348
,A344,A341;
                    then x in rng ff /\ rng g by A347;
                    hence thesis by A243;
                  end;
                  suppose
A349:               rng f /\ rng Col(G1,width G1) <> {};
A350:               f is_sequence_on G1 by A174,A246,GOBOARD1:def 9;
                    then consider t be FinSequence of TOP-REAL 2 such that
A351:               rng t c= rng f and
A352:               t/.1 in rng Col(G1,1) & t/.len t in rng Col(G1,
width G1) & 1<=len t & t is_sequence_on G1 by A340,A349,GOBOARD1:36;
                    consider tt be FinSequence of TOP-REAL 2 such that
A353:               tt/.1 in rng Col(D,1) & tt/.len tt in rng Col(D,
                    width D) & 1<=len tt & tt is_sequence_on D and
A354:               rng tt c= rng t by A3,A342,A352,GOBOARD1:35;
A355:               Seg len Lma = dom Lma & len Lma=width G1 by FINSEQ_1:def 3
,MATRIX_0:def 7;
                    reconsider lg=len g-1 as Element of NAT by A32,INT_1:5;
                    defpred P[Nat] means $1 in dom g & g/.$1 in rng Line(G1,mi
                    );
                    defpred R[Nat] means $1 in dom g & g/.$1 in rng Line(G1,ma
                    );
A356:               lg<=len g by XREAL_1:43;
A357:               now
                      set x = the Element of rng g /\ rng Line(G1,mi);
                      x in rng g by A39,XBOOLE_0:def 4;
                      then consider n being Element of NAT such that
A358:                 n in dom g & x=g/.n by PARTFUN2:2;
                      reconsider n as Nat;
                      take n;
                      thus P[n] by A39,A358,XBOOLE_0:def 4;
                    end;
                    consider pf be Nat such that
A359:               P [pf] & for n be Nat st P[n] holds pf<=n from
                    NAT_1:sch 5(A357);
                    defpred PP[Nat] means
$1 in dom f implies for m
                    st m in dom G1 & f/.$1 in rng Line(G1,m) holds mi<=m;
                    reconsider C = Col(G1,width G1), Li= Line(G1,mi), La= Line
                    (G1,ma) as FinSequence of TOP-REAL 2;
A360:               lg+ 1=len g;
A361:               now
                      set x = the Element of rng g /\ rng Line(G1,ma);
                      x in rng g by A41,XBOOLE_0:def 4;
                      then consider n being Element of NAT such that
A362:                 n in dom g & x=g/.n by PARTFUN2:2;
                      reconsider n as Nat;
                      take n;
                      thus R[n] by A41,A362,XBOOLE_0:def 4;
                    end;
                    consider pl be Nat such that
A363:               R [pl] & for n be Nat st R[n] holds pl<=n from
                    NAT_1:sch 5(A361);
                    reconsider pf,pl as Nat;
A364:               1<=pf by A359,FINSEQ_3:25;
                    consider K2 be Element of NAT such that
A365:               K2 in dom Lma and
A366:               g/.pl=Lma/.K2 by A363,PARTFUN2:2;
                    reconsider CK2 = Col(G1,K2) as FinSequence of TOP-REAL 2;
                    consider K1 be Element of NAT such that
A367:               K1 in dom Li and
A368:               g/.pf=Li/.K1 by A359,PARTFUN2:2;
                    reconsider CK1 = Col(G1,K1) as FinSequence of TOP-REAL 2;
                    deffunc F(Nat) = G1*($1,K1);
                    consider h1 be FinSequence of TOP-REAL 2 such that
A369:               len h1 = l1 & for n being Nat st n in dom h1
                    holds h1/.n=F(n) from FINSEQ_4:sch 2;
A370:               for k st PP[k] holds PP[k+1]
                    proof
                      let k such that
A371:                 PP[k];
                      assume
A372:                 k+1 in dom f;
                      let m such that
A373:                 m in dom G1 & f/.(k+1) in rng Line(G1,m);
                      now
                        per cases;
                        suppose
A374:                     k=0;
                          w+1=ma1 & 1 in Seg l by A158,FINSEQ_1:1;
                          then f/.1=f1/.ma1 & f1/.ma1=g1/.ma1 by A170,A155;
                          then f/.(k+1) in rng Li by A49,A50,A374,PARTFUN2:2;
                          hence thesis by A39,A373,GOBOARD1:3;
                        end;
                        suppose
                          k<>0;
                          then 0<k;
                          then
A375:                     0+1<=k by NAT_1:13;
                          k+1<=len f by A372,FINSEQ_3:25;
                          then
A376:                     k<=len f by NAT_1:13;
                          then
A377:                     k in dom f by A375,FINSEQ_3:25;
                          then consider i1,i2 be Nat such that
A378:                     [i1,i2] in Indices G1 and
A379:                     f/.k=G1*(i1,i2) by A174;
A380:                     i2 in Seg width G1 by A96,A378,ZFMISC_1:87;
                          consider j1,j2 be Nat such that
A381:                     [ j1,j2] in Indices G1 and
A382:                     f/.(k+1)=G1*(j1,j2) by A174,A372;
                          reconsider Lj1 = Line(G1,j1) as FinSequence of
                          TOP-REAL 2;
A383:                     j2 in Seg width G1 by A96,A381,ZFMISC_1:87;
A384:                     Seg len Lj1 = dom Lj1 & len Lj1=width G1 by
FINSEQ_1:def 3,MATRIX_0:def 7;
                          then Lj1/.j2 = Lj1.j2 by A383,PARTFUN1:def 6;
                          then f/.(k+1)=Lj1/.j2 by A382,A383,MATRIX_0:def 7;
                          then
A385:                     f/.(k+1) in rng Lj1 by A383,A384,PARTFUN2:2;
A386:                     j1 in dom G1 by A96,A381,ZFMISC_1:87;
                          reconsider Li1 = Line(G1,i1) as FinSequence of
                          TOP-REAL 2;
A387:                     i1 in dom G1 by A96,A378,ZFMISC_1:87;
A388:                     Seg len Li1 = dom Li1 & len Li1=width G1 by
FINSEQ_1:def 3,MATRIX_0:def 7;
                          then
A389:                     Li1/.i2 = Li1.i2 by A380,PARTFUN1:def 6;
                          then f/.k=Li1/.i2 by A379,A380,MATRIX_0:def 7;
                          then
A390:                     f/.k in rng Li1 by A380,A388,PARTFUN2:2;
                          then
A391:                     mi<=i1 by A371,A375,A376,A387,FINSEQ_3:25;
                          now
                            per cases by A391,XXREAL_0:1;
                            suppose
A392:                         mi=i1;
                              g1/.(w+k) = f1/.(w+k) by A170,A157,A377
                                .= f/.k by A155,A157,A377
                                .= Li1/.i2 by A379,A380,A389,MATRIX_0:def 7;
                              then g1/.(w+k) in rng Line(G1,mi) by A380,A388
,A392,PARTFUN2:2;
                              then
A393:                         w+k<=ma1 by A46,A170,A157,A377;
                              w+1<=w+k by A375,XREAL_1:7;
                              then w+k=ma1 by A393,XXREAL_0:1;
                              then
A394:                         ma1+1=w+(k+1);
                              mi+1<=ma by A128,NAT_1:13;
                              then
A395:                         mi+1<=len G1 by A48,XXREAL_0:2;
                              1<=mi+1 by A42,NAT_1:13;
                              then
A396:                         mi+1 in dom G1 by A395,FINSEQ_3:25;
                              f /.(k+1)=f1/.(w+(k+1)) & f1/.(w+(k+1))=g1
                              /.(w+(k+1)) by A170,A155,A157,A372;
                              then f/.(k+1) in rng Line(G1,mi+1) by A4,A6,A9
,A39,A46,A394,A396,GOBOARD1:28;
                              then m=mi+1 by A373,A396,GOBOARD1:3;
                              hence thesis by NAT_1:11;
                            end;
                            suppose
A397:                         mi<i1;
                              now
                                per cases by A350,A372,A377,A387,A390,
GOBOARD1:27;
                                suppose
                                  f/.(k+1) in rng Line(G1,i1);
                                  hence thesis by A373,A387,A397,GOBOARD1:3;
                                end;
                                suppose
                                  for l be Nat st f/.(k+1
) in rng Line(G1,l) & l in dom G1 holds |.i1-l.|=1;
                                  then
A398:                             |.i1-j1.|=1 by A386,A385;
                                  now
                                    per cases by A398,SEQM_3:41;
                                    suppose
A399:                                 j1<i1 & i1=j1+1;
                                      then mi<=i1-1 by A397,NAT_1:13;
                                      hence thesis by A373,A386,A385,A399,
GOBOARD1:3;
                                    end;
                                    suppose
                                      i1<j1 & j1=i1+1;
                                      then mi<j1 by A397,XXREAL_0:2;
                                      hence thesis by A373,A386,A385,GOBOARD1:3
;
                                    end;
                                  end;
                                  hence thesis;
                                end;
                              end;
                              hence thesis;
                            end;
                          end;
                          hence thesis;
                        end;
                      end;
                      hence thesis;
                    end;
A400:               Seg len Li = dom Li & len Li=width G1 by FINSEQ_1:def 3
,MATRIX_0:def 7;
A401:               now
                      let n;
A402:                 l1<=mi by XREAL_1:43;
                      assume
A403:                 n in dom h1;
                      then
A404:                 1<=n by FINSEQ_3:25;
                      n<=l1 by A369,A403,FINSEQ_3:25;
                      then
A405:                 n<=mi by A402,XXREAL_0:2;
                      mi<= len G1 by A39,FINSEQ_3:25;
                      then n<=len G1 by A405,XXREAL_0:2;
                      hence n in dom G1 by A404,FINSEQ_3:25;
                    end;
A406:               now
                      let n;
                      assume
A407:                 n in dom h1;
                       reconsider i=n,K1 as Nat;
                      take i,K1;
                      n in dom G1 by A401,A407;
                      hence
                      [i,K1] in Indices G1 & h1/.n=G1*(i,K1) by A96,A367,A400
,A369,A407,ZFMISC_1:87;
                    end;
A408:               now
                      let n;
                      assume that
A409:                 n in dom h1 and
A410:                 n+1 in dom h1;
                      n+1 in dom G1 by A401,A410;
                      then
A411:                 [n+1,K1 ] in Indices G1 by A96,A367,A400,ZFMISC_1:87;
                      let i1,i2,j1,j2 be Nat;
                      assume that
A412:                 [i1,i2] in Indices G1 and
A413:                 [j1,j2] in Indices G1 and
A414:                 h1/.n=G1*(i1,i2) and
A415:                 h1/.(n+1)=G1*(j1,j2);
                      h1/.(n+1)=G1*(n+1,K1) by A369,A410;
                      then
A416:                 j1=n+1 & j2=K1 by A411,A413,A415,GOBOARD1:5;
                      n in dom G1 by A401,A409;
                      then
A417:                 [n,K1] in Indices G1 by A96,A367,A400,ZFMISC_1:87;
                      h1/.n=G1*(n,K1) by A369,A409;
                      then i1=n & i2=K1 by A417,A412,A414,GOBOARD1:5;
                      hence |.i1-j1.|+|.i2-j2.|= |.n-n+-1.|+0 by A416,
ABSVALUE:2
                        .= |.1.| by COMPLEX1:52
                        .= 1 by ABSVALUE:def 1;
                    end;
A418:               pf<=len g by A359,FINSEQ_3:25;
A419:               Lma/.K2 = Lma.K2 by A365,PARTFUN1:def 6;
                    then
A420:               g/.pl=G1*(ma, K2) by A365,A366,A355,MATRIX_0:def 7;
                    deffunc F(Nat) = G1*(ma+$1,K2);
                    consider h2 be FinSequence of TOP-REAL 2 such that
A421:               len h2=l2 & for n being Nat st n in dom h2 holds
                    h2/.n=F(n) from FINSEQ_4:sch 2;
A422:               PP[0] by FINSEQ_3:25;
A423:               for n holds PP[n] from NAT_1:sch 2(A422,A370);
A424:               rng h1 /\ rng tt = {}
                    proof
                      set x = the Element of rng h1 /\ rng tt;
                      assume
A425:                 not thesis;
                      then x in rng h1 by XBOOLE_0:def 4;
                      then consider i2 be Element of NAT such that
A426:                 i2 in dom h1 and
A427:                 h1/.i2=x by PARTFUN2:2;
                      Seg len h1 = dom h1 by FINSEQ_1:def 3;
                      then
A428:                 l1<mi & i2<=l1 by A369,A426,FINSEQ_1:1,XREAL_1:44;
                      i2 in dom G1 by A401,A426;
                      then
A429:                 [i2,K1] in Indices G1 by A96,A367,A400,ZFMISC_1:87;
                      x in rng tt by A425,XBOOLE_0:def 4;
                      then x in rng t by A354;
                      then consider i1 be Element of NAT such that
A430:                 i1 in dom f and
A431:                 f/.i1=x by A351,PARTFUN2:2;
                      consider n1,n2 be Nat such that
A432:                 [n1,n2] in Indices G1 and
A433:                 f/.i1=G1*(n1,n2) by A174,A430;
                      reconsider Ln1 = Line(G1,n1) as FinSequence of TOP-REAL
                      2;
A434:                 n2 in Seg width G1 by A96,A432,ZFMISC_1:87;
A435:                 Seg len Ln1 = dom Ln1 & len Ln1=width G1 by
FINSEQ_1:def 3,MATRIX_0:def 7;
                      then Ln1/.n2 = Ln1.n2 by A434,PARTFUN1:def 6;
                      then f/.i1=Ln1/.n2 by A433,A434,MATRIX_0:def 7;
                      then
A436:                 f/.i1 in rng Ln1 by A434,A435,PARTFUN2:2;
                      n1 in dom G1 by A96,A432,ZFMISC_1:87;
                      then
A437:                 mi<=n1 by A423,A430,A436;
                      x=G1*(i2,K1) by A369,A426,A427;
                      then i2=n1 by A431,A432,A433,A429,GOBOARD1:5;
                      hence contradiction by A437,A428,XXREAL_0:2;
                    end;
                    defpred PP[Nat] means
         $1 in dom f implies for m
                    st m in dom G1 & f/.$1 in rng Line(G1,m) holds m<=ma;
A438:               for k st PP[k] holds PP[k+1]
                    proof
                      let k such that
A439:                 PP[k];
                      assume
A440:                 k+1 in dom f;
                      let m such that
A441:                 m in dom G1 & f/.(k+1) in rng Line(G1,m);
                      now
                        per cases;
                        suppose
A442:                     k=0;
                          1 in Seg l by A158,FINSEQ_1:1;
                          then f/.1=f1/.(w+1) & f1/.(w+1)=g1/.(w+1) by A170
,A155;
                          then f/.(k+1) in rng Li by A49,A50,A442,PARTFUN2:2;
                          hence thesis by A39,A127,A441,GOBOARD1:3;
                        end;
                        suppose
A443:                     k<>0;
A444:                     k+1<= len f by A177,A440,FINSEQ_1:1;
                          then
A445:                     k<=len f by NAT_1:13;
                          consider j1,j2 be Nat such that
A446:                     [ j1,j2] in Indices G1 and
A447:                     f/.(k+1)=G1*(j1,j2) by A174,A440;
                          reconsider Lj1 = Line(G1,j1) as FinSequence of
                          TOP-REAL 2;
A448:                     j2 in Seg width G1 by A96,A446,ZFMISC_1:87;
A449:                     Seg len Lj1 = dom Lj1 & len Lj1=width G1 by
FINSEQ_1:def 3,MATRIX_0:def 7;
                          then Lj1/.j2 = Lj1.j2 by A448,PARTFUN1:def 6;
                          then f/.(k+1)=Lj1/.j2 by A447,A448,MATRIX_0:def 7;
                          then
A450:                     f/.(k+1) in rng Lj1 by A448,A449,PARTFUN2:2;
A451:                     j1 in dom G1 by A96,A446,ZFMISC_1:87;
                          then
A452:                     j1=m by A441,A450,GOBOARD1:3;
                          0<k by A443;
                          then
A453:                     0+1<=k by NAT_1:13;
                          then
A454:                     k in dom f by A445,FINSEQ_3:25;
                          then consider i1,i2 be Nat such that
A455:                     [i1,i2] in Indices G1 and
A456:                     f/.k=G1*(i1,i2) by A174;
                          reconsider Li1 = Line(G1,i1) as FinSequence of
                          TOP-REAL 2;
A457:                     i2 in Seg width G1 by A96,A455,ZFMISC_1:87;
A458:                     Seg len Li1 = dom Li1 & len Li1=width G1 by
FINSEQ_1:def 3,MATRIX_0:def 7;
                          then Li1/.i2 = Li1.i2 by A457,PARTFUN1:def 6;
                          then f/.k=Li1/.i2 by A456,A457,MATRIX_0:def 7;
                          then
A459:                     f/.k in rng Li1 by A457,A458,PARTFUN2:2;
A460:                     i1 in dom G1 by A96,A455,ZFMISC_1:87;
                          then
A461:                     i1<=ma by A439,A453,A445,A459,FINSEQ_3:25;
                          now
                            per cases by A461,XXREAL_0:1;
                            case
A462:                         ma=i1;
A463:                         w+1<=w+k by A453,XREAL_1:7;
A464:                         f /.k=f1/.(w+k) & f1/.(w+k)=g1/.(w+k) by A170
,A155,A157,A454;
                              then ma1 <> w+k by A39,A41,A46,A95,A459,A462,
GOBOARD1:3;
                              then ma1<w+k by A463,XXREAL_0:1;
                              then
A465:                         mi1<=w+k by A130,A170,A157,A454,A459,A462,A464;
                              w+k<=mi1 by A155,A156,A445,XREAL_1:7;
                              then w+k=mi1 by A465,XXREAL_0:1;
                              hence contradiction by A155,A444,NAT_1:13;
                            end;
                            case
A466:                         i1<ma;
                              now
                                per cases by A350,A440,A454,A460,A459,
GOBOARD1:27;
                                suppose
                                  f/.(k+1) in rng Line(G1,i1);
                                  hence thesis by A441,A460,A466,GOBOARD1:3;
                                end;
                                suppose
                                  for l be Nat st f/.(k+1
) in rng Line(G1,l) & l in dom G1 holds |.i1-l.|=1;
                                  then
A467:                             |.i1-j1.|=1 by A451,A450;
                                  now
                                    per cases by A467,SEQM_3:41;
                                    suppose
                                      j1<i1 & i1=j1+1;
                                      hence thesis by A452,A466,XXREAL_0:2;
                                    end;
                                    suppose
                                      i1<j1 & j1=i1+1;
                                      hence thesis by A452,A466,NAT_1:13;
                                    end;
                                  end;
                                  hence thesis;
                                end;
                              end;
                              hence thesis;
                            end;
                          end;
                          hence thesis;
                        end;
                      end;
                      hence thesis;
                    end;
A468:               Seg len h1 = dom h1 by FINSEQ_1:def 3;
A469:               now
                      let n;
A470:                 n<=n+ma by NAT_1:11;
                      assume
A471:                 n in dom h2;
                      then n<=l2 by A421,FINSEQ_3:25;
                      then
A472:                 ma+n<=ma+l2 by XREAL_1:7;
                      1<=n by A471,FINSEQ_3:25;
                      then 1<=n+ma by A470,XXREAL_0:2;
                      hence ma+n in dom G1 by A472,FINSEQ_3:25;
                    end;
A473:               now
                      let n;
                      assume
A474:                 n in dom h2;
                       reconsider m=ma+n,K2 as Nat;
                      take m,K2;
                      ma+n in dom G1 by A469,A474;
                      hence
                      [m,K2] in Indices G1 & h2/.n=G1*(m,K2) by A96,A365,A355
,A421,A474,ZFMISC_1:87;
                    end;
A475:               now
                      let n;
                      assume that
A476:                 n in dom h2 and
A477:                 n+1 in dom h2;
                      ma+(n+1) in dom G1 by A469,A477;
                      then
A478:                 [ma+ n+1,K2] in Indices G1 by A96,A365,A355,ZFMISC_1:87;
                      let i1,i2,j1,j2 be Nat;
                      assume that
A479:                 [i1,i2] in Indices G1 and
A480:                 [j1,j2] in Indices G1 and
A481:                 h2/.n=G1*(i1,i2) and
A482:                 h2/.(n+1)=G1*(j1,j2);
                      h2/.(n+1)=G1*(ma+(n+1),K2) by A421,A477;
                      then
A483:                 j1=ma+n+1 & j2=K2 by A478,A480,A482,GOBOARD1:5;
                      ma +n in dom G1 by A469,A476;
                      then
A484:                 [ma+n,K2] in Indices G1 by A96,A365,A355,ZFMISC_1:87;
                      h2/.n=G1*(ma+n,K2) by A421,A476;
                      then i1=ma+n & i2=K2 by A484,A479,A481,GOBOARD1:5;
                      hence |.i1-j1.|+|.i2-j2.|= |.ma+n-(ma+n)+-1.|+0 by
A483,ABSVALUE:2
                        .= |.1.| by COMPLEX1:52
                        .= 1 by ABSVALUE:def 1;
                    end;
A485:               Seg len h2 = dom h2 by FINSEQ_1:def 3;
A486:               pl<=len g by A363,FINSEQ_3:25;
A487:               1<=pl by A363,FINSEQ_3:25;
A488:               pl <> pf by A39,A41,A95,A359,A363,GOBOARD1:3;
                    now
                      per cases by A488,XXREAL_0:1;
                      suppose
                        pf<pl;
                        then pf<len g by A486,XXREAL_0:2;
                        then 1<len g by A364,XXREAL_0:2;
                        hence 1+1<=len g by NAT_1:13;
                      end;
                      suppose
                        pf>pl;
                        then pl<len g by A418,XXREAL_0:2;
                        then 1<len g by A487,XXREAL_0:2;
                        hence 1+1<=len g by NAT_1:13;
                      end;
                    end;
                    then 1<=len g - 1 by XREAL_1:19;
                    then
A489:               lg in dom g by A356,FINSEQ_3:25;
A490:               PP[0] by FINSEQ_3:25;
A491:               for n holds PP[n] from NAT_1:sch 2(A490,A438);
A492:               rng h2 /\ rng tt = {}
                    proof
                      set x = the Element of rng h2 /\ rng tt;
                      assume
A493:                 not thesis;
                      then x in rng h2 by XBOOLE_0:def 4;
                      then consider i2 be Element of NAT such that
A494:                 i2 in dom h2 and
A495:                 h2/.i2=x by PARTFUN2:2;
                      0+1<=i2 by A494,FINSEQ_3:25;
                      then
A496:                 0<i2;
                      ma+i2 in dom G1 by A469,A494;
                      then
A497:                 [ma+i2,K2] in Indices G1 by A96,A365,A355,ZFMISC_1:87;
                      x in rng tt by A493,XBOOLE_0:def 4;
                      then x in rng t by A354;
                      then consider i1 be Element of NAT such that
A498:                 i1 in dom f and
A499:                 f/.i1=x by A351,PARTFUN2:2;
                      consider n1,n2 be Nat such that
A500:                 [n1,n2] in Indices G1 and
A501:                 f/.i1=G1*(n1,n2) by A174,A498;
                      reconsider Ln1 = Line(G1,n1) as FinSequence of TOP-REAL
                      2;
A502:                 n2 in Seg width G1 by A96,A500,ZFMISC_1:87;
A503:                 Seg len Ln1 = dom Ln1 & len Ln1=width G1 by
FINSEQ_1:def 3,MATRIX_0:def 7;
                      then Ln1/.n2 = Ln1.n2 by A502,PARTFUN1:def 6;
                      then f/.i1=Ln1/.n2 by A501,A502,MATRIX_0:def 7;
                      then
A504:                 f/.i1 in rng Ln1 by A502,A503,PARTFUN2:2;
A505:                 n1 in dom G1 by A96,A500,ZFMISC_1:87;
                      x=G1*(ma+i2,K2) by A421,A494,A495;
                      then ma+i2=n1 by A499,A500,A501,A497,GOBOARD1:5;
                      hence contradiction by A491,A498,A505,A504,A496,
XREAL_1:29;
                    end;
                    1<=len g by A13,GOBOARD1:22;
                    then
A506:               len g in dom g by FINSEQ_3:25;
A507:               dom g c= dom g2 by A23,A24,A16,A17,FINSEQ_1:5;
                    now
                      consider i2 be Element of NAT such that
A508:                 i2 in dom C and
A509:                 C/.i2=g/.len g by A28,PARTFUN2:2;
A510:                 dom C = Seg len C by FINSEQ_1:def 3
                        .= Seg len G1 by MATRIX_0:def 8
                        .= dom G1 by FINSEQ_1:def 3;
                      then
A511:                 [i2,width G1] in Indices G1 by A73,A96,A508,ZFMISC_1:87;
                      C/.i2 = C.i2 by A508,PARTFUN1:def 6;
                      then
A512:                 g/.len g=G1*(i2,width G1) by A508,A509,A510,
MATRIX_0:def 8;
                      assume
A513:                 pl=len g;
                      consider n1,n2 be Nat such that
A514:                 [n1,n2] in Indices G1 and
A515:                 g/.lg=G1*(n1,n2) by A15,A489,GOBOARD1:def 9;
A516:                 n1 in dom G1 by A96,A514,ZFMISC_1:87;
A517:                 n2 in Seg width G1 by A96,A514,ZFMISC_1:87;
                      [ma,K2] in Indices G1 by A41,A96,A365,A355,ZFMISC_1:87;
                      then i2=ma by A420,A513,A512,A511,GOBOARD1:5;
                      then
A518:                 |.n1-ma.|+|.n2- width G1.|=1 by A15,A489,A506,A360,A512
,A511,A514,A515,GOBOARD1:def 9;
                      now
                        per cases by A518,SEQM_3:42;
                        suppose
A519:                     |.n1-ma.|=1 & n2=width G1;
A520:                     dom C = Seg len C by FINSEQ_1:def 3
                            .= Seg len G1 by MATRIX_0:def 8
                            .= dom G1 by FINSEQ_1:def 3;
                          then C/.n1 = C.n1 by A516,PARTFUN1:def 6;
                          then g/.lg=C/.n1 by A515,A516,A519,MATRIX_0:def 8;
                          then
A521:                     g/.lg in rng C by A516,A520,PARTFUN2:2;
                          g/.lg=g2/.lg by A13,A24,A17,A489,FINSEQ_4:71;
                          hence contradiction by A13,A17,A489,A507,A521,
XREAL_1:44;
                        end;
                        suppose
A522:                     |.n2-width G1.|=1 & n1=ma;
                          len Lma = width G1 by MATRIX_0:def 7;
                          then
A523:                     n2 in dom Lma by A517,FINSEQ_1:def 3;
                          then La/.n2 = Lma.n2 by PARTFUN1:def 6;
                          then g/.lg=Lma/.n2 by A515,A517,A522,MATRIX_0:def 7;
                          then g/.lg in rng Lma by A523,PARTFUN2:2;
                          hence contradiction by A363,A489,A513,XREAL_1:44;
                        end;
                      end;
                      hence contradiction;
                    end;
                    then
A524:               pl<len g by A486,XXREAL_0:1;
                    len C=len G1 by MATRIX_0:def 8;
                    then
A525:               dom C = Seg len G1 by FINSEQ_1:def 3
                      .= dom G1 by FINSEQ_1:def 3;
A526:               Li/.K1 = Li.K1 by A367,PARTFUN1:def 6;
                    then
A527:               g/.pf=G1*(mi, K1) by A367,A368,A400,MATRIX_0:def 7;
                    now
                      consider i2 be Element of NAT such that
A528:                 i2 in dom C and
A529:                 C/.i2=g/.len g by A28,PARTFUN2:2;
                      C/.i2 = C.i2 by A528,PARTFUN1:def 6;
                      then
A530:                 g/.len g=G1*(i2,width G1) by A525,A528,A529,
MATRIX_0:def 8;
A531:                 [i2,width G1] in Indices G1 by A73,A96,A525,A528,
ZFMISC_1:87;
                      assume
A532:                 pf=len g;
                      consider n1,n2 be Nat such that
A533:                 [n1,n2] in Indices G1 and
A534:                 g/.lg=G1*(n1,n2) by A15,A489,GOBOARD1:def 9;
A535:                 n1 in dom G1 by A96,A533,ZFMISC_1:87;
A536:                 n2 in Seg width G1 by A96,A533,ZFMISC_1:87;
                      [mi,K1] in Indices G1 by A39,A96,A367,A400,ZFMISC_1:87;
                      then i2=mi by A527,A532,A530,A531,GOBOARD1:5;
                      then
A537:                 |.n1-mi.|+|.n2- width G1.|=1 by A15,A489,A506,A360,A530
,A531,A533,A534,GOBOARD1:def 9;
                      now
                        per cases by A537,SEQM_3:42;
                        suppose
A538:                     |.n1-mi.|=1 & n2=width G1;
A539:                     dom C = Seg len C by FINSEQ_1:def 3
                            .= Seg len G1 by MATRIX_0:def 8
                            .= dom G1 by FINSEQ_1:def 3;
                          then C/.n1 = C.n1 by A535,PARTFUN1:def 6;
                          then g/.lg=C/.n1 by A534,A535,A538,MATRIX_0:def 8;
                          then
A540:                     g/.lg in rng C by A535,A539,PARTFUN2:2;
                          g/.lg=g2/.lg by A13,A24,A17,A489,FINSEQ_4:71;
                          hence contradiction by A13,A17,A489,A507,A540,
XREAL_1:44;
                        end;
                        suppose
A541:                     |.n2-width G1.|=1 & n1=mi;
                          len Li = width G1 by MATRIX_0:def 7;
                          then
A542:                     n2 in dom Li by A536,FINSEQ_1:def 3;
                          then Li/.n2 = Li.n2 by PARTFUN1:def 6;
                          then g/.lg=Li/.n2 by A534,A536,A541,MATRIX_0:def 7;
                          then g/.lg in rng Li by A542,PARTFUN2:2;
                          hence contradiction by A359,A489,A532,XREAL_1:44;
                        end;
                      end;
                      hence contradiction;
                    end;
                    then
A543:               pf<len g by A418,XXREAL_0:1;
                    now
                      per cases by A488,XXREAL_0:1;
                      suppose
A544:                   pf<pl;
                        pl<=pl+1 by NAT_1:11;
                        then reconsider LL=pl+1-pf as Element of NAT by A544,
INT_1:5,XXREAL_0:2;
                        reconsider w1=pf-1 as Element of NAT by A364,INT_1:5;
                        set F1=g|pl;
                        defpred P[Nat, Element of TOP-REAL 2] means $2 = F1/.(
                        w1+$1);
A545:                   for n being Nat st n in Seg LL ex p st P[n,p];
                        consider F be FinSequence of TOP-REAL 2 such that
A546:                   len F = LL & for n being Nat st n in Seg LL
                        holds P[n,F/.n] from FINSEQ_4:sch 1(A545);
                        set hf=h1^F;
                        set FF = h1^F^h2;
A547:                   Seg len F = dom F by FINSEQ_1:def 3;
A548:                   for n st n in Seg LL holds F1/.(w1+n)=g/.(w1+n)
                        & w1+n in dom g
                        proof
                          let n such that
A549:                     n in Seg LL;
                          n<=LL by A549,FINSEQ_1:1;
                          then n+pf<=LL+ pf by XREAL_1:7;
                          then
A550:                     n+pf-1<=pl by XREAL_1:20;
                          1<=n by A549,FINSEQ_1:1;
                          then 0+1<=w1+n by XREAL_1:7;
                          then w1+n in Seg pl by A550,FINSEQ_1:1;
                          hence thesis by A363,FINSEQ_4:71;
                        end;
A551:                   rng F c= rng g2
                        proof
                          let x be object;
                          assume x in rng F;
                          then consider n being Element of NAT such that
A552:                     n in dom F and
A553:                     x=F/.n by PARTFUN2:2;
                          F/.n= F1/.(w1+n) by A546,A547,A552;
                          then
A554:                     F/.n=g/.(w1+n) by A548,A546,A547,A552;
                          w1+n in dom g by A548,A546,A547,A552;
                          then x in rng g by A553,A554,PARTFUN2:2;
                          hence thesis by A18;
                        end;
                        pf+1<=pl by A544,NAT_1:13;
                        then pf+1<=pl+1 by NAT_1:13;
                        then
A555:                   1<=LL by XREAL_1:19;
A556:                   now
                          let i1,i2,j1,j2 be Nat;
                          assume that
A557:                     [ i1,i2] in Indices G1 and
A558:                     [j1,j2] in Indices G1 and
A559:                     hf/.len hf=G1*( i1,i2) and
A560:                     h2/.1=G1*(j1,j2) and
                          len hf in dom hf and
A561:                     1 in dom h2;
                          ma+1 in dom G1 by A469,A561;
                          then
A562:                     [
ma+1,K2] in Indices G1 by A96,A365,A355,ZFMISC_1:87;
A563:                     [ma,K2] in Indices G1 by A41,A96,A365,A355,
ZFMISC_1:87;
A564:                     len F in dom F by A546,A547,A555,FINSEQ_1:1;
                          hf/.len hf=hf/.(len h1+len F) by FINSEQ_1:22
                            .= F/.len F by A564,FINSEQ_4:69
                            .= F1/.(w1+LL) by A546,A547,A564
                            .= G1*(ma,K2) by A420,A548,A546,A547,A564;
                          then
A565:                     i1=ma & i2=K2 by A557,A559,A563,GOBOARD1:5;
                          h2/.1=G1*(ma+1,K2) by A421,A561;
                          then j1=ma+1 & j2=K2 by A558,A560,A562,GOBOARD1:5;
                          hence |.i1-j1.|+|.i2-j2.|=|.ma-(ma+1).|+0 by A565,
ABSVALUE:2
                            .=|.-(ma+1 -ma).|
                            .=|.1.| by COMPLEX1:52
                            .=1 by ABSVALUE:def 1;
                        end;
A566:                   rng FF = rng(h1^F) \/ rng h2 by FINSEQ_1:31
                          .= rng h1 \/ rng F \/ rng h2 by FINSEQ_1:31;
A567:                   for k st k in Seg width G1 & rng F /\ rng Col(G1
                        ,k)={} holds rng FF /\ rng Col(G1,k)={}
                        proof
A568:                     len Col(G1,K2)=len G1 by MATRIX_0:def 8;
A569:                     len Col(G1,K1)=len G1 by MATRIX_0:def 8;
                          let k;
                          assume that
A570:                     k in Seg width G1 and
A571:                     rng F /\ rng Col(G1,k)={};
                          set gg=Col(G1,k);
                          assume
A572:                     rng FF /\ rng gg <> {};
                          set x = the Element of rng FF /\ rng gg;
                          rng FF = rng F \/ (rng h1 \/ rng h2) by A566,
XBOOLE_1:4;
                          then
A573:                     rng FF /\ rng gg = {} \/ (rng h1 \/ rng h2) /\
                          rng gg by A571,XBOOLE_1:23
                            .= rng h1 /\ rng gg \/ rng h2 /\ rng gg by
XBOOLE_1:23;
                          now
                            per cases by A572,A573,XBOOLE_0:def 3;
                            suppose
A574:                         x in rng h1 /\ rng gg;
                              then x in rng h1 by XBOOLE_0:def 4;
                              then consider i being Element of NAT such that
A575:                         i in dom h1 and
A576:                         x=h1/.i by PARTFUN2:2;
A577:                         x=G1*(i,K1) by A369,A575,A576;
                              reconsider y=x as Point of TOP-REAL 2 by A576;
A578:                         Lmi/.K1 = Lmi.K1 by A367,PARTFUN1:def 6;
A579:                         x in rng gg by A574,XBOOLE_0:def 4;
A580:                         dom CK1 = Seg len G1 by A569,FINSEQ_1:def 3
                                .= dom G1 by FINSEQ_1:def 3;
                              then
A581:                         CK1/.mi = CK1.mi by A39,PARTFUN1:def 6;
A582:                         i in dom CK1 by A401,A575,A580;
                              CK1/.i = CK1.i by A401,A575,A580,PARTFUN1:def 6;
                              then y=CK1/.i by A577,A580,A582,MATRIX_0:def 8;
                              then y in rng CK1 by A582,PARTFUN2:2;
                              then
A583:                         K1=k by A367,A400,A570,A579,GOBOARD1:4;
A584:                         1 in Seg LL by A555,FINSEQ_1:1;
                              then F/.1=F1/.(w1+1) & F1/.(w1+1)=g/.(w1+1) by
A548,A546;
                              then F/.1= CK1/.mi by A39,A367,A368,A400,A578
,A581,MATRIX_0:42;
                              then
A585:                         F/.1 in rng Col(G1,k) by A39,A580,A583,PARTFUN2:2
;
                              F/.1 in rng F by A546,A547,A584,PARTFUN2:2;
                              hence contradiction by A571,A585,XBOOLE_0:def 4;
                            end;
                            suppose
A586:                         x in rng h2 /\ rng gg;
                              then x in rng h2 by XBOOLE_0:def 4;
                              then consider i being Element of NAT such that
A587:                         i in dom h2 and
A588:                         x=h2/.i by PARTFUN2:2;
A589:                         x=G1*(ma+i,K2) by A421,A587,A588;
                              reconsider y=x as Point of TOP-REAL 2 by A588;
A590:                         Lma /.K2 = Lma.K2 by A365,PARTFUN1:def 6;
A591:                         x in rng gg by A586,XBOOLE_0:def 4;
A592:                         dom CK2 = Seg len G1 by A568,FINSEQ_1:def 3
                                .= dom G1 by FINSEQ_1:def 3;
                              then
A593:                         CK2/.ma = CK2.ma by A41,PARTFUN1:def 6;
A594:                         ma+i in dom CK2 by A469,A587,A592;
                              CK2/.(ma+i) = CK2.(ma+i) by A469,A587,A592,
PARTFUN1:def 6;
                              then y=CK2/.(ma+i) by A589,A592,A594,
MATRIX_0:def 8;
                              then y in rng CK2 by A594,PARTFUN2:2;
                              then
A595:                         K2=k by A365,A355,A570,A591,GOBOARD1:4;
A596:                         LL in Seg LL by A555,FINSEQ_1:1;
                              then F/.LL=F1/.(w1+LL) & F1/.(w1+LL)=g/.(w1+LL
                              ) by A548,A546;
                              then F/.LL= CK2/.ma by A41,A365,A366,A355,A590
,A593,MATRIX_0:42;
                              then
A597:                         F/.LL in rng Col(G1,k) by A41,A592,A595,
PARTFUN2:2;
                              F/.LL in rng F by A546,A547,A596,PARTFUN2:2;
                              hence contradiction by A571,A597,XBOOLE_0:def 4;
                            end;
                          end;
                          hence contradiction;
                        end;
                        rng F /\ rng C = {}
                        proof
                          reconsider w=w1 as Nat;
                          set x = the Element of rng F /\ rng C;
                          assume
A598:                     not thesis;
                          then
A599:                     x in rng C by XBOOLE_0:def 4;
                          x in rng F by A598,XBOOLE_0:def 4;
                          then consider i1 be Element of NAT such that
A600:                     i1 in dom F and
A601:                     F/.i1=x by PARTFUN2:2;
A602:                     Seg len F = dom F by FINSEQ_1:def 3;
                          then i1<=LL by A546,A600,FINSEQ_1:1;
                          then
A603:                     w+i1<=w+LL by XREAL_1:7;
A604:                     w1+i1 in dom g by A548,A546,A600,A602;
                          then
A605:                     w+i1 in dom g2 by A13,A24,A17,FINSEQ_4:71;
                          F /.i1=F1/.(w1+i1) & F1/.(w1+i1)= g/.(w1+i1)
                          by A548,A546,A600,A602;
                          then g2/.(w+i1) in rng C by A13,A24,A17,A599,A601
,A604,FINSEQ_4:71;
                          then m<=w+i1 by A13,A605;
                          hence contradiction by A17,A524,A603,XXREAL_0:2;
                        end;
                        then rng FF /\ rng Col(G1,width G1) = {} by A73,A567;
                        then
A606:                   rng FF misses rng Col(G1,width G1) by XBOOLE_0:def 7;
                        now
                          reconsider w=w1 as Nat;
                          let n;
                          assume
A607:                     n in dom F;
                          then w1+n in dom g by A548,A546,A547;
                          then consider i,j such that
A608:                     [i,j] in Indices G1 & g/.(w+n)=G1*(i,j) by A15,
GOBOARD1:def 9;
                          take i,j;
                          F/.n= F1/.(w1+n) by A546,A547,A607;
                          hence [i,j] in Indices G1 & F/.n=G1*(i,j) by A548
,A546,A547,A607,A608;
                        end;
                        then for n st n in dom(h1^F) ex i,j st [i,j] in
                        Indices G1 & (h1^F)/.n=G1*(i,j) by A406,GOBOARD1:23;
                        then
A609:                   for n st n in dom FF ex i,j st [i,j] in Indices
                        G1 & FF/.n=G1*(i,j) by A473,GOBOARD1:23;
A610:                   now
A611:                     [mi,K1] in Indices G1 by A39,A96,A367,A400,
ZFMISC_1:87;
                          let i1,i2,j1,j2 be Nat;
                          assume that
A612:                     [i1,i2] in Indices G1 and
A613:                     [j1,j2] in Indices G1 and
A614:                     h1/.len h1=G1*(i1,i2) and
A615:                     F/.1=G1*(j1,j2) and
A616:                     len h1 in dom h1 and
A617:                     1 in dom F;
                          F/.1=F1/.(w1+1) by A546,A547,A617;
                          then F/.1=g/.(w1+1) by A548,A546,A547,A617
                            .= G1*(mi,K1) by A367,A368,A400,A526,MATRIX_0:def 7
;
                          then
A618:                     j1=mi & j2=K1 by A613,A615,A611,GOBOARD1:5;
                          l1 in dom G1 by A369,A401,A616;
                          then
A619:                     [l1,K1] in Indices G1 by A96,A367,A400,ZFMISC_1:87;
                          h1/.len h1=G1*(l1,K1) by A369,A616;
                          then i1=l1 & i2=K1 by A612,A614,A619,GOBOARD1:5;
                          hence |.i1-j1.|+|.i2-j2.|= |.mi-1-mi.|+0 by A618,
ABSVALUE:2
                            .= |.-1.|
                            .= |.1.| by COMPLEX1:52
                            .= 1 by ABSVALUE:def 1;
                        end;
                        now
                          let n;
                          assume that
A620:                     n in dom F and
A621:                     n+1 in dom F;
                          F/.n=F1/.(w1+n) by A546,A547,A620;
                          then
A622:                     F/.n=g/.(w1+n) by A548,A546,A547,A620;
                          F/.(n+1)=F1/.(w1+(n+1)) by A546,A547,A621;
                          then
A623:                     w1+n+1 in dom g & F/.(n+1)=g/.(w1+n+1) by A548,A546
,A547,A621;
                          let i1,i2,j1,j2 be Nat;
                          assume
A624:                     [i1,i2] in Indices G1 & [j1,j2] in Indices
                          G1 & F/.n=G1*(i1,i2) & F/. (n+1)=G1*(j1,j2);
                          w1+n in dom g by A548,A546,A547,A620;
                          hence |.i1-j1.|+|.i2-j2.|=1 by A15,A622,A623,A624,
GOBOARD1:def 9;
                        end;
                        then for n st n in dom(h1^F) & n+1 in dom(h1^F) for
i1,i2,j1,j2 be Nat st [i1,i2] in Indices G1 & [j1,j2] in Indices G1
& (h1^F)/.n=G1*(i1,i2) & (h1^F)/.(n+1)=G1*(j1,j2) holds |.i1-j1.|+|.i2-j2.|=1
                        by A408,A610,GOBOARD1:24;
                        then for n st n in dom FF & n+1 in dom FF for i1,i2,
j1,j2 be Nat st [i1,i2] in Indices G1 & [j1,j2] in Indices G1 & FF/.
n=G1*(i1,i2) & FF/.(n+1)=G1*(j1,j2) holds |.i1-j1.|+|.i2-j2.|=1 by A475,A556,
GOBOARD1:24;
                        then FF is_sequence_on G1 by A609,GOBOARD1:def 9;
                        then
A625:                   FF is_sequence_on D by A73,A152,A606,GOBOARD1:25;
                        set x = the Element of rng FF /\ rng tt;
A626:                   0+1<=len F+(len h1+len h2) by A546,A555,XREAL_1:7;
A627:                   now
                          per cases;
                          suppose
A628:                       mi=1;
A629:                       pf in Seg pl by A364,A544,FINSEQ_1:1;
A630:                       1 in Seg LL by A555,FINSEQ_1:1;
                            h1 = {} by A369,A628;
                            then FF=F^h2 by FINSEQ_1:34;
                            then FF/.1=F/.1 by A546,A547,A630,FINSEQ_4:68
                              .= F1/.(w1+1) by A546,A630
                              .= g/.pf by A363,A629,FINSEQ_4:71;
                            hence FF/.1 in rng Line(G1,1) by A359,A628;
                          end;
                          suppose
A631:                       mi<>1;
                            1<=mi by A39,FINSEQ_3:25;
                            then 1<mi by A631,XXREAL_0:1;
                            then 1+1<=mi by NAT_1:13;
                            then
A632:                       1<=l1 by XREAL_1:19;
                            then
A633:                       1 in Seg l1 by FINSEQ_1:1;
                            len Line(G1,1)=width G1 by MATRIX_0:def 7;
                            then
A634:                       K1 in dom L1 by A367,A400,FINSEQ_1:def 3;
                            then
A635:                       L1/.K1 = L1.K1 by PARTFUN1:def 6;
                            len(h1^F)=len h1 + len F & 0<=len F by FINSEQ_1:22;
                            then 0+1<=len(h1^F) by A369,A632,XREAL_1:7;
                            then 1 in dom(h1^F) by FINSEQ_3:25;
                            then FF/.1=(h1^F)/.1 by FINSEQ_4:68
                              .=h1/.1 by A369,A468,A633,FINSEQ_4:68
                              .=G1*(1,K1) by A369,A468,A633
                              .=L1/.K1 by A367,A400,A635,MATRIX_0:def 7;
                            hence FF/.1 in rng Line(G1,1) by A634,PARTFUN2:2;
                          end;
                        end;
A636:                   w1+LL= pl;
A637:                   now
                          per cases;
                          suppose
A638:                       ma=len G1;
                            1<=pl by A24,A363,FINSEQ_1:1;
                            then
A639:                       pl in Seg pl by FINSEQ_1:1;
A640:                       len F in dom F by A546,A555,FINSEQ_3:25;
                            h2 = {} by A421,A638;
                            then
A641:                       FF=h1^F by FINSEQ_1:34;
                            then FF/.len FF=FF/.(len h1+len F) by FINSEQ_1:22
                              .= F/.LL by A546,A641,A640,FINSEQ_4:69
                              .= F1/.pl by A546,A547,A636,A640
                              .= g/.pl by A363,A639,FINSEQ_4:71;
                            hence FF/.len FF in rng Line(G1,len G1) by A363
,A638;
                          end;
                          suppose
A642:                       ma<>len G1;
                            ma<=len G1 by A41,FINSEQ_3:25;
                            then ma<len G1 by A642,XXREAL_0:1;
                            then ma+1<=len G1 by NAT_1:13;
                            then
A643:                       1<=l2 by XREAL_1:19;
                            then
A644:                       l2 in Seg l2 by FINSEQ_1:1;
                            len Line(G1,len G1)=width G1 by MATRIX_0:def 7;
                            then
A645:                       K2 in dom Ll by A365,A355,FINSEQ_1:def 3;
                            then
A646:                       Ll/.K2 = Ll.K2 by PARTFUN1:def 6;
A647:                       len h2 in dom h2 by A421,A643,FINSEQ_3:25;
                            FF/.len FF=FF/.(len(h1^F)+len h2) by FINSEQ_1:22
                              .=h2/.l2 by A421,A647,FINSEQ_4:69
                              .=G1*(ma+l2,K2) by A421,A485,A644
                              .=Ll/.K2 by A365,A355,A646,MATRIX_0:def 7;
                            hence FF/.len FF in rng Line(G1,len G1) by A645,
PARTFUN2:2;
                          end;
                        end;
                        rng tt c= rng f by A351,A354;
                        then
A648:                   rng tt c= rng g1 by A178;
A649:                   len FF=len(h1^F)+len h2 by FINSEQ_1:22
                          .=len h1+len F+len h2 by FINSEQ_1:22;
                        then 1 in dom FF by A626,FINSEQ_3:25;
                        then
A650:                   FF/. 1 in rng Line(D,1) by A73,A26,A152,A627,A606,
MATRIX_0:75;
                        len FF in dom FF by A649,A626,FINSEQ_3:25;
                        then
A651:                   FF/.len FF in rng Line(D,len D) by A73,A27,A152,A153
,A637,A606,MATRIX_0:75;
                        width D=k by A3,A73,MATRIX_0:63;
                        then rng FF /\ rng tt <> {} by A2,A92,A353,A649,A626
,A625,A650,A651,A341;
                        then
A652:                   x in rng FF /\ rng tt;
                        rng tt /\ rng FF = ((rng h1 \/ rng F) /\ rng tt
                        ) \/ {} by A492,A566,XBOOLE_1:23
                          .= {} \/ rng F /\ rng tt by A424,XBOOLE_1:23
                          .= rng tt /\ rng F;
                        then rng FF /\ rng tt c= rng g1 /\ rng g2 by A648,A551,
XBOOLE_1:27;
                        hence thesis by A652;
                      end;
                      suppose
A653:                   pl<pf;
                        pf<=pf+1 by NAT_1:11;
                        then reconsider LL=pf+1-pl as Element of NAT by A653,
INT_1:5,XXREAL_0:2;
                        set F1=g|pf;
                        defpred P[Nat,Element of TOP-REAL 2] means for k st k
                        = pf+1-$1 holds $2 = F1/.k;
A654:                   for n,k st n in Seg LL & k = pf+1-n holds F1/.k
                        =g/.k & k in dom g
                        proof
                          let n,k;
                          assume that
A655:                     n in Seg LL and
A656:                     k = pf+1-n;
A657:                     n<=LL by A655,FINSEQ_1:1;
                          1<=n by A655,FINSEQ_1:1;
                          then
A658:                     pf+1-n<=pf+1-1 by XREAL_1:13;
                          LL<=pf+1-0 by XREAL_1:13;
                          then reconsider w=pf+1-n as Element of NAT by A657,
INT_1:5,XXREAL_0:2;
                          pf+1-LL<=pf+1-n by A657,XREAL_1:13;
                          then 1<=pf+1-n by A487,XXREAL_0:2;
                          then w in Seg pf by A658,FINSEQ_1:1;
                          hence thesis by A359,A656,FINSEQ_4:71;
                        end;
A659:                   for n st n in Seg LL holds pf+1-n is Element of NAT
                        proof
                          let n;
A660:                     LL<=pf+1-0 by XREAL_1:13;
                          assume n in Seg LL;
                          then n<=LL by FINSEQ_1:1;
                          hence thesis by A660,INT_1:5,XXREAL_0:2;
                        end;
A661:                   for n being Nat st n in Seg LL ex p st P[n,p]
                        proof
                          let n be Nat;
                          assume
A662:                     n in Seg LL;
                          then reconsider
                          k = pf+1-n as Nat by A659;
                          take g/.k;
                          thus thesis by A654,A662;
                        end;
                        consider F be FinSequence of TOP-REAL 2 such that
A663:                   len F = LL & for n being Nat st n in Seg LL
                        holds P[n,F/.n] from FINSEQ_4:sch 1(A661);
                        set hf=h1^F;
                        set FF = h1^F^h2;
A664:                   Seg len F = dom F by FINSEQ_1:def 3;
A665:                   rng F c= rng g2
                        proof
                          let x be object;
                          assume x in rng F;
                          then consider n being Element of NAT such that
A666:                     n in dom F and
A667:                     x=F/.n by PARTFUN2:2;
                          reconsider u = pf+1-n as Nat by A659,A663
,A664,A666;
                          F/.n=F1/.u by A663,A664,A666;
                          then pf+1-n in dom g & F/.n=g/.u by A654,A663,A664
,A666;
                          then x in rng g by A667,PARTFUN2:2;
                          hence thesis by A18;
                        end;
                        pl+1<=pf by A653,NAT_1:13;
                        then pl+1<=pf+1 by NAT_1:13;
                        then
A668:                   1<=LL by XREAL_1:19;
A669:                   now
                          reconsider u = pf+1-LL as Nat;
                          let i1,i2,j1,j2 be Nat;
                          assume that
A670:                     [ i1,i2] in Indices G1 and
A671:                     [j1,j2] in Indices G1 and
A672:                     hf/.len hf=G1*(i1,i2) and
A673:                     h2/.1=G1*(j1,j2) and
                          len hf in dom hf and
A674:                     1 in dom h2;
                          ma+1 in dom G1 by A469,A674;
                          then
A675:                     [ma+1,K2] in Indices G1 by A96,A365,A355,ZFMISC_1:87;
A676:                     [ma,K2] in Indices G1 by A41,A96,A365,A355,
ZFMISC_1:87;
A677:                     len F in dom F by A663,A668,FINSEQ_3:25;
                          hf/.len hf=hf/.(len h1+len F) by FINSEQ_1:22
                            .= F/.len F by A677,FINSEQ_4:69
                            .= F1/.u by A663,A664,A677
                            .= g/.u by A654,A663,A664,A677
                            .= G1*(ma,K2) by A365,A366,A355,A419,MATRIX_0:def 7
;
                          then
A678:                     i1=ma & i2=K2 by A670,A672,A676,GOBOARD1:5;
                          h2/.1=G1*(ma+1,K2) by A421,A674;
                          then j1=ma+1 & j2=K2 by A671,A673,A675,GOBOARD1:5;
                          hence |.i1-j1.|+|.i2-j2.|=|.ma-(ma+1).|+0 by A678,
ABSVALUE:2
                            .=|.-(ma+1 -ma).|
                            .=|.1.| by COMPLEX1:52
                            .=1 by ABSVALUE:def 1;
                        end;
                        now
                          let n;
                          assume
A679:                     n in dom F;
                          then reconsider
                          w=pf+1-n as Nat by A659,A663,A664;
A680:                     F/.n=F1/.w by A663,A664,A679;
                          then pf+1-n in dom g by A654,A663,A664,A679;
                          then consider i,j such that
A681:                     [i,j] in Indices G1 & g/.w=G1*(i,j) by A15,
GOBOARD1:def 9;
                          take i,j;
                          thus [i,j] in Indices G1 & F/.n=G1*(i,j) by A654,A663
,A664,A679,A680,A681;
                        end;
                        then for n st n in dom(h1^F) ex i,j st [i,j] in
                        Indices G1 & (h1^F)/.n=G1*(i,j) by A406,GOBOARD1:23;
                        then
A682:                   for n st n in dom FF ex i,j st [i,j] in Indices
                        G1 & FF/.n=G1*(i,j) by A473,GOBOARD1:23;
                        set x = the Element of rng FF /\ rng tt;
A683:                   0+1<=len F+(len h1+len h2) by A663,A668,XREAL_1:7;
A684:                   rng FF = rng(h1^F) \/ rng h2 by FINSEQ_1:31
                          .= rng h1 \/ rng F \/ rng h2 by FINSEQ_1:31;
A685:                   for k st k in Seg width G1 & rng F /\ rng Col(
                        G1,k)={} holds rng FF /\ rng Col(G1,k)={}
                        proof
A686:                     len Col(G1,K2)=len G1 by MATRIX_0:def 8;
A687:                     len Col(G1,K1)=len G1 by MATRIX_0:def 8;
                          let k;
                          assume that
A688:                     k in Seg width G1 and
A689:                     rng F /\ rng Col(G1,k)={};
                          set gg=Col(G1,k);
                          assume
A690:                     rng FF /\ rng gg <> {};
                          set x = the Element of rng FF /\ rng gg;
                          rng FF = rng F \/ (rng h1 \/ rng h2) by A684,
XBOOLE_1:4;
                          then
A691:                     rng FF /\ rng gg = {} \/ (rng h1 \/ rng h2)
                          /\ rng gg by A689,XBOOLE_1:23
                            .= rng h1 /\ rng gg \/ rng h2 /\ rng gg by
XBOOLE_1:23;
                          now
                            per cases by A690,A691,XBOOLE_0:def 3;
                            suppose
A692:                         x in rng h1 /\ rng gg;
                              then
A693:                         x in rng gg by XBOOLE_0:def 4;
A694:                         1 in Seg LL by A668,FINSEQ_1:1;
                              pf+1-1 = pf;
                              then
A695:                         F/.1=F1/.pf & F1/.pf=g/.pf by A654,A663,A694;
                              x in rng h1 by A692,XBOOLE_0:def 4;
                              then consider i being Element of NAT such that
A696:                         i in dom h1 and
A697:                         x=h1/.i by PARTFUN2:2;
A698:                         x=G1*(i,K1) by A369,A696,A697;
                              reconsider y=x as Point of TOP-REAL 2 by A697;
A699:                         Lmi/.K1 = Lmi.K1 by A367,PARTFUN1:def 6;
A700:                         dom CK1 = Seg len G1 by A687,FINSEQ_1:def 3
                                .= dom G1 by FINSEQ_1:def 3;
                              then
A701:                         i in dom CK1 by A401,A696;
                              CK1/.i = CK1.i by A401,A696,A700,PARTFUN1:def 6;
                              then y=CK1/.i by A698,A700,A701,MATRIX_0:def 8;
                              then y in rng CK1 by A701,PARTFUN2:2;
                              then
A702:                         K1=k by A367,A400,A688,A693,GOBOARD1:4;
                              CK1/.mi = CK1.mi by A39,A700,PARTFUN1:def 6;
                              then F/.1=CK1/.mi by A39,A367,A368,A400,A699,A695
,MATRIX_0:42;
                              then
A703:                         F/.1 in rng Col(G1,k) by A39,A700,A702,PARTFUN2:2
;
                              F/.1 in rng F by A663,A664,A694,PARTFUN2:2;
                              hence contradiction by A689,A703,XBOOLE_0:def 4;
                            end;
                            suppose
A704:                         x in rng h2 /\ rng gg;
                              then x in rng h2 by XBOOLE_0:def 4;
                              then consider i being Element of NAT such that
A705:                         i in dom h2 and
A706:                         x=h2/.i by PARTFUN2:2;
A707:                         x=G1*(ma+i,K2) by A421,A705,A706;
                              reconsider y=x as Point of TOP-REAL 2 by A706;
A708:                         x in rng gg by A704,XBOOLE_0:def 4;
                              reconsider u = pf+1-LL as Nat;
A709:                         Lma /.K2 = Lma.K2 by A365,PARTFUN1:def 6;
A710:                         dom CK2 = Seg len G1 by A686,FINSEQ_1:def 3
                                .= dom G1 by FINSEQ_1:def 3;
                              then
A711:                         CK2/.ma = CK2.ma by A41,PARTFUN1:def 6;
A712:                         ma+i in dom CK2 by A469,A705,A710;
                              CK2/.(ma+i) = CK2.(ma+i) by A469,A705,A710,
PARTFUN1:def 6;
                              then y=CK2/.(ma+i) by A707,A710,A712,
MATRIX_0:def 8;
                              then y in rng CK2 by A712,PARTFUN2:2;
                              then
A713:                         K2=k by A365,A355,A688,A708,GOBOARD1:4;
A714:                         LL in Seg LL by A668,FINSEQ_1:1;
                              then F/.LL=F1/.u & F1/.u=g/.u by A654,A663;
                              then F/.LL= CK2/.ma by A41,A365,A366,A355,A709
,A711,MATRIX_0:42;
                              then
A715:                         F/.LL in rng Col(G1,k) by A41,A710,A713,
PARTFUN2:2;
                              F/.LL in rng F by A663,A664,A714,PARTFUN2:2;
                              hence contradiction by A689,A715,XBOOLE_0:def 4;
                            end;
                          end;
                          hence contradiction;
                        end;
                        rng F /\ rng C = {}
                        proof
                          set x = the Element of rng F /\ rng C;
                          assume
A716:                     not thesis;
                          then
A717:                     x in rng C by XBOOLE_0:def 4;
                          x in rng F by A716,XBOOLE_0:def 4;
                          then consider i1 be Element of NAT such that
A718:                     i1 in dom F and
A719:                     F/.i1=x by PARTFUN2:2;
                          reconsider w=pf+1-i1 as Nat by A659,A663
,A664,A718;
                          1<=i1 by A718,FINSEQ_3:25;
                          then
A720:                     w<=pf+1-1 by XREAL_1:13;
A721:                     w in dom g by A654,A663,A664,A718;
                          then
A722:                     w in dom g2 by A13,A24,A17,FINSEQ_4:71;
                          F/.i1=F1/.w & F1/.w=g/.w by A654,A663,A664,A718;
                          then g2/.w in rng C by A13,A24,A17,A717,A719,A721,
FINSEQ_4:71;
                          then m<=w by A13,A722;
                          hence contradiction by A17,A543,A720,XXREAL_0:2;
                        end;
                        then rng FF /\ rng Col(G1,width G1) = {} by A73,A685;
                        then
A723:                   rng FF misses rng Col(G1,width G1) by XBOOLE_0:def 7;
A724:                   now
                          let n;
                          assume that
A725:                     n in dom F and
A726:                     n+1 in dom F;
                          reconsider w3=pf+1-n, w2=pf+1-(n+1) as Element of
                          NAT by A659,A663,A664,A725,A726;
                          F/.n=F1/.w3 by A663,A664,A725;
                          then
A727:                     pf+1-n in dom g & F/.n =g/.w3 by A654,A663,A664,A725;
                          F/.(n+1)=F1/.w2 by A663,A664,A726;
                          then
A728:                     pf+1-(n+1) in dom g & F/.(n+1)=g/.w2 by A654,A663
,A664,A726;
                          let i1,i2,j1,j2 be Nat;
                          assume
A729:                     [i1,i2] in Indices G1 & [j1,j2] in
                          Indices G1 & F/.n=G1*(i1,i2) & F/. (n+1)=G1*(j1,j2);
                          w2+1=pf+1-n;
                          hence 1=|.j1-i1.|+|.j2-i2.| by A15,A727,A728,A729,
GOBOARD1:def 9
                            .= |.-(i1-j1).|+|.-(i2-j2).|
                            .= |.i1-j1.|+|.-(i2-j2).| by COMPLEX1:52
                            .= |.i1-j1.|+|.i2-j2.| by COMPLEX1:52;
                        end;
A730:                   pf+1-1 = pf;
A731:                   now
                          per cases;
                          suppose
A732:                       mi=1;
A733:                       pf in Seg pf by A364,FINSEQ_1:1;
A734:                       1 in Seg LL by A668,FINSEQ_1:1;
                            h1 = {} by A369,A732;
                            then FF=F^h2 by FINSEQ_1:34;
                            then FF/.1=F/.1 by A663,A664,A734,FINSEQ_4:68
                              .= F1/.pf by A663,A730,A734
                              .= g/.pf by A359,A733,FINSEQ_4:71;
                            hence FF/.1 in rng Line(G1,1) by A359,A732;
                          end;
                          suppose
A735:                       mi<>1;
                            1<=mi by A39,FINSEQ_3:25;
                            then 1<mi by A735,XXREAL_0:1;
                            then 1+1<=mi by NAT_1:13;
                            then
A736:                       1<=l1 by XREAL_1:19;
                            then
A737:                       1 in Seg l1 by FINSEQ_1:1;
                            len Line(G1,1)=width G1 by MATRIX_0:def 7;
                            then
A738:                       K1 in dom L1 by A367,A400,FINSEQ_1:def 3;
                            then
A739:                       L1/.K1 = L1.K1 by PARTFUN1:def 6;
                            len(h1^F)=len h1 + len F & 0<=len F by FINSEQ_1:22;
                            then 0+1<=len(h1^F) by A369,A736,XREAL_1:7;
                            then 1 in dom(h1^F) by FINSEQ_3:25;
                            then FF/.1=(h1^F)/.1 by FINSEQ_4:68
                              .=h1/.1 by A369,A468,A737,FINSEQ_4:68
                              .=G1*(1,K1) by A369,A468,A737
                              .=L1/.K1 by A367,A400,A739,MATRIX_0:def 7;
                            hence FF/.1 in rng Line(G1,1) by A738,PARTFUN2:2;
                          end;
                        end;
                        rng tt c= rng f by A351,A354;
                        then
A740:                   rng tt c= rng g1 by A178;
                        now
A741:                     [mi,K1] in Indices G1 by A39,A96,A367,A400,
ZFMISC_1:87;
                          reconsider w4 = pf + 1 - 1 as Nat;
                          let i1,i2,j1,j2 be Nat;
                          assume that
A742:                     [i1,i2] in Indices G1 and
A743:                     [j1,j2] in Indices G1 and
A744:                     h1/.len h1=G1*(i1,i2) and
A745:                     F/.1=G1*(j1,j2) and
A746:                     len h1 in dom h1 and
A747:                     1 in dom F;
                          F/.1 = F1/.w4 by A663,A664,A747
                            .= g/.w4 by A654,A663,A664,A747
                            .= G1*(mi,K1) by A367,A368,A400,A526,MATRIX_0:def 7
;
                          then
A748:                     j1=mi & j2=K1 by A743,A745,A741,GOBOARD1:5;
                          l1 in dom G1 by A369,A401,A746;
                          then
A749:                     [l1,K1] in Indices G1 by A96,A367,A400,ZFMISC_1:87;
                          h1/.len h1=G1*(l1,K1) by A369,A746;
                          then i1=l1 & i2=K1 by A742,A744,A749,GOBOARD1:5;
                          hence |.i1-j1.|+|.i2-j2.|= |.mi-1-mi.|+0 by A748,
ABSVALUE:2
                            .= |.-1.|
                            .= |.1.| by COMPLEX1:52
                            .= 1 by ABSVALUE:def 1;
                        end;
                        then for n st n in dom(h1^F) & n+1 in dom(h1^F) for
i1,i2,j1,j2 be Nat st [i1,i2] in Indices G1 & [j1,j2] in Indices G1
& (h1^F)/.n=G1*(i1,i2) & (h1^F)/.(n+1)=G1*(j1,j2) holds |.i1-j1.|+|.i2-j2.|=1
                        by A408,A724,GOBOARD1:24;
                        then for n st n in dom FF & n+1 in dom FF for m,k,i,
j st [m,k] in Indices G1 & [i,j] in Indices G1 & FF/.n=G1*(m,k) & FF/.(n+1)=G1*
(i,j) holds |.m-i.|+|.k-j.|=1 by A475,A669,GOBOARD1:24;
                        then FF is_sequence_on G1 by A682,GOBOARD1:def 9;
                        then
A750:                   FF is_sequence_on D by A73,A152,A723,GOBOARD1:25;
A751:                   len FF=len(h1^F)+len h2 by FINSEQ_1:22
                          .=len h1+len F+len h2 by FINSEQ_1:22;
                        then 1 in dom FF by A683,FINSEQ_3:25;
                        then
A752:                   FF/. 1 in rng Line(D,1) by A73,A26,A152,A731,A723,
MATRIX_0:75;
A753:                   now
                          per cases;
                          suppose
A754:                       ma=len G1;
A755:                       pl in Seg pf by A487,A653,FINSEQ_1:1;
A756:                       pf+1-(pf+1-pl) = pl;
A757:                       len F in dom F by A663,A668,FINSEQ_3:25;
                            h2 = {} by A421,A754;
                            then
A758:                       FF=h1^F by FINSEQ_1:34;
                            then FF/.len FF=FF/.(len h1+len F) by FINSEQ_1:22
                              .= F/.LL by A663,A758,A757,FINSEQ_4:69
                              .= F1/.pl by A663,A664,A756,A757
                              .= g/.pl by A359,A755,FINSEQ_4:71;
                            hence FF/.len FF in rng Line(G1,len G1) by A363
,A754;
                          end;
                          suppose
A759:                       ma<>len G1;
                            ma<=len G1 by A41,FINSEQ_3:25;
                            then ma<len G1 by A759,XXREAL_0:1;
                            then ma+1<=len G1 by NAT_1:13;
                            then
A760:                       1<=l2 by XREAL_1:19;
                            then
A761:                       l2 in Seg l2 by FINSEQ_1:1;
                            len Line(G1,len G1)=width G1 by MATRIX_0:def 7;
                            then
A762:                       K2 in dom Ll by A365,A355,FINSEQ_1:def 3;
                            then
A763:                       Ll/.K2 = Ll.K2 by PARTFUN1:def 6;
A764:                       len h2 in dom h2 by A421,A760,FINSEQ_3:25;
                            FF/.len FF=FF/.(len(h1^F)+len h2) by FINSEQ_1:22
                              .=h2/.l2 by A421,A764,FINSEQ_4:69
                              .=G1*(ma+l2,K2) by A421,A485,A761
                              .=Ll/.K2 by A365,A355,A763,MATRIX_0:def 7;
                            hence FF/.len FF in rng Line(G1,len G1) by A762,
PARTFUN2:2;
                          end;
                        end;
                        len FF in dom FF by A751,A683,FINSEQ_3:25;
                        then
A765:                   FF/.len FF in rng Line(D,len D) by A73,A27,A152,A153
,A753,A723,MATRIX_0:75;
                        width D=k by A3,A73,MATRIX_0:63;
                        then rng FF /\ rng tt <> {} by A2,A92,A353,A751,A683
,A750,A752,A765,A341;
                        then
A766:                   x in rng FF /\ rng tt;
                        rng tt /\ rng FF = ((rng h1 \/ rng F) /\ rng tt
                        ) \/ {} by A492,A684,XBOOLE_1:23
                          .= {} \/ rng F /\ rng tt by A424,XBOOLE_1:23
                          .= rng tt /\ rng F;
                        then rng FF /\ rng tt c= rng g1 /\ rng g2 by A740,A665,
XBOOLE_1:27;
                        hence thesis by A766;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                hence thesis;
              end;
            end;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
A767: P[0];
A768: for k holds P[k] from NAT_1:sch 2(A767,A1);
A769: now
    let k;
    let G1,g1,g2;
    assume k=width G1 & k>0 & 1<=len g1 & 1<=len g2 & g1 is_sequence_on G1
& g2 is_sequence_on G1 & g1/.1 in rng Line(G1,1) & g1/.len g1 in rng Line(G1,
    len G1) & g2/.1 in rng Col(G1,1) & g2/.len g2 in rng Col(G1,width G1);
    then rng g1 /\ rng g2 <> {} by A768;
    hence rng g1 meets rng g2 by XBOOLE_0:def 7;
  end;
  width G <> 0 by MATRIX_0:def 10;
  then
A770: width G > 0;
  assume 1<=len f1 & 1<=len f2 & f1 is_sequence_on G & f2 is_sequence_on G &
f1/.1 in rng Line(G,1) & f1/.len f1 in rng Line(G,len G) & f2/.1 in rng Col(G,
  1) & f2/.len f2 in rng Col(G,width G);
  hence thesis by A769,A770;
end;
