reserve p,p1,p2,q for Point of TOP-REAL 2,
  f,g,g1,g2 for FinSequence of TOP-REAL 2,
  n,m,i,j,k for Nat,
  G for Go-board,
  x for set;

theorem Th1:
  (for n st n in dom f ex i,j st [i,j] in Indices G & f/.n=G*(i,j))
& f is one-to-one unfolded s.n.c. special implies ex g st g is_sequence_on G &
g is one-to-one unfolded s.n.c. special & L~f = L~g & f/.1 = g/.1 & f/.len f =
  g/.len g & len f <= len g
proof
  defpred P[Nat] means for f st len f = $1 & (for n st n in dom f
ex i,j st [i,j] in Indices G & f/.n=G*(i,j)) & f is one-to-one unfolded s.n.c.
special ex g st g is_sequence_on G & g is one-to-one unfolded s.n.c. special &
  L~f=L~g & f/.1=g/.1 & f/.len f=g/.len g & len f<=len g;
A1: for k st P[k] holds P[k+1]
  proof
    let k such that
A2: P[k];
    let f such that
A3: len f=k+1 and
A4: for n st n in dom f ex i,j st [i,j] in Indices G & f/.n=G*(i,j) and
A5: f is one-to-one and
A6: f is unfolded and
A7: f is s.n.c. and
A8: f is special;
    per cases;
    suppose
A9:   k=0;
      take g=f;
A10:  dom f = {1} by A3,A9,FINSEQ_1:2,def 3;
      now
        let n;
        assume that
A11:    n in dom g and
A12:    n+1 in dom g;
        n=1 by A10,A11,TARSKI:def 1;
        hence for i1,i2,j1,j2 be Nat st [i1,i2] in Indices G & [j1,
j2] in Indices G & g/.n=G*(i1,i2) & g/.(n+1)=G*(j1,j2)
   holds |.i1-j1.|+|.i2-j2.|=1 by A10,A12,TARSKI:def 1;
      end;
      hence g is_sequence_on G by A4,GOBOARD1:def 9;
      thus thesis by A5,A6,A7,A8;
    end;
    suppose
A13:  k<>0;
A14:  len (f|k)=k by A3,FINSEQ_1:59,NAT_1:11;
      set f1=f|k;
A15:  f1 is unfolded by A3,A6,A13,Lm1;
A16:  f1 is s.n.c. by A7,GOBOARD2:7;
      f1 = f|Seg k by FINSEQ_1:def 16;
      then
A17:  f1 is one-to-one by A5,FUNCT_1:52;
A18:  dom G = Seg len G by FINSEQ_1:def 3;
      1<=len f by A3,NAT_1:11;
      then
A19:  k+1 in dom f by A3,FINSEQ_3:25;
      then consider j1,j2 be Nat such that
A20:  [j1,j2] in Indices G and
A21:  f/.(k+1)=G*(j1,j2) by A4;
A22:  Indices G = [:dom G,Seg width G:] by MATRIX_0:def 4;
      then
A23:  j1 in dom G by A20,ZFMISC_1:87;
A24:  0+1<=k by A13,NAT_1:13;
      then
A25:  1 in Seg k by FINSEQ_1:1;
A26:  k<=k+1 by NAT_1:11;
      then
A27:  k in dom f by A3,A24,FINSEQ_3:25;
      then consider i1,i2 be Nat such that
A28:  [i1,i2] in Indices G and
A29:  f/.k=G*(i1,i2) by A4;
      reconsider l1 = Line(G,i1), c1 = Col(G,i2) as FinSequence of TOP-REAL 2;
      set x1 = X_axis(l1), y1 = Y_axis(l1), x2 = X_axis(c1), y2 = Y_axis(c1);
A30:  dom y1=Seg len y1 & len y1=len l1 by FINSEQ_1:def 3,GOBOARD1:def 2;
A31:  dom(f|k)=Seg len(f|k) by FINSEQ_1:def 3;
A32:  f1 is special
      proof
        let n be Nat;
        assume that
A33:    1<=n and
A34:    n+1 <= len f1;
        n+1 in dom f1 by A33,A34,SEQ_4:134;
        then
A35:    f1/.(n+1)=f/.(n+1) by A27,A14,A31,FINSEQ_4:71;
        len f1 <=len f by A3,A26,FINSEQ_1:59;
        then
A36:    n+1 <= len f by A34,XXREAL_0:2;
        n in dom f1 by A33,A34,SEQ_4:134;
        then f1/.n=f/.n by A27,A14,A31,FINSEQ_4:71;
        hence thesis by A8,A33,A35,A36;
      end;
      now
        let n;
        assume
A37:    n in dom f1;
        then n in dom f by A27,A14,A31,FINSEQ_4:71;
        then consider i,j such that
A38:    [i,j] in Indices G & f/.n=G*(i,j) by A4;
        take i,j;
        thus [i,j] in Indices G & f1/.n=G*(i,j) by A27,A14,A31,A37,A38,
FINSEQ_4:71;
      end;
      then consider g1 such that
A39:  g1 is_sequence_on G and
A40:  g1 is one-to-one and
A41:  g1 is unfolded and
A42:  g1 is s.n.c. and
A43:  g1 is special and
A44:  L~g1=L~f1 and
A45:  g1/.1=f1/.1 and
A46:  g1/.len g1=f1/.len f1 and
A47:  len f1<=len g1 by A2,A14,A17,A15,A16,A32;
A48:  for n st n in dom g1 & n+1 in dom g1 holds for m,k,i,j st [m,k] in
Indices G & [i,j] in Indices G & g1/.n = G*(m,k) & g1/.(n+1) = G*(i,j) holds
      |.m-i.|+|.k-j.| = 1 by A39,GOBOARD1:def 9;
A49:  1<k implies rng g1 c= L~f1
      proof
        assume 1<k;
        then
A50:    1+1<=k by NAT_1:13;
        let x be object;
        assume x in rng g1;
        then ex n being Element of NAT st n in dom g1 & g1/.n=x by PARTFUN2:2;
        hence thesis by A14,A44,A47,A50,GOBOARD1:1,XXREAL_0:2;
      end;
A51:  k in Seg k by A24,FINSEQ_1:1;
A52:  k=1 implies L~g1={} & rng g1 = {f/.k}
      proof
A53:    g1/.len g1=f/.k by A27,A14,A51,A46,FINSEQ_4:71;
        assume
A54:    k=1;
        hence L~g1={} by A14,A44,TOPREAL1:22;
        then
A55:    len g1=1 or len g1=0 by TOPREAL1:22;
A56:    rng g1 c= {f/.k}
        proof
          let x be object;
          assume x in rng g1;
          then consider n being Element of NAT such that
A57:      n in dom g1 and
A58:      g1/.n=x by PARTFUN2:2;
          n in Seg len g1 by A57,FINSEQ_1:def 3;
          then n=len g1 by A55,FINSEQ_1:2,TARSKI:def 1;
          hence thesis by A53,A58,TARSKI:def 1;
        end;
        1<=len g1 by A3,A47,A54,FINSEQ_1:59;
        then len g1 in dom g1 by FINSEQ_3:25;
        then f/.k in rng g1 by A53,PARTFUN2:2;
        then {f/.k} c= rng g1 by ZFMISC_1:31;
        hence thesis by A56;
      end;
A59:  len c1 = len G by MATRIX_0:def 8;
      then
A60:  dom c1 = Seg len G by FINSEQ_1:def 3
        .= dom G by FINSEQ_1:def 3;
A61:  dom y2=Seg len y2 & len y2=len c1 by FINSEQ_1:def 3,GOBOARD1:def 2;
A62:  dom x1=Seg len x1 & len x1=len l1 by FINSEQ_1:def 3,GOBOARD1:def 1;
A63:  dom x2= Seg len x2 by FINSEQ_1:def 3;
A64:  len x2=len c1 by GOBOARD1:def 1;
      then
A65:  dom c1 = Seg len x2 by FINSEQ_1:def 3
        .= dom x2 by FINSEQ_1:def 3;
A66:  i1 in dom G by A28,A22,ZFMISC_1:87;
      then
A67:  x1 is constant by GOBOARD1:def 4;
A68:  i2 in Seg width G by A28,A22,ZFMISC_1:87;
      then
A69:  x2 is increasing by GOBOARD1:def 7;
A70:  y2 is constant by A68,GOBOARD1:def 5;
A71:  y1 is increasing by A66,GOBOARD1:def 6;
A72:  len l1=width G by MATRIX_0:def 7;
      then
A73:  Seg width G = dom l1 by FINSEQ_1:def 3;
A74:  j2 in Seg width G by A20,A22,ZFMISC_1:87;
A75:  for n st n in dom g1 ex m,k st [m,k] in Indices G & g1/.n=G*(m,k)
      by A39,GOBOARD1:def 9;
      now
        per cases by A8,A27,A28,A29,A19,A20,A21,GOBOARD2:11;
        suppose
A76:      i1=j1;
          set ppi = G*(i1,i2), pj = G*(i1,j2);
          now
            per cases by XXREAL_0:1;
            case
A77:          i2>j2;
              l1/.i2=l1.i2 by A68,A73,PARTFUN1:def 6;
              then
A78:          l1/.i2=ppi by A68,MATRIX_0:def 7;
              then
A79:          y1.i2=ppi`2 by A68,A30,A72,GOBOARD1:def 2;
              l1/.j2 = l1.j2 by A74,A73,PARTFUN1:def 6;
              then
A80:          l1/.j2=pj by A74,MATRIX_0:def 7;
              then
A81:          y1.j2=pj`2 by A74,A30,A72,GOBOARD1:def 2;
              then
A82:          pj`2<ppi`2 by A68,A74,A71,A30,A72,A77,A79,SEQM_3:def 1;
              reconsider l=i2-j2 as Element of NAT by A77,INT_1:5;
              defpred P1[Nat,set] means for m st m=i2-$1 holds $2=G*(i1,m);
              set lk={w where w is Point of TOP-REAL 2: w`1=ppi`1 & pj`2<=w`2
              & w`2<= ppi`2};
A83:          ppi=|[ppi`1,ppi`2]| by EUCLID:53;
A84:          now
                let n;
                assume n in Seg l;
                then
A85:            n<=l by FINSEQ_1:1;
                l<=i2 by XREAL_1:43;
                then reconsider w=i2-n as Element of NAT by A85,INT_1:5
,XXREAL_0:2;
                i2-n<=i2 & i2<=width G by A68,FINSEQ_1:1,XREAL_1:43;
                then
A86:            w<=width G by XXREAL_0:2;
A87:            1<=j2 by A74,FINSEQ_1:1;
                i2-l<=i2-n by A85,XREAL_1:13;
                then 1<=w by A87,XXREAL_0:2;
                then w in Seg width G by A86,FINSEQ_1:1;
                hence
                i2-n is Element of NAT & [i1,i2-n] in Indices G & i2-n in
                Seg width G by A22,A66,ZFMISC_1:87;
              end;
A88:          now
                let n be Nat;
                assume n in Seg l;
                then reconsider m=i2-n as Element of NAT by A84;
                take p=G*(i1,m);
                thus P1[n,p];
              end;
              consider g2 such that
A89:          len g2=l & for n being Nat st n in Seg l holds P1[n,g2
              /.n] from FINSEQ_4:sch 1(A88);
              take g=g1^g2;
A90:          dom g2 = Seg l by A89,FINSEQ_1:def 3;
              now
                let n;
                assume
A91:            n in dom g2;
                then i2-n is Element of NAT by A84,A90;
                then reconsider m=i2-n as Nat;
                take k=i1,m;
                thus [k,m] in Indices G & g2/.n=G*(k,m) by A84,A89,A90,A91;
              end;
              then
A92:          for n st n in dom g ex i,j st [i,j] in Indices G & g/.n=G*
              (i,j) by A75,GOBOARD1:23;
A93:          dom g2=Seg len g2 by FINSEQ_1:def 3;
A94:          x1.i2=ppi`1 by A68,A62,A72,A78,GOBOARD1:def 1;
A95:          now
                let n,p;
                assume that
A96:            n in dom g2 and
A97:            g2/.n=p;
                reconsider n1=i2-n as Element of NAT by A84,A90,A96;
                n<=len g2 by A96,FINSEQ_3:25;
                then
A98:            i2-len g2<= n1 by XREAL_1:13;
                set pn = G*(i1,n1);
A99:            g2/.n=pn by A89,A93,A96;
A100:           i2-n in Seg width G by A84,A89,A93,A96;
                then
A101:           x1.n1=x1.i2 by A68,A67,A62,A72,SEQM_3:def 10;
                l1/.n1=l1.n1 by A73,A100,PARTFUN1:def 6;
                then
A102:           l1/.n1=pn by A100,MATRIX_0:def 7;
                then
A103:           y1.n1=pn`2 by A30,A72,A100,GOBOARD1:def 2;
                x1.n1=pn`1 by A62,A72,A100,A102,GOBOARD1:def 1;
                hence
                p`1=ppi`1 & pj`2<=p`2 & p`2<=ppi`2 by A68,A74,A71,A30,A72,A89
,A79,A81,A94,A97,A100,A99,A98,A101,A103,SEQ_4:137,XREAL_1:43;
                dom l1 = Seg len l1 by FINSEQ_1:def 3;
                hence p in rng l1 by A72,A97,A100,A99,A102,PARTFUN2:2;
                1<=n by A96,FINSEQ_3:25;
                then n1<i2 by XREAL_1:44;
                hence p`2<ppi`2 by A68,A71,A30,A72,A79,A97,A100,A99,A103,
SEQM_3:def 1;
              end;
A104:         g2 is special
              proof
                let n be Nat;
                set p = g2/.n;
                assume
A105:           1<=n & n+1 <= len g2;
                then n in dom g2 by SEQ_4:134;
                then
A106:           p`1=ppi`1 by A95;
                n+1 in dom g2 by A105,SEQ_4:134;
                hence thesis by A95,A106;
              end;
A107:         now
                let n,m,p,q;
                assume that
A108:           n in dom g2 and
A109:           m in dom g2 and
A110:           n<m and
A111:           g2/.n=p & g2/.m=q;
A112:           i2-n in Seg width G by A84,A90,A108;
                reconsider n1=i2-n, m1=i2-m as Element of NAT by A84,A90,A108
,A109;
                set pn = G*(i1,n1), pm = G*(i1,m1);
A113:           m1<n1 by A110,XREAL_1:15;
                l1/.n1=l1.n1 by A73,A84,A90,A108,PARTFUN1:def 6;
                then l1/.n1=pn by A112,MATRIX_0:def 7;
                then
A114:           y1.n1=pn`2 by A30,A72,A112,GOBOARD1:def 2;
A115:           i2-m in Seg width G by A84,A90,A109;
                l1/.m1 = l1.m1 by A73,A84,A90,A109,PARTFUN1:def 6;
                then l1/.m1= pm by A115,MATRIX_0:def 7;
                then
A116:           y1.m1=pm`2 by A30,A72,A115,GOBOARD1:def 2;
                g2/.n=pn & g2/.m=pm by A89,A90,A108,A109;
                hence q`2<p`2 by A71,A30,A72,A111,A112,A115,A113,A114,A116,
SEQM_3:def 1;
              end;
              for n,m st m>n+1 & n in dom g2 & n+1 in dom g2 & m in dom
              g2 & m+1 in dom g2 holds LSeg(g2,n) misses LSeg(g2,m)
              proof
                let n,m;
                assume that
A117:           m>n+1 and
A118:           n in dom g2 and
A119:           n+1 in dom g2 and
A120:           m in dom g2 and
A121:           m+1 in dom g2 and
A122:           LSeg(g2,n) /\ LSeg(g2,m) <> {};
                reconsider p1=g2/.n, p2=g2/.(n+1), q1=g2/.m, q2=g2/.(m+1) as
                Point of TOP-REAL 2;
A123:           p1`1=ppi`1 & p2`1= ppi`1 by A95,A118,A119;
                n<n+1 by NAT_1:13;
                then
A124:           p2`2<p1`2 by A107,A118,A119;
                set lp = {w where w is Point of TOP-REAL 2: w`1=ppi`1 & p2`2<=
w`2 & w`2<=p1`2}, lq = {w where w is Point of TOP-REAL 2: w`1=ppi`1 & q2`2<=w`2
                & w`2<=q1`2};
A125:           p1=|[p1`1,p1`2]| & p2=|[p2`1,p2`2]| by EUCLID:53;
                m<m+1 by NAT_1:13;
                then
A126:           q2`2<q1`2 by A107,A120,A121;
A127:           q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]| by EUCLID:53;
                set x = the Element of LSeg(g2,n) /\ LSeg(g2,m);
A128:           x in LSeg(g2,n) by A122,XBOOLE_0:def 4;
A129:           q1`1= ppi`1 & q2`1=ppi`1 by A95,A120,A121;
A130:           x in LSeg(g2,m) by A122,XBOOLE_0:def 4;
                1 <= m & m+1<= len g2 by A120,A121,FINSEQ_3:25;
                then LSeg(g2,m) = LSeg(q2,q1) by TOPREAL1:def 3
                  .=lq by A126,A129,A127,TOPREAL3:9;
                then
A131:           ex tm be Point of TOP-REAL 2 st tm=x & tm`1=ppi`1 & q2`2
                <=tm`2 & tm`2<=q1`2 by A130;
                1 <= n & n+1 <= len g2 by A118,A119,FINSEQ_3:25;
                then LSeg(g2,n) = LSeg(p2,p1) by TOPREAL1:def 3
                  .=lp by A124,A123,A125,TOPREAL3:9;
                then
A132:           ex tn be Point of TOP-REAL 2 st tn=x & tn`1=ppi`1 & p2`2
                <=tn`2 & tn`2<=p1`2 by A128;
                q1`2<p2`2 by A107,A117,A119,A120;
                hence contradiction by A132,A131,XXREAL_0:2;
              end;
              then
A133:         g2 is s.n.c. by GOBOARD2:1;
A134:         not f/.k in L~g2
              proof
                set ls={LSeg(g2,m): 1<=m & m+1 <= len g2};
                assume f/.k in L~g2;
                then consider X be set such that
A135:           f/.k in X and
A136:           X in ls by TARSKI:def 4;
                consider m such that
A137:           X=LSeg(g2,m) and
A138:           1<=m & m+1 <= len g2 by A136;
                reconsider q1=g2/.m, q2=g2/.(m+1) as Point of TOP-REAL 2;
A139:           m in dom g2 by A138,SEQ_4:134;
                then
A140:           q1`1=ppi`1 by A95;
                set lq={w where w is Point of TOP-REAL 2: w`1=ppi`1 & q2`2<=w
                `2 & w`2<=q1`2};
A141:           q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]| by EUCLID:53;
A142:           m+1 in dom g2 by A138,SEQ_4:134;
                then
A143:           q2`1=ppi`1 by A95;
                m<m+1 by NAT_1:13;
                then
A144:           q2`2<q1`2 by A107,A139,A142;
                LSeg(g2,m)=LSeg(q2,q1) by A138,TOPREAL1:def 3
                  .=lq by A140,A143,A144,A141,TOPREAL3:9;
                then ex p st p=f/.k & p`1=ppi`1 & q2`2<=p`2 & p`2<=q1`2 by A135
,A137;
                hence contradiction by A29,A95,A139;
              end;
              x1.j2 =pj`1 by A74,A62,A72,A80,GOBOARD1:def 1;
              then
A145:         ppi`1=pj`1 by A68,A74,A67,A62,A72,A94,SEQM_3:def 10;
              now
                let n,m be Element of NAT;
                assume that
A146:           n in dom g2 & m in dom g2 and
A147:           n<>m;
                reconsider n1=i2-n, m1=i2-m as Element of NAT by A84,A90,A146;
A148:           g2/.n=G*(i1,n1) & g2/.m=G*(i1,m1) by A89,A90,A146;
                assume
A149:           g2/.n=g2/.m;
                [i1,i2-n] in Indices G & [i1,i2-m] in Indices G by A84,A90,A146
;
                then n1=m1 by A148,A149,GOBOARD1:5;
                hence contradiction by A147;
              end;
              then for n,m being Element of NAT
      st n in dom g2 & m in dom g2 & g2/.n = g2/.m holds
              n = m;
              then
A150:         g2 is one-to-one by PARTFUN2:9;
              reconsider m1=i2-l as Element of NAT by ORDINAL1:def 12;
A151:         pj=|[pj`1,pj`2]| by EUCLID:53;
A152:         LSeg(f,k)=LSeg(pj,ppi) by A3,A24,A29,A21,A76,TOPREAL1:def 3
                .= lk by A82,A145,A83,A151,TOPREAL3:9;
A153:         rng g2 c= LSeg(f,k)
              proof
                let x be object;
                assume x in rng g2;
                then consider n being Element of NAT such that
A154:           n in dom g2 and
A155:           g2/.n=x by PARTFUN2:2;
                reconsider n1=i2-n as Element of NAT by A84,A89,A93,A154;
                set pn = G*(i1,n1);
A156:           g2/.n=pn by A89,A93,A154;
                then
A157:           pn`2<=ppi`2 by A95,A154;
                pn`1=ppi`1 & pj`2<=pn`2 by A95,A154,A156;
                hence thesis by A152,A155,A156,A157;
              end;
A158:         now
                let n;
                assume that
A159:           n in dom g2 and
A160:           n+1 in dom g2;
                reconsider m1=i2-n,m2=i2-(n+1) as Element of NAT by A84,A90
,A159,A160;
                let l1,l2,l3,l4 be Nat;
                assume that
A161:           [l1,l2] in Indices G and
A162:           [l3,l4] in Indices G and
A163:           g2/.n=G*(l1,l2) and
A164:           g2/.(n+1)=G*(l3,l4);
                [i1,i2-(n+1)] in Indices G & g2/.(n+1)=G*(i1,m2) by A84,A89,A90
,A160;
                then
A165:           l3=i1 & l4=m2 by A162,A164,GOBOARD1:5;
                [i1,i2-n] in Indices G & g2/.n=G*(i1,m1) by A84,A89,A90,A159;
                then l1=i1 & l2=m1 by A161,A163,GOBOARD1:5;
                hence |.l1-l3.|+|.l2-l4.|= 0+|.i2-n-(i2-(n+1)).| by A165,
ABSVALUE:2
                  .= 1 by ABSVALUE:def 1;
              end;
              now
                let l1,l2,l3,l4 be Nat;
                assume that
A166:           [l1,l2] in Indices G and
A167:           [l3,l4] in Indices G and
A168:           g1/.len g1=G*(l1,l2) and
A169:           g2/.1=G*(l3,l4) and
                len g1 in dom g1 and
A170:           1 in dom g2;
                reconsider m1=i2-1 as Element of NAT by A84,A90,A170;
                [i1,i2-1] in Indices G & g2/.1=G*(i1,m1) by A84,A89,A90,A170;
                then
A171:           l3=i1 & l4=m1 by A167,A169,GOBOARD1:5;
                f1/.len f1=f/.k by A27,A14,A51,FINSEQ_4:71;
                then l1=i1 & l2=i2 by A46,A28,A29,A166,A168,GOBOARD1:5;
                hence |.l1-l3.|+|.l2-l4.|=0+|.i2-(i2-1).| by A171,ABSVALUE:2
                  .=1 by ABSVALUE:def 1;
              end;
              then for n st n in dom g & n+1 in dom g holds for m,k,i,j st [m
,k] in Indices G & [i,j] in Indices G & g/.n=G*(m,k) & g/.(n+1)=G*(i,j) holds
              |.m-i.|+|.k-j.|=1 by A48,A158,GOBOARD1:24;
              hence g is_sequence_on G by A92,GOBOARD1:def 9;
A172:         LSeg(f,k)=LSeg(ppi,pj) by A3,A24,A29,A21,A76,TOPREAL1:def 3;
A173:         not f/.k in rng g2
              proof
                assume f/.k in rng g2;
                then consider n being Element of NAT such that
A174:           n in dom g2 and
A175:           g2/.n=f/.k by PARTFUN2:2;
                reconsider n1=i2-n as Element of NAT by A84,A89,A93,A174;
                [i1,i2-n] in Indices G & g2/.n=G*(i1,n1) by A84,A89,A93,A174;
                then
A176:           n1=i2 by A28,A29,A175,GOBOARD1:5;
                0<n by A93,A174,FINSEQ_1:1;
                hence contradiction by A176;
              end;
              rng g1 /\ rng g2 = {}
              proof
                set x = the Element of rng g1 /\ rng g2;
                assume
A177:           not thesis;
                then
A178:           x in rng g2 by XBOOLE_0:def 4;
A179:           x in rng g1 by A177,XBOOLE_0:def 4;
                now
                  per cases by A24,XXREAL_0:1;
                  suppose
                    k=1;
                    hence contradiction by A52,A173,A179,A178,TARSKI:def 1;
                  end;
                  suppose
                    1<k;
                    then x in L~f1 /\ LSeg(f,k) & L~f1 /\ LSeg(f,k)={f/.k} by
A3,A6,A7,A49,A153,A179,A178,GOBOARD2:4,XBOOLE_0:def 4;
                    hence contradiction by A173,A178,TARSKI:def 1;
                  end;
                end;
                hence contradiction;
              end;
              then rng g1 misses rng g2;
              hence g is one-to-one by A40,A150,FINSEQ_3:91;
A180:         for n st 1<=n & n+2 <= len g2 holds LSeg(g2,n) /\ LSeg(g2,
              n+1) = {g2/.(n+1)}
              proof
                let n;
                assume that
A181:           1<=n and
A182:           n+2 <= len g2;
A183:           n+1 in dom g2 by A181,A182,SEQ_4:135;
                then g2/.(n+1) in rng g2 by PARTFUN2:2;
                then g2/.(n+1) in lk by A152,A153;
                then consider u1 be Point of TOP-REAL 2 such that
A184:           g2/.(n+1)=u1 and
A185:           u1`1=ppi`1 and
                pj`2<=u1`2 and
                u1`2<=ppi`2;
A186:           n+2 in dom g2 by A181,A182,SEQ_4:135;
                then g2/.(n+2) in rng g2 by PARTFUN2:2;
                then g2/.(n+2) in lk by A152,A153;
                then consider u2 be Point of TOP-REAL 2 such that
A187:           g2/.(n+2)=u2 and
A188:           u2`1=ppi`1 and
                pj`2<=u2`2 and
                u2`2<=ppi`2;
                n+(1+1) = n+1+1 & 1 <= n+1 by NAT_1:11;
                then
A189:           LSeg(g2,n+1)=LSeg(u1,u2) by A182,A184,A187,TOPREAL1:def 3;
                n+1<n+1+1 by NAT_1:13;
                then
A190:           u2`2<u1`2 by A107,A183,A186,A184,A187;
A191:           n in dom g2 by A181,A182,SEQ_4:135;
                then g2/.n in rng g2 by PARTFUN2:2;
                then g2/.n in lk by A152,A153;
                then consider u be Point of TOP-REAL 2 such that
A192:           g2/.n=u and
A193:           u`1=ppi`1 and
                pj`2<=u`2 and
                u`2<=ppi`2;
                n+1 <= n+2 by XREAL_1:6;
                then n+1 <= len g2 by A182,XXREAL_0:2;
                then
A194:           LSeg(g2,n)=LSeg(u,u1) by A181,A192,A184,TOPREAL1:def 3;
                set lg = {w where w is Point of TOP-REAL 2: w`1=ppi`1 & u2`2<=
                w`2 & w`2<=u`2};
                n<n+1 by NAT_1:13;
                then
A195:           u1`2<u`2 by A107,A191,A183,A192,A184;
                then
A196:           u1 in lg by A185,A190;
                u=|[u`1,u`2]| & u2=|[u2`1,u2`2 ]| by EUCLID:53;
                then LSeg(g2/.n,g2/.(n+2))=lg by A192,A193,A187,A188,A190,A195,
TOPREAL3:9,XXREAL_0:2;
                hence thesis by A192,A184,A187,A194,A189,A196,TOPREAL1:8;
              end;
              thus g is unfolded
              proof
                let n be Nat;
                assume that
A197:           1<=n and
A198:           n+2 <= len g;
A199:           n+1+1<=len g by A198;
A200:           n+(1+1)=n+1+1;
A201:           n<=n+1 by NAT_1:11;
                n+1<=n+1+1 by NAT_1:11;
                then
A202:           n+1 <= len g by A198,XXREAL_0:2;
A203:           len g=len g1+len g2 by FINSEQ_1:22;
                n+2-len g1 = n-len g1 +2;
                then
A204:           n-len g1 + 2 <= len g2 by A198,A203,XREAL_1:20;
A205:           1 <= n+1 & n+1+1 = n+(1+1) by NAT_1:11;
                per cases;
                suppose
A206:             n+2 <= len g1;
A207:             n+(1+1)=n+1+1;
A208:             n+1 in dom g1 by A197,A206,SEQ_4:135;
                  then
A209:             g/.(n+1)=g1/.(n+1) by FINSEQ_4:68;
                  n in dom g1 by A197,A206,SEQ_4:135;
                  then
A210:             LSeg(g1,n)=LSeg(g,n) by A208,TOPREAL3:18;
                  n+2 in dom g1 by A197,A206,SEQ_4:135;
                  then LSeg(g1,n+1)=LSeg(g,n+1) by A208,A207,TOPREAL3:18;
                  hence thesis by A41,A197,A206,A210,A209;
                end;
                suppose
                  len g1 < n+2;
                  then len g1+1<=n+2 by NAT_1:13;
                  then
A211:             len g1<=n+2-1 by XREAL_1:19;
                  now
                    per cases;
                    suppose
A212:                 len g1=n+1;
                      now
                        1<len g1 by A197,A212,NAT_1:13;
                        then
A213:                   1+1<=len g1 by NAT_1:13;
                        assume k=1;
                        hence contradiction by A52,A213,TOPREAL1:23;
                      end;
                      then 1<k by A24,XXREAL_0:1;
                      then
A214:                 L~f1 /\ LSeg(f,k)={f/.k} by A3,A6,A7,GOBOARD2:4;
                      g/.(n+1) in LSeg(g,n) & g/.(n+1) in LSeg(g,n+1) by A197
,A198,A202,A205,TOPREAL1:21;
                      then g/.(n+1) in LSeg(g,n) /\ LSeg(g,n+1) by
XBOOLE_0:def 4;
                      then
A215:                 {g/.(n+1)} c= LSeg(g,n) /\ LSeg(g,n+1) by ZFMISC_1:31;
A216:                 1<=len g-len g1 by A199,A212,XREAL_1:19;
                      then 1 in dom g2 by A203,FINSEQ_3:25;
                      then
A217:                 g2/.1 in rng g2 by PARTFUN2:2;
                      then g2/.1 in lk by A152,A153;
                      then consider u1 be Point of TOP-REAL 2 such that
A218:                 g2/.1=u1 and
                      u1`1=ppi`1 and
                      pj`2<=u1`2 and
                      u1`2<=ppi`2;
                      ppi in LSeg(ppi,pj) by RLTOPSP1:68;
                      then
A219:                 LSeg(ppi,u1) c= LSeg(f,k) by A172,A153,A217,A218,
TOPREAL1:6;
                      1<=n+1 by NAT_1:11;
                      then
A220:                 n+1 in dom g1 by A212,FINSEQ_3:25;
                      then
A221:                 g/.(n+1)=f1/.len f1 by A46,A212,FINSEQ_4:68
                        .= ppi by A27,A14,A51,A29,FINSEQ_4:71;
                      n in dom g1 by A197,A201,A212,FINSEQ_3:25;
                      then
A222:                 LSeg(g,n)=LSeg(g1,n) by A220,TOPREAL3:18;
                      g/.(n+2)=g2/.1 by A200,A203,A212,A216,SEQ_4:136;
                      then
A223:                 LSeg(g,n+1)=LSeg(ppi,u1 ) by A198,A205,A221,A218,
TOPREAL1:def 3;
                      LSeg(g1,n) c= L~f1 by A44,TOPREAL3:19;
                      then LSeg(g,n) /\ LSeg(g,n+1) c= {g /.(n+1)} by A29,A214
,A222,A221,A219,A223,XBOOLE_1:27;
                      hence thesis by A215;
                    end;
                    suppose
                      len g1<>n+1;
                      then len g1<n+1 by A211,XXREAL_0:1;
                      then
A224:                 len g1<=n by NAT_1:13;
                      then reconsider n1=n-len g1 as Element of NAT by INT_1:5;
                      now
                        per cases;
                        suppose
A225:                     len g1=n;
                          then 1 <= len g2 by A202,A203,XREAL_1:6;
                          then
A226:                     g/.(n+1)=g2/.1 by A225,SEQ_4:136;
A227:                     0+2<=len g2 by A198,A203,A225,XREAL_1:6;
                          then 1<=len g2 by XXREAL_0:2;
                          then
A228:                     1 in dom g2 by FINSEQ_3:25;
                          then g2/.1 in rng g2 by PARTFUN2:2;
                          then g2/.1 in lk by A152,A153;
                          then consider u1 be Point of TOP-REAL 2 such that
A229:                     g2/.1=u1 and
A230:                     u1`1=ppi`1 and
                          pj`2<=u1`2 and
A231:                     u1`2<=ppi`2;
A232:                     2 in dom g2 by A227,FINSEQ_3:25;
                          then g2/.2 in rng g2 by PARTFUN2:2;
                          then g2/.2 in lk by A152,A153;
                          then consider u2 be Point of TOP-REAL 2 such that
A233:                     g2 /.2=u2 and
A234:                     u2`1=ppi`1 and
                          pj`2<=u2`2 and
A235:                     u2`2<=ppi`2;
                          set lg = {w where w is Point of TOP-REAL 2 : w`1=ppi
                          `1 & u2`2<=w`2 & w`2<=ppi`2};
                          u2`2<u1`2 by A107,A228,A232,A229,A233;
                          then
A236:                     u1 in lg by A230,A231;
                          u2=|[u2`1,u2`2]| by EUCLID:53;
                          then
A237:                     LSeg
(ppi,g2/.2)=lg by A83,A233,A234,A235,TOPREAL3:9;
                          1<=len g1 by A24,A14,A47,XXREAL_0:2;
                          then len g1 in dom g1 by FINSEQ_3:25;
                          then g/.n=f1/.len f1 by A46,A225,FINSEQ_4:68
                            .= ppi by A27,A14,A51,A29,FINSEQ_4:71;
                          then
A238:                     LSeg(g,n)=LSeg(ppi,u1) by A197,A202,A226,A229,
TOPREAL1:def 3;
                          2 <= len g2 by A198,A203,A225,XREAL_1:6;
                          then g/.(n+2)=g2/.2 by A225,SEQ_4:136;
                          then
LSeg(g,n+1)=LSeg(u1,u2) by A198,A205,A226,A229,A233,TOPREAL1:def 3;
                          hence thesis by A226,A229,A233,A236,A238,A237,
TOPREAL1:8;
                        end;
                        suppose
                          len g1<>n;
                          then
A239:                     len g1<n by A224,XXREAL_0:1;
                          then len g1+1<=n by NAT_1:13;
                          then
A240:                     1<=n1 by XREAL_1:19;
                          n1 + len g1 = n;
                          then
A241:                     LSeg(g,n)=LSeg(g2,n1) by A202,A239,GOBOARD2:5;
A242:                     n+1 = n1+1+len g1;
                          n1 + 1 + len g1 = n + 1;
                          then n1+1 <= len g2 by A202,A203,XREAL_1:6;
                          then
A243:                     g/.(n+1)=g2/.(n1+1) by A242,NAT_1:11,SEQ_4:136;
                          len g1<n+1 by A201,A239,XXREAL_0:2;
                          then LSeg (g,n+1)=LSeg(g2,n1+1) by A199,A242,
GOBOARD2:5;
                          hence thesis by A180,A204,A241,A243,A240;
                        end;
                      end;
                      hence thesis;
                    end;
                  end;
                  hence thesis;
                end;
              end;
A244:         L~g2 c= LSeg(f,k)
              proof
                let x be object;
                set ls={LSeg(g2,m): 1<=m & m+1 <= len g2};
                assume x in L~g2;
                then consider X be set such that
A245:           x in X and
A246:           X in ls by TARSKI:def 4;
                consider m such that
A247:           X=LSeg(g2,m) and
A248:           1<=m & m+1 <= len g2 by A246;
                reconsider q1=g2/.m, q2=g2/.(m+1) as Point of TOP-REAL 2;
A249:           LSeg(g2,m)=LSeg(q1,q2) by A248,TOPREAL1:def 3;
                m+1 in dom g2 by A248,SEQ_4:134;
                then
A250:           g2/.(m+1) in rng g2 by PARTFUN2:2;
                m in dom g2 by A248,SEQ_4:134;
                then g2/.m in rng g2 by PARTFUN2:2;
                then LSeg(q1,q2) c= LSeg(ppi,pj) by A172,A153,A250,TOPREAL1:6;
                hence thesis by A172,A245,A247,A249;
              end;
A251:         L~g1 /\ L~g2 = {}
              proof
                per cases;
                suppose
                  k=1;
                  hence thesis by A52;
                end;
                suppose
                  k<>1;
                  then 1<k by A24,XXREAL_0:1;
                  then L~g1 /\ LSeg(f,k)={f/.k} by A3,A6,A7,A44,GOBOARD2:4;
                  then
A252:             L~g1 /\ L~g2 c= {f/.k} by A244,XBOOLE_1:26;
                  now
                    set x = the Element of L~g1 /\ L~g2;
                    assume L~g1 /\ L~g2 <> {};
                    then x in {f/.k} & x in L~g2 by A252,XBOOLE_0:def 4;
                    hence contradiction by A134,TARSKI:def 1;
                  end;
                  hence thesis;
                end;
              end;
              for n,m st m>n+1 & n in dom g & n+1 in dom g & m in dom g
              & m+1 in dom g holds LSeg(g,n) misses LSeg(g,m)
              proof
A253:           1<=len g1 by A24,A14,A47,XXREAL_0:2;
                then len g1 in dom g1 by FINSEQ_3:25;
                then
A254:           g/.len g1=g1/.len g1 by FINSEQ_4:68
                  .= ppi by A27,A14,A51,A46,A29,FINSEQ_4:71;
                reconsider qq=g2/.1 as Point of TOP-REAL 2;
                set l1 = {LSeg(g1,i): 1<=i & i+1 <= len g1}, l2 = {LSeg(g2,j):
                1<=j & j+1 <= len g2};
                let n,m;
                assume that
A255:           m>n+1 and
A256:           n in dom g and
A257:           n+1 in dom g and
A258:           m in dom g and
A259:           m+1 in dom g;
A260:           1<=n by A256,FINSEQ_3:25;
                j2+1<=i2 by A77,NAT_1:13;
                then
A261:           1<=l by XREAL_1:19;
                then
A262:           1 in dom g2 by A89,FINSEQ_3:25;
                then
A263:           qq`1=ppi`1 & qq`2<ppi`2 by A95;
A264:           g/.(len g1+1)=qq by A89,A261,SEQ_4:136;
A265:           pj`2<=qq`2 by A95,A262;
A266:           m+1<=len g by A259,FINSEQ_3:25;
A267:           1<=m+1 by A259,FINSEQ_3:25;
A268:           1<=n+1 by A257,FINSEQ_3:25;
A269:           n+1<=len g by A257,FINSEQ_3:25;
A270:           qq=|[qq`1,qq`2]| by EUCLID:53;
A271:           1<=m by A258,FINSEQ_3:25;
                set ql={z where z is Point of TOP-REAL 2: z`1=ppi`1 & qq`2<=z
                `2 & z`2<=ppi`2};
A272:           n<=n+1 by NAT_1:11;
A273:           len g=len g1+len g2 by FINSEQ_1:22;
                then len g1+1<=len g by A89,A261,XREAL_1:7;
                then
A274:           LSeg (g,len g1)=LSeg(qq,ppi) by A253,A254,A264,TOPREAL1:def 3
                  .= ql by A83,A263,A270,TOPREAL3:9;
A275:           m<=m+1 by NAT_1:11;
                then
A276:           n+1<=m+1 by A255,XXREAL_0:2;
                now
                  per cases;
                  suppose
A277:               m+1<=len g1;
                    then m<=len g1 by A275,XXREAL_0:2;
                    then
A278:               m in dom g1 by A271,FINSEQ_3:25;
                    m+1 in dom g1 by A267,A277,FINSEQ_3:25;
                    then
A279:               LSeg(g,m)=LSeg(g1,m) by A278,TOPREAL3:18;
A280:               n+1<=len g1 by A276,A277,XXREAL_0:2;
                    then n<=len g1 by A272,XXREAL_0:2;
                    then
A281:               n in dom g1 by A260,FINSEQ_3:25;
                    n+1 in dom g1 by A268,A280,FINSEQ_3:25;
                    then LSeg(g,n)=LSeg(g1,n) by A281,TOPREAL3:18;
                    hence thesis by A42,A255,A279;
                  end;
                  suppose
                    len g1<m+1;
                    then
A282:               len g1<=m by NAT_1:13;
                    then reconsider m1=m-len g1 as Element of NAT by INT_1:5;
                    now
                      per cases;
                      suppose
A283:                   m=len g1;
A284:                   LSeg(g,m) c= LSeg(f,k)
                        proof
                          let x be object;
                          assume x in LSeg(g,m);
                          then consider px be Point of TOP-REAL 2 such that
A285:                     px=x & px`1=ppi`1 and
A286:                     qq`2<=px`2 and
A287:                     px`2<= ppi`2 by A274,A283;
                          pj`2<=px`2 by A265,A286,XXREAL_0:2;
                          hence thesis by A152,A285,A287;
                        end;
                        n<=len g1 by A255,A272,A283,XXREAL_0:2;
                        then
A288:                   n in dom g1 by A260,FINSEQ_3:25;
                        now
                          1<len g1 by A255,A268,A283,XXREAL_0:2;
                          then
A289:                     1+1<=len g1 by NAT_1:13;
                          assume k=1;
                          hence contradiction by A52,A289,TOPREAL1:23;
                        end;
                        then 1<k by A24,XXREAL_0:1;
                        then
A290:                   L~f1 /\ LSeg(f,k) ={f/.k} by A3,A6,A7,GOBOARD2:4;
A291:                   n+1 in dom g1 by A255,A268,A283,FINSEQ_3:25;
                        then
A292:                   LSeg(g,n)=LSeg(g1,n) by A288,TOPREAL3:18;
                        then LSeg(g,n) in l1 by A255,A260,A283;
                        then LSeg(g,n) c= L~f1 by A44,ZFMISC_1:74;
                        then
A293:                   LSeg(g,n) /\ LSeg(g,m) c= {f/.k} by A290,A284,
XBOOLE_1:27;
                        now
                          set x = the Element of LSeg(g,n) /\ LSeg(g,m);
                          assume
A294:                     LSeg(g,n)/\ LSeg(g,m)<>{};
                          then
A295:                     x in LSeg(g,n) by XBOOLE_0:def 4;
                          x in {f/.k} by A293,A294;
                          then
A296:                     x=f/.k by TARSKI:def 1;
                          f/.k=g1/.len g1 by A27,A14,A51,A46,FINSEQ_4:71;
                          hence contradiction by A40,A41,A42,A255,A283,A288
,A291,A292,A295,A296,GOBOARD2:2;
                        end;
                        hence thesis;
                      end;
                      suppose
                        m<>len g1;
                        then
A297:                   len g1<m by A282,XXREAL_0:1;
                        then len g1+1<= m by NAT_1:13;
                        then
A298:                   1<=m1 by XREAL_1:19;
                        m+1 = m1+1 +len g1;
                        then
A299:                   m1+1 <= len g2 by A266,A273,XREAL_1:6;
                        m = m1+len g1;
                        then
A300:                   LSeg(g,m)=LSeg(g2,m1) by A266,A297,GOBOARD2:5;
                        then LSeg(g,m) in l2 by A298,A299;
                        then
A301:                   LSeg(g,m) c= L~g2 by ZFMISC_1:74;
                        now
                          per cases;
                          suppose
A302:                       n+1<=len g1;
                            then n<=len g1 by A272,XXREAL_0:2;
                            then
A303:                       n in dom g1 by A260,FINSEQ_3:25;
                            n+1 in dom g1 by A268,A302,FINSEQ_3:25;
                            then LSeg(g,n)=LSeg(g1,n) by A303,TOPREAL3:18;
                            then LSeg(g,n) in l1 by A260,A302;
                            then LSeg(g,n) c= L~g1 by ZFMISC_1:74;
                            then LSeg(g,n) /\ LSeg(g,m) = {} by A251,A301,
XBOOLE_1:3,27;
                            hence thesis;
                          end;
                          suppose
                            len g1<n+1;
                            then
A304:                       len g1<=n by NAT_1:13;
                            then reconsider n1=n-len g1 as Element of NAT by
INT_1:5;
A305:                       n - len g1 + 1 = n + 1 - len g1;
A306:                       n = n1 + len g1;
                            now
                              per cases;
                              suppose
A307:                           len g1=n;
                                now
                                  reconsider q1=g2/.m1, q2=g2/.(m1+1) as Point
                                  of TOP-REAL 2;

set x = the Element of LSeg(g,n) /\ LSeg(g,m );
                                  set q1l={v where v is Point of TOP-REAL 2: v
                                  `1=ppi`1 & q2`2<=v`2 & v`2<=q1`2};
A308:                             q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]| by
EUCLID:53;
                                  assume
A309:                             LSeg(g,n) /\ LSeg(g,m)<>{};
                                  then
A310:                             x in LSeg(g,m) by XBOOLE_0:def 4;
                                  x in LSeg(g,n) by A309,XBOOLE_0:def 4;
                                  then
A311:                             ex qx be Point of TOP-REAL 2 st qx=x &
qx`1=ppi`1 & qq`2<=qx`2 & qx`2<=ppi`2 by A274,A307;
A312:                             m1 in dom g2 by A298,A299,SEQ_4:134;
                                  then
A313:                             q1 `1=ppi`1 by A95;
A314:                             m1+1 in dom g2 by A298,A299,SEQ_4:134;
                                  then
A315:                             q2`1=ppi`1 by A95;
                                  m1<m1+1 by NAT_1:13;
                                  then
A316:                             q2`2<q1`2 by A107,A312,A314;
                                  LSeg(g2,m1)=LSeg(q2,q1) by A298,A299,
TOPREAL1:def 3
                                    .=q1l by A313,A315,A316,A308,TOPREAL3:9;
                                  then
A317:                             ex qy be Point of TOP-REAL 2 st qy=x &
qy`1=ppi`1 & q2`2<=qy`2 & qy`2<=q1`2 by A300,A310;
                                  m1 > n1 + 1 & n1 + 1 >= 1 by A255,A305,
NAT_1:11,XREAL_1:9;
                                  then m1 > 1 by XXREAL_0:2;
                                  then q1`2 < qq`2 by A107,A262,A312;
                                  hence contradiction by A311,A317,XXREAL_0:2;
                                end;
                                hence thesis;
                              end;
                              suppose
                                n<>len g1;
                                then len g1<n by A304,XXREAL_0:1;
                                then
A318:                           LSeg(g,n)=LSeg(g2,n1) by A269,A306,GOBOARD2:5;
                                m1>n1+1 by A255,A305,XREAL_1:9;
                                hence thesis by A133,A300,A318;
                              end;
                            end;
                            hence thesis;
                          end;
                        end;
                        hence thesis;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                hence thesis;
              end;
              hence g is s.n.c. by GOBOARD2:1;
              now
                set p=g1/.len g1, q=g2/.1;
                j2+1<=i2 by A77,NAT_1:13;
                then 1<=l by XREAL_1:19;
                then 1 in dom g2 by A90,FINSEQ_1:1;
                then q`1=ppi`1 by A95;
                hence p`1=q`1 or p`2=q`2 by A27,A14,A51,A46,A29,FINSEQ_4:71;
              end;
              hence g is special by A43,A104,GOBOARD2:8;
              thus L~g=L~f
              proof
                set lg = {LSeg(g,i): 1<=i & i+1 <= len g}, lf = {LSeg(f,j): 1
                <=j & j+1 <= len f};
A319:           len g = len g1 + len g2 by FINSEQ_1:22;
A320:           now
                  let j;
                  assume that
A321:             len g1<=j and
A322:             j<=len g;
                  reconsider w = j-len g1 as Element of NAT by A321,INT_1:5;
                  let p such that
A323:             p=g/.j;
                  now
                    per cases;
                    suppose
A324:                 j=len g1;
                      1<=len g1 by A24,A14,A47,XXREAL_0:2;
                      then len g1 in dom g1 by FINSEQ_3:25;
                      then
A325:                 g/.len g1 = f1/.len f1 by A46,FINSEQ_4:68
                        .= G*(i1,i2) by A27,A14,A51,A29,FINSEQ_4:71;
                      hence p`1=G*(i1,i2)`1 by A323,A324;
                      thus G*(i1,j2)`2<=p`2 & p`2<=G*(i1,i2)`2 by A68,A74,A71
,A30,A72,A77,A79,A81,A323,A324,A325,SEQM_3:def 1;
                      dom l1 = Seg len l1 by FINSEQ_1:def 3;
                      hence p in rng l1 by A68,A72,A78,A323,A324,A325,
PARTFUN2:2;
                    end;
                    suppose
                      j<>len g1;
                      then len g1 < j by A321,XXREAL_0:1;
                      then len g1 + 1<=j by NAT_1:13;
                      then
A326:                 1<=w by XREAL_1:19;
A327:                 w<=len g2 by A319,A322,XREAL_1:20;
                      then
A328:                 w in dom g2 by A326,FINSEQ_3:25;
                      w + len g1 = j;
                      then g/.j=g2/.w by A326,A327,SEQ_4:136;
                      hence p`1=ppi`1 & pj`2<=p`2 & p`2<= ppi`2 & p in rng l1
                      by A95,A323,A328;
                    end;
                  end;
                  hence p`1=ppi`1 & pj`2<=p`2 & p`2<=ppi`2 & p in rng l1;
                end;
                thus L~g c= L~f
                proof
                  let x be object;
                  assume x in L~g;
                  then consider X be set such that
A329:             x in X and
A330:             X in lg by TARSKI:def 4;
                  consider i such that
A331:             X=LSeg(g,i) and
A332:             1<=i and
A333:             i+1 <= len g by A330;
                  per cases;
                  suppose
A334:               i+1 <= len g1;
                    i<=i+1 by NAT_1:11;
                    then i<=len g1 by A334,XXREAL_0:2;
                    then
A335:               i in dom g1 by A332,FINSEQ_3:25;
                    1<=i+1 by NAT_1:11;
                    then i+1 in dom g1 by A334,FINSEQ_3:25;
                    then X=LSeg(g1,i) by A331,A335,TOPREAL3:18;
                    then X in {LSeg(g1,j): 1<=j & j+1 <= len g1} by A332,A334;
                    then
A336:               x in L~f1 by A44,A329,TARSKI:def 4;
                    L~f1 c= L~f by TOPREAL3:20;
                    hence thesis by A336;
                  end;
                  suppose
A337:               i+1 > len g1;
                    reconsider q1=g/.i, q2=g/.(i+1) as Point of TOP-REAL 2;
A338:               i<=len g by A333,NAT_1:13;
A339:               len g1<=i by A337,NAT_1:13;
                    then
A340:               q1`1=ppi`1 by A320,A338;
A341:               q1`2<=ppi`2 by A320,A339,A338;
A342:               pj`2<=q1`2 by A320,A339,A338;
                    q2`1=ppi`1 by A320,A333,A337;
                    then
A343:               q2=|[q1`1,q2`2 ]| by A340,EUCLID:53;
A344:               q2`2<=ppi`2 by A320,A333,A337;
A345:               q1=|[q1`1,q1`2]| & LSeg(g,i)=LSeg(q2,q1) by A332,A333,
EUCLID:53,TOPREAL1:def 3;
A346:               pj`2<= q2`2 by A320,A333,A337;
                    now
                      per cases by XXREAL_0:1;
                      suppose
                        q1`2>q2`2;
                        then LSeg(g,i)={p2: p2`1=q1`1 & q2`2<=p2`2 & p2`2<=q1
                        `2} by A343,A345,TOPREAL3:9;
                        then consider p2 such that
A347:                   p2 =x & p2`1=q1`1 and
A348:                   q2`2<=p2`2 & p2`2<=q1`2 by A329,A331;
                        pj`2<=p2`2 & p2`2<=ppi`2 by A341,A346,A348,XXREAL_0:2;
                        then
A349:                   x in LSeg(f,k) by A152,A340,A347;
                        LSeg(f,k) in lf by A3,A24;
                        hence thesis by A349,TARSKI:def 4;
                      end;
                      suppose
                        q1`2=q2`2;
                        then LSeg(g,i)={q1} by A343,A345,RLTOPSP1:70;
                        then x=q1 by A329,A331,TARSKI:def 1;
                        then
A350:                   x in LSeg(f,k) by A152,A340,A342,A341;
                        LSeg(f,k) in lf by A3,A24;
                        hence thesis by A350,TARSKI:def 4;
                      end;
                      suppose
                        q1`2<q2`2;
                        then LSeg(g,i)= {p1: p1`1=q1`1 & q1`2<=p1`2 & p1`2<=
                        q2`2} by A343,A345,TOPREAL3:9;
                        then consider p2 such that
A351:                   p2 =x & p2`1=q1`1 and
A352:                   q1`2<=p2`2 & p2`2<=q2`2 by A329,A331;
                        pj`2<=p2`2 & p2`2<=ppi`2 by A342,A344,A352,XXREAL_0:2;
                        then
A353:                   x in LSeg(f,k) by A152,A340,A351;
                        LSeg(f,k) in lf by A3,A24;
                        hence thesis by A353,TARSKI:def 4;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                let x be object;
                assume x in L~f;
                then
A354:           x in L~f1 \/ LSeg(f,k) by A3,A13,GOBOARD2:3;
                per cases by A354,XBOOLE_0:def 3;
                suppose
A355:             x in L~f1;
                  L~g1 c= L~g by GOBOARD2:6;
                  hence thesis by A44,A355;
                end;
                suppose
                  x in LSeg(f,k);
                  then consider p1 such that
A356:             p1=x and
A357:             p1`1=ppi`1 and
A358:             pj`2<=p1`2 and
A359:             p1`2<=ppi`2 by A152;
                  defpred P2[Nat] means len g1<=$1 & $1<=len g & for q st q=g
                  /.$1 holds q`2>=p1`2;
A360:             now
                    reconsider n=len g1 as Nat;
                    take n;
                    thus P2[n]
                    proof
                      thus len g1<=n & n<=len g by A319,XREAL_1:31;
                      1<=len g1 by A24,A14,A47,XXREAL_0:2;
                      then
A361:                 len g1 in dom g1 by FINSEQ_3:25;
                      let q;
                      assume q=g/.n;
                      then q=f1/.len f1 by A46,A361,FINSEQ_4:68
                        .=G*(i1,i2) by A27,A14,A51,A29,FINSEQ_4:71;
                      hence thesis by A359;
                    end;
                  end;
A362:             for n be Nat holds P2[n] implies n<=len g;
                  consider ma be Nat such that
A363:             P2[ma] & for n be Nat st P2[n] holds n<=ma from
                  NAT_1:sch 6 (A362,A360);
                  reconsider ma as Element of NAT by ORDINAL1:def 12;
                  now
                    per cases;
                    suppose
A364:                 ma=len g;
                      j2+1<=i2 by A77,NAT_1:13;
                      then
A365:                 1<=l by XREAL_1:19;
                      then len g1+1<=ma by A89,A319,A364,XREAL_1:7;
                      then
A366:                 len g1<=ma-1 by XREAL_1:19;
                      then 0+1<=ma by XREAL_1:19;
                      then reconsider m1=ma-1 as Element of NAT by INT_1:5;
                      reconsider q=g/.m1 as Point of TOP-REAL 2;
A367:                 ma-1<=len g by A364,XREAL_1:43;
                      then
A368:                 q`1=ppi`1 by A320,A366;
A369:                 pj`2<=q`2 by A320,A367,A366;
                      set lq={e where e is Point of TOP-REAL 2: e`1=ppi`1 & pj
                      `2<=e`2 & e`2<=q`2};
A370:                 i2-l=j2;
A371:                 l in dom g2 by A89,A365,FINSEQ_3:25;
                      then
A372:                 g/.ma=g2/.l by A89,A319,A364,FINSEQ_4:69
                        .= pj by A89,A90,A371,A370;
                      then p1`2<=pj`2 by A363;
                      then
A373:                 p1`2=pj`2 by A358,XXREAL_0:1;
                      1<=len g1 by A24,A14,A47,XXREAL_0:2;
                      then
A374:                 1<=m1 by A366,XXREAL_0:2;
A375:                 m1+1=ma;
                      then q=|[q`1,q`2]| & LSeg(g,m1)=LSeg(pj,q) by A364,A372
,A374,EUCLID:53,TOPREAL1:def 3;
                      then LSeg(g,m1)=lq by A145,A151,A368,A369,TOPREAL3:9;
                      then
A376:                 p1 in LSeg(g,m1) by A357,A373,A369;
                      LSeg(g,m1) in lg by A364,A374,A375;
                      hence thesis by A356,A376,TARSKI:def 4;
                    end;
                    suppose
                      ma<>len g;
                      then ma<len g by A363,XXREAL_0:1;
                      then
A377:                 ma+1<=len g by NAT_1:13;
                      reconsider qa=g/.ma, qa1=g/.(ma+1) as Point of TOP-REAL
                      2;
                      set lma = {p2: p2`1=ppi`1 & qa1`2<=p2`2 & p2`2<=qa`2};
A378:                 qa1=|[qa1 `1, qa1 `2]| by EUCLID:53;
A379:                 p1`2<=qa`2 by A363;
A380:                 len g1<=ma+1 by A363,NAT_1:13;
                      then
A381:                 qa1 `1 = ppi`1 by A320,A377;
A382:                 now
                        assume p1`2<=qa1`2;
                        then for q holds q=g/.(ma+1) implies p1`2<=q`2;
                        then ma+1<=ma by A363,A377,A380;
                        hence contradiction by XREAL_1:29;
                      end;
A383:                 qa`1=ppi`1 & qa=|[qa`1,qa`2]| by A320,A363,EUCLID:53;
A384:                 1<=ma by A24,A14,A47,A363,NAT_1:13;
                      then LSeg(g,ma)=LSeg(qa1,qa) by A377,TOPREAL1:def 3
                        .= lma by A379,A382,A381,A383,A378,TOPREAL3:9
,XXREAL_0:2;
                      then
A385:                 x in LSeg(g,ma) by A356,A357,A379,A382;
                      LSeg(g,ma) in lg by A377,A384;
                      hence thesis by A385,TARSKI:def 4;
                    end;
                  end;
                  hence thesis;
                end;
              end;
A386:         len g=len g1 + len g2 by FINSEQ_1:22;
              1<=len g1 by A24,A14,A47,XXREAL_0:2;
              then 1 in dom g1 by FINSEQ_3:25;
              hence g/.1=f1/.1 by A45,FINSEQ_4:68
                .=f/.1 by A27,A25,FINSEQ_4:71;
              j2+1<=i2 by A77,NAT_1:13;
              then
A387:         1<=l by XREAL_1:19;
              then
A388:         l in dom g2 by A90,FINSEQ_1:1;
              hence g/.len g=g2/.l by A89,A386,FINSEQ_4:69
                .=G*(i1,m1) by A89,A90,A388
                .=f/.len f by A3,A21,A76;
              thus len f<=len g by A3,A14,A47,A89,A387,A386,XREAL_1:7;
            end;
            case
A389:         i2=j2;
              k<>k+1;
              hence contradiction by A5,A27,A29,A19,A21,A76,A389,PARTFUN2:10;
            end;
            case
A390:         i2<j2;
              l1/.i2=l1.i2 by A68,A73,PARTFUN1:def 6;
              then
A391:         l1/.i2=ppi by A68,MATRIX_0:def 7;
              then
A392:         y1.i2=ppi`2 by A68,A30,A72,GOBOARD1:def 2;
              l1/.j2=l1.j2 by A74,A73,PARTFUN1:def 6;
              then
A393:         l1/.j2=pj by A74,MATRIX_0:def 7;
              then
A394:         y1.j2=pj`2 by A74,A30,A72,GOBOARD1:def 2;
              then
A395:         ppi`2<pj`2 by A68,A74,A71,A30,A72,A390,A392,SEQM_3:def 1;
              reconsider l=j2-i2 as Element of NAT by A390,INT_1:5;
              deffunc F(Nat) = G*(i1,i2+$1);
              consider g2 such that
A396:         len g2=l & for n being Nat st n in dom g2 holds g2/.n=
              F(n) from FINSEQ_4:sch 2;
              take g=g1^g2;
A397:         dom g2 = Seg len g2 by FINSEQ_1:def 3;
A398:         now
                let n;
A399:           n<=i2+n by NAT_1:11;
                assume
A400:           n in Seg l;
                then n<=l by FINSEQ_1:1;
                then
A401:           i2+n<=l+i2 by XREAL_1:7;
                j2<=width G by A74,FINSEQ_1:1;
                then
A402:           i2+n<=width G by A401,XXREAL_0:2;
                1<=n by A400,FINSEQ_1:1;
                then 1<=i2+n by A399,XXREAL_0:2;
                hence i2+n in Seg width G by A402,FINSEQ_1:1;
                hence [i1,i2+n] in Indices G by A22,A66,ZFMISC_1:87;
              end;
              now
                let n such that
A403:           n in dom g2;
                take m=i1,k=i2+n;
                thus [m,k] in Indices G & g2/.n=G*(m,k) by A396,A398,A397,A403;
              end;
              then
A404:         for n st n in dom g ex i,j st [i,j] in Indices G & g/.n=G*
              (i,j) by A75,GOBOARD1:23;
A405:         x1.i2=ppi`1 by A68,A62,A72,A391,GOBOARD1:def 1;
A406:         now
                let n,p;
                assume that
A407:           n in dom g2 and
A408:           g2/.n=p;
A409:           g2/.n=G*(i1,i2+n) by A396,A407;
                set n1=i2+n;
                set pn = G*(i1,n1);
A410:           i2+n in Seg width G by A396,A398,A397,A407;
                then
A411:           x1.n1=x1.i2 by A68,A67,A62,A72,SEQM_3:def 10;
                l1/.n1=l1.n1 by A73,A396,A398,A397,A407,PARTFUN1:def 6;
                then
A412:           l1/.n1=pn by A410,MATRIX_0:def 7;
                then
A413:           y1.n1=pn`2 by A30,A72,A410,GOBOARD1:def 2;
                n<=len g2 by A397,A407,FINSEQ_1:1;
                then
A414:           n1<=i2+len g2 by XREAL_1:7;
                x1.n1=pn`1 by A62,A72,A410,A412,GOBOARD1:def 1;
                hence
                p`1=ppi`1 & ppi`2<=p`2 & p`2<=pj`2 by A68,A74,A71,A30,A72,A396
,A392,A394,A405,A408,A409,A410,A414,A411,A413,SEQ_4:137,XREAL_1:31;
                dom l1 = Seg len l1 by FINSEQ_1:def 3;
                hence p in rng l1 by A72,A408,A409,A410,A412,PARTFUN2:2;
                1<=n by A397,A407,FINSEQ_1:1;
                then i2<n1 by XREAL_1:29;
                hence p`2>ppi`2 by A68,A71,A30,A72,A392,A408,A409,A410,A413,
SEQM_3:def 1;
              end;
A415:         g2 is special
              proof
                let n be Nat;
                set p = g2/.n;
                assume
A416:           1<=n & n+1 <= len g2;
                then n in dom g2 by SEQ_4:134;
                then
A417:           p`1=ppi`1 by A406;
                n+1 in dom g2 by A416,SEQ_4:134;
                hence thesis by A406,A417;
              end;
              now
                let n,m be Element of NAT;
                assume that
A418:           n in dom g2 & m in dom g2 and
A419:           n<>m;
A420:           g2/.n=G*(i1,i2+n) & g2/.m=G*(i1,i2+m) by A396,A418;
                assume
A421:           g2/.n=g2/.m;
                [ i1,i2+n] in Indices G & [i1,i2+m] in Indices G by A396,A398
,A397,A418;
                then i2+n=i2+m by A420,A421,GOBOARD1:5;
                hence contradiction by A419;
              end;
              then for n,m being Element of NAT
st n in dom g2 & m in dom g2 & g2/.n = g2/.m holds
              n = m;
              then
A422:         g2 is one-to-one by PARTFUN2:9;
              set lk={w where w is Point of TOP-REAL 2: w`1=ppi`1 & ppi`2<=w`2
              & w`2<= pj`2};
A423:         ppi=|[ppi`1,ppi`2]| by EUCLID:53;
A424:         now
                let n,m,p,q;
                assume that
A425:           n in dom g2 and
A426:           m in dom g2 and
A427:           n<m and
A428:           g2/.n=p & g2/.m=q;
A429:           i2+n in Seg width G by A396,A398,A397,A425;
                set n1=i2+n, m1=i2+m;
                set pn = G*(i1,n1), pm = G*(i1,m1);
A430:           n1<m1 by A427,XREAL_1:8;
                l1/.n1=l1.n1 by A73,A396,A398,A397,A425,PARTFUN1:def 6;
                then l1/.n1=pn by A429,MATRIX_0:def 7;
                then
A431:           y1.n1=pn`2 by A30,A72,A429,GOBOARD1:def 2;
A432:           i2+m in Seg width G by A396,A398,A397,A426;
                l1/.m1 = l1.m1 by A73,A396,A398,A397,A426,PARTFUN1:def 6;
                then l1/.m1=pm by A432,MATRIX_0:def 7;
                then
A433:           y1.m1=pm`2 by A30,A72,A432,GOBOARD1:def 2;
                g2/.n=G*(i1,i2+n) & g2/.m=G*(i1,i2+m) by A396,A425,A426;
                hence p`2<q`2 by A71,A30,A72,A428,A429,A432,A430,A431,A433,
SEQM_3:def 1;
              end;
              for n,m st m>n+1 & n in dom g2 & n+1 in dom g2 & m in dom
              g2 & m+1 in dom g2 holds LSeg(g2,n) misses LSeg(g2,m)
              proof
                let n,m;
                assume that
A434:           m>n+1 and
A435:           n in dom g2 and
A436:           n+1 in dom g2 and
A437:           m in dom g2 and
A438:           m+1 in dom g2 and
A439:           LSeg(g2,n) /\ LSeg(g2,m) <> {};
                reconsider p1=g2/.n, p2=g2/.(n+1), q1=g2/.m, q2=g2/.(m+1) as
                Point of TOP-REAL 2;
A440:           p1`1=ppi`1 & p2`1= ppi`1 by A406,A435,A436;
                n<n+1 by NAT_1:13;
                then
A441:           p1`2<p2`2 by A424,A435,A436;
                set lp = {w where w is Point of TOP-REAL 2: w`1=ppi`1 & p1`2<=
w`2 & w`2<=p2`2}, lq = {w where w is Point of TOP-REAL 2: w`1=ppi`1 & q1`2<=w`2
                & w`2<=q2`2};
A442:           p1=|[p1`1,p1`2]| & p2=|[p2`1,p2`2]| by EUCLID:53;
                m<m+1 by NAT_1:13;
                then
A443:           q1`2<q2`2 by A424,A437,A438;
A444:           q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]| by EUCLID:53;
                set x = the Element of LSeg(g2,n) /\ LSeg(g2,m);
A445:           x in LSeg(g2,n) by A439,XBOOLE_0:def 4;
A446:           q1`1= ppi`1 & q2`1=ppi`1 by A406,A437,A438;
A447:           x in LSeg(g2,m) by A439,XBOOLE_0:def 4;
                1 <= m & m+1<= len g2 by A437,A438,FINSEQ_3:25;
                then LSeg(g2,m) = LSeg(q1,q2) by TOPREAL1:def 3
                  .=lq by A443,A446,A444,TOPREAL3:9;
                then
A448:           ex tm be Point of TOP-REAL 2 st tm=x & tm`1=ppi`1 & q1`2
                <=tm`2 & tm`2<=q2`2 by A447;
                1 <= n & n+1 <= len g2 by A435,A436,FINSEQ_3:25;
                then LSeg(g2,n) = LSeg(p1,p2) by TOPREAL1:def 3
                  .=lp by A441,A440,A442,TOPREAL3:9;
                then
A449:           ex tn be Point of TOP-REAL 2 st tn=x & tn`1=ppi`1 & p1`2
                <=tn`2 & tn`2<=p2`2 by A445;
                p2`2<q1`2 by A424,A434,A436,A437;
                hence contradiction by A449,A448,XXREAL_0:2;
              end;
              then
A450:         g2 is s.n.c. by GOBOARD2:1;
A451:         not f/.k in L~g2
              proof
                set ls={LSeg(g2,m): 1<=m & m+1 <= len g2};
                assume f/.k in L~g2;
                then consider X be set such that
A452:           f/.k in X and
A453:           X in ls by TARSKI:def 4;
                consider m such that
A454:           X=LSeg(g2,m) and
A455:           1<=m & m+1 <= len g2 by A453;
                reconsider q1=g2/.m, q2=g2/.(m+1) as Point of TOP-REAL 2;
A456:           m in dom g2 by A455,SEQ_4:134;
                then
A457:           q1`1=ppi`1 by A406;
                set lq={w where w is Point of TOP-REAL 2: w`1=ppi`1 & q1`2<=w
                `2 & w`2<=q2`2};
A458:           q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]| by EUCLID:53;
A459:           m+1 in dom g2 by A455,SEQ_4:134;
                then
A460:           q2`1=ppi`1 by A406;
                m<m+1 by NAT_1:13;
                then
A461:           q1`2<q2`2 by A424,A456,A459;
                LSeg(g2,m)=LSeg(q1,q2) by A455,TOPREAL1:def 3
                  .=lq by A457,A460,A461,A458,TOPREAL3:9;
                then ex p st p=f/.k & p`1=ppi`1 & q1`2<=p`2 & p`2<=q2`2 by A452
,A454;
                hence contradiction by A29,A406,A456;
              end;
              x1.j2 =pj`1 by A74,A62,A72,A393,GOBOARD1:def 1;
              then
A462:         ppi`1=pj`1 by A68,A74,A67,A62,A72,A405,SEQM_3:def 10;
A463:         now
                let n;
                assume that
A464:           n in dom g2 and
A465:           n+1 in dom g2;
                let l1,l2,l3,l4 be Nat;
                assume that
A466:           [l1,l2] in Indices G and
A467:           [l3,l4] in Indices G and
A468:           g2/.n=G*(l1,l2) and
A469:           g2/.(n+1)=G*(l3,l4);
                g2/.(n+1)=G*(i1,i2+(n+1)) & [i1,i2+(n+1)] in Indices G
                by A396,A398,A397,A465;
                then
A470:           l3=i1 & l4=i2+(n+1) by A467,A469,GOBOARD1:5;
                g2/.n=G*(i1,i2+n) & [i1,i2+n] in Indices G by A396,A398,A397
,A464;
                then l1=i1 & l2=i2+n by A466,A468,GOBOARD1:5;
                hence |.l1-l3.|+|.l2-l4.|= 0+|.i2+n-(i2+(n+1)).| by A470,
ABSVALUE:2
                  .= |.-1.|
                  .= |.1.| by COMPLEX1:52
                  .= 1 by ABSVALUE:def 1;
              end;
              now
                let l1,l2,l3,l4 be Nat;
                assume that
A471:           [ l1,l2] in Indices G and
A472:           [l3,l4] in Indices G and
A473:           g1/.len g1=G*(l1,l2) and
A474:           g2/.1=G*(l3,l4) and
                len g1 in dom g1 and
A475:           1 in dom g2;
                g2/.1=G* (i1,i2+1) & [i1,i2+1] in Indices G by A396,A398,A397
,A475;
                then
A476:           l3=i1 & l4=i2+1 by A472,A474,GOBOARD1:5;
                f1/.len f1=f/.k by A27,A14,A51,FINSEQ_4:71;
                then l1=i1 & l2=i2 by A46,A28,A29,A471,A473,GOBOARD1:5;
                hence |.l1-l3.|+|.l2-l4.|=0+|.i2-(i2+1).| by A476,ABSVALUE:2
                  .=|.i2-i2+-1.|
                  .=|.1.| by COMPLEX1:52
                  .=1 by ABSVALUE:def 1;
              end;
              then for n st n in dom g & n+1 in dom g holds for m,k,i,j st [m
,k] in Indices G & [i,j] in Indices G & g/.n=G*(m,k) & g/.(n+1)=G*(i,j) holds
              |.m-i.|+|.k-j.|=1 by A48,A463,GOBOARD1:24;
              hence g is_sequence_on G by A404,GOBOARD1:def 9;
A477:         pj=|[pj`1,pj `2]| by EUCLID:53;
A478:         LSeg(f,k)=LSeg(ppi,pj) by A3,A24,A29,A21,A76,TOPREAL1:def 3
                .= lk by A395,A462,A423,A477,TOPREAL3:9;
A479:         rng g2 c= LSeg(f,k)
              proof
                let x be object;
                assume x in rng g2;
                then consider n being Element of NAT such that
A480:           n in dom g2 and
A481:           g2/.n=x by PARTFUN2:2;
                set pn = G*(i1,(i2+n));
A482:           g2/.n=G*(i1,i2+n) by A396,A480;
                then
A483:           pn`2<=pj`2 by A406,A480;
                pn`1=ppi`1 & ppi`2<=pn`2 by A406,A480,A482;
                hence thesis by A478,A481,A482,A483;
              end;
A484:         not f/.k in rng g2
              proof
                assume f/.k in rng g2;
                then consider n being Element of NAT such that
A485:           n in dom g2 and
A486:           g2/.n=f/.k by PARTFUN2:2;
A487:           0<n by A485,FINSEQ_3:25;
A488:           g2/.n=G*(i1,i2+n) by A396,A485;
                dom g2 = Seg len g2 by FINSEQ_1:def 3;
                then [i1,i2+n] in Indices G by A396,A398,A485;
                then i2+n=i2 by A28,A29,A486,A488,GOBOARD1:5;
                hence contradiction by A487;
              end;
              rng g1 /\ rng g2 = {}
              proof
                set x = the Element of rng g1 /\ rng g2;
                assume
A489:           not thesis;
                then
A490:           x in rng g2 by XBOOLE_0:def 4;
A491:           x in rng g1 by A489,XBOOLE_0:def 4;
                now
                  per cases by A24,XXREAL_0:1;
                  suppose
                    k=1;
                    hence contradiction by A52,A484,A491,A490,TARSKI:def 1;
                  end;
                  suppose
                    1<k;
                    then x in L~f1 /\ LSeg(f,k) & L~f1 /\ LSeg(f,k)={f/.k} by
A3,A6,A7,A49,A479,A491,A490,GOBOARD2:4,XBOOLE_0:def 4;
                    hence contradiction by A484,A490,TARSKI:def 1;
                  end;
                end;
                hence contradiction;
              end;
              then rng g1 misses rng g2;
              hence g is one-to-one by A40,A422,FINSEQ_3:91;
A492:         LSeg(f,k)=LSeg(ppi,pj) by A3,A24,A29,A21,A76,TOPREAL1:def 3;
A493:         for n st 1<=n & n+2 <= len g2 holds LSeg(g2,n) /\ LSeg(g2,
              n+1) = {g2/.(n+1)}
              proof
                let n;
                assume that
A494:           1<=n and
A495:           n+2 <= len g2;
A496:           n+1 in dom g2 by A494,A495,SEQ_4:135;
                then g2/.(n+1) in rng g2 by PARTFUN2:2;
                then g2/.(n+1) in lk by A478,A479;
                then consider u1 be Point of TOP-REAL 2 such that
A497:           g2/.(n+1)=u1 and
A498:           u1`1=ppi`1 and
                ppi`2<=u1`2 and
                u1`2<=pj`2;
A499:           n+2 in dom g2 by A494,A495,SEQ_4:135;
                then g2/.(n+2) in rng g2 by PARTFUN2:2;
                then g2/.(n+2) in lk by A478,A479;
                then consider u2 be Point of TOP-REAL 2 such that
A500:           g2/.(n+2)=u2 and
A501:           u2`1=ppi`1 and
                ppi`2<=u2`2 and
                u2`2<=pj`2;
                1<= n+1 & n+1+1 = n+(1+1) by NAT_1:11;
                then
A502:           LSeg(g2,n+1)=LSeg( u1,u2) by A495,A497,A500,TOPREAL1:def 3;
                n+1<n+1+1 by NAT_1:13;
                then
A503:           u1`2<u2`2 by A424,A496,A499,A497,A500;
A504:           n in dom g2 by A494,A495,SEQ_4:135;
                then g2/.n in rng g2 by PARTFUN2:2;
                then g2/.n in lk by A478,A479;
                then consider u be Point of TOP-REAL 2 such that
A505:           g2/.n=u and
A506:           u`1=ppi`1 and
                ppi`2<=u`2 and
                u`2<=pj`2;
                n+1 <= n+2 by XREAL_1:6;
                then n+1 <= len g2 by A495,XXREAL_0:2;
                then
A507:           LSeg(g2,n)=LSeg(u,u1) by A494,A505,A497,TOPREAL1:def 3;
                set lg = {w where w is Point of TOP-REAL 2: w`1=ppi`1 & u`2<=w
                `2 & w`2<=u2`2};
                n<n+1 by NAT_1:13;
                then
A508:           u`2<u1`2 by A424,A504,A496,A505,A497;
                then
A509:           u1 in lg by A498,A503;
                u=|[u`1,u`2]| & u2=|[u2`1,u2`2 ]| by EUCLID:53;
                then LSeg(g2/.n,g2/.(n+2))=lg by A505,A506,A500,A501,A503,A508,
TOPREAL3:9,XXREAL_0:2;
                hence thesis by A505,A497,A500,A507,A502,A509,TOPREAL1:8;
              end;
              thus g is unfolded
              proof
                let n be Nat;
                assume that
A510:           1<=n and
A511:           n+2 <= len g;
A512:           n+1+1<=len g by A511;
                n+1<=n+1+1 by NAT_1:11;
                then
A513:           n+1 <= len g by A511,XXREAL_0:2;
A514:           len g=len g1+len g2 by FINSEQ_1:22;
                n+2-len g1 = n-len g1 +2;
                then
A515:           n-len g1 + 2 <= len g2 by A511,A514,XREAL_1:20;
A516:           1<= n+1 by NAT_1:11;
A517:           n<=n+1 by NAT_1:11;
A518:           n+(1+1)=n+1+1;
                per cases;
                suppose
A519:             n+2 <= len g1;
A520:             n+(1+1)=n+1+1;
A521:             n+1 in dom g1 by A510,A519,SEQ_4:135;
                  then
A522:             g/.(n+1)=g1/.(n+1) by FINSEQ_4:68;
                  n in dom g1 by A510,A519,SEQ_4:135;
                  then
A523:             LSeg(g1,n)=LSeg(g,n) by A521,TOPREAL3:18;
                  n+2 in dom g1 by A510,A519,SEQ_4:135;
                  then LSeg(g1,n+1)=LSeg(g,n+1) by A521,A520,TOPREAL3:18;
                  hence thesis by A41,A510,A519,A523,A522;
                end;
                suppose
                  len g1 < n+2;
                  then len g1+1<=n+2 by NAT_1:13;
                  then
A524:             len g1<=n+2-1 by XREAL_1:19;
                  now
                    per cases;
                    suppose
A525:                 len g1=n+1;
                      now
                        1<len g1 by A510,A525,NAT_1:13;
                        then
A526:                   1+1<=len g1 by NAT_1:13;
                        assume k=1;
                        hence contradiction by A52,A526,TOPREAL1:23;
                      end;
                      then 1<k by A24,XXREAL_0:1;
                      then
A527:                 L~f1 /\ LSeg(f,k)={f/.k} by A3,A6,A7,GOBOARD2:4;
                      g/.(n+1) in LSeg(g,n) & g/.(n+1) in LSeg(g,n+1) by A510
,A511,A516,A513,A518,TOPREAL1:21;
                      then g/.(n+1) in LSeg(g,n) /\ LSeg(g,n+1) by
XBOOLE_0:def 4;
                      then
A528:                 {g/.(n+1)} c= LSeg(g,n) /\ LSeg(g,n+1) by ZFMISC_1:31;
A529:                 1<=len g-len g1 by A512,A525,XREAL_1:19;
                      then 1 in dom g2 by A514,FINSEQ_3:25;
                      then
A530:                 g2/.1 in rng g2 by PARTFUN2:2;
                      then g2/.1 in lk by A478,A479;
                      then consider u1 be Point of TOP-REAL 2 such that
A531:                 g2/.1=u1 and
                      u1`1=ppi`1 and
                      ppi`2<=u1`2 and
                      u1`2<=pj`2;
                      ppi in LSeg(ppi,pj) by RLTOPSP1:68;
                      then
A532:                 LSeg(ppi,u1) c= LSeg(f,k) by A492,A479,A530,A531,
TOPREAL1:6;
                      1<=n+1 by NAT_1:11;
                      then
A533:                 n+1 in dom g1 by A525,FINSEQ_3:25;
                      then
A534:                 g/.(n+1)=f1/.len f1 by A46,A525,FINSEQ_4:68
                        .= ppi by A27,A14,A51,A29,FINSEQ_4:71;
                      n in dom g1 by A510,A517,A525,FINSEQ_3:25;
                      then
A535:                 LSeg(g,n)=LSeg(g1,n) by A533,TOPREAL3:18;
                      g/.(n+2)=g2/.1 by A518,A514,A525,A529,SEQ_4:136;
                      then
A536:                 LSeg(g,n+1)=LSeg(ppi,u1 ) by A511,A516,A518,A534,A531,
TOPREAL1:def 3;
                      LSeg(g1,n) c= L~f1 by A44,TOPREAL3:19;
                      then LSeg(g,n) /\ LSeg(g,n+1) c= {g /.(n+1)} by A29,A527
,A535,A534,A532,A536,XBOOLE_1:27;
                      hence thesis by A528;
                    end;
                    suppose
                      len g1<>n+1;
                      then len g1<n+1 by A524,XXREAL_0:1;
                      then
A537:                 len g1<=n by NAT_1:13;
                      then reconsider n1=n-len g1 as Element of NAT by INT_1:5;
                      now
                        per cases;
                        suppose
A538:                     len g1=n;
                          then
A539:                     2 <= len g2 by A511,A514,XREAL_1:6;
                          then 1 <= len g2 by XXREAL_0:2;
                          then
A540:                     g/.(n+1)=g2/.1 by A538,SEQ_4:136;
                          1<=len g2 by A539,XXREAL_0:2;
                          then
A541:                     1 in dom g2 by FINSEQ_3:25;
                          then g2/.1 in rng g2 by PARTFUN2:2;
                          then g2/.1 in lk by A478,A479;
                          then consider u1 be Point of TOP-REAL 2 such that
A542:                     g2/.1=u1 and
A543:                     u1`1=ppi`1 & ppi`2<=u1`2 and
                          u1`2<=pj`2;
                          1<=len g1 by A24,A14,A47,XXREAL_0:2;
                          then len g1 in dom g1 by FINSEQ_3:25;
                          then g/.n=f1/.len f1 by A46,A538,FINSEQ_4:68
                            .= ppi by A27,A14,A51,A29,FINSEQ_4:71;
                          then
A544:                     LSeg(g,n)=LSeg(ppi,u1) by A510,A513,A540,A542,
TOPREAL1:def 3;
A545:                     2 in dom g2 by A539,FINSEQ_3:25;
                          then g2/.2 in rng g2 by PARTFUN2:2;
                          then g2/.2 in lk by A478,A479;
                          then consider u2 be Point of TOP-REAL 2 such that
A546:                     g2 /.2=u2 and
A547:                     u2`1=ppi`1 & ppi`2<=u2`2 and
                          u2`2<=pj`2;
                          set lg = {w where w is Point of TOP-REAL 2 : w`1=ppi
                          `1 & ppi`2<=w`2 & w`2<=u2`2};
                          u1`2<u2`2 by A424,A541,A545,A542,A546;
                          then u2=|[u2`1,u2`2]| & u1 in lg by A543,EUCLID:53;
                          then
A548:                     u1 in LSeg(ppi,u2) by A423,A547,TOPREAL3:9;
                          g/.(n+2)=g2/.2 by A538,A539,SEQ_4:136;
                          then LSeg(g,n+1)=LSeg(u1,u2) by A511,A516,A518,A540
,A542,A546,TOPREAL1:def 3;
                          hence thesis by A540,A542,A544,A548,TOPREAL1:8;
                        end;
                        suppose
                          len g1<>n;
                          then
A549:                     len g1<n by A537,XXREAL_0:1;
                          then len g1+1<=n by NAT_1:13;
                          then
A550:                     1<=n1 by XREAL_1:19;
                          n1 + len g1 = n;
                          then
A551:                     LSeg(g,n)=LSeg(g2,n1) by A513,A549,GOBOARD2:5;
A552:                     n1+1+len g1 = n+1;
                          then n1+1<=len g2 by A513,A514,XREAL_1:6;
                          then
A553:                     g/.(n+1)=g2/.(n1+1) by A552,NAT_1:11,SEQ_4:136;
                          len g1<n+1 by A517,A549,XXREAL_0:2;
                          then LSeg(g,n+1)=LSeg(g2,n1+1) by A512,A552,
GOBOARD2:5;
                          hence thesis by A493,A515,A551,A553,A550;
                        end;
                      end;
                      hence thesis;
                    end;
                  end;
                  hence thesis;
                end;
              end;
A554:         L~g2 c= LSeg(f,k)
              proof
                let x be object;
                set ls={LSeg(g2,m): 1<=m & m+1 <= len g2};
                assume x in L~g2;
                then consider X be set such that
A555:           x in X and
A556:           X in ls by TARSKI:def 4;
                consider m such that
A557:           X=LSeg(g2,m) and
A558:           1<=m & m+1 <= len g2 by A556;
                reconsider q1=g2/.m, q2=g2/.(m+1) as Point of TOP-REAL 2;
A559:           LSeg(g2,m)=LSeg(q1,q2) by A558,TOPREAL1:def 3;
                m+1 in dom g2 by A558,SEQ_4:134;
                then
A560:           g2/.(m+1) in rng g2 by PARTFUN2:2;
                m in dom g2 by A558,SEQ_4:134;
                then g2/.m in rng g2 by PARTFUN2:2;
                then LSeg(q1,q2) c= LSeg(ppi,pj) by A492,A479,A560,TOPREAL1:6;
                hence thesis by A492,A555,A557,A559;
              end;
A561:         L~g1 /\ L~g2 = {}
              proof
                per cases;
                suppose
                  k=1;
                  hence thesis by A52;
                end;
                suppose
                  k<>1;
                  then 1<k by A24,XXREAL_0:1;
                  then L~g1 /\ LSeg(f,k)={f/.k} by A3,A6,A7,A44,GOBOARD2:4;
                  then
A562:             L~g1 /\ L~g2 c= {f/.k} by A554,XBOOLE_1:26;
                  now
                    set x = the Element of L~g1 /\ L~g2;
                    assume L~g1 /\ L~g2 <> {};
                    then x in {f/.k} & x in L~g2 by A562,XBOOLE_0:def 4;
                    hence contradiction by A451,TARSKI:def 1;
                  end;
                  hence thesis;
                end;
              end;
              for n,m st m>n+1 & n in dom g & n+1 in dom g & m in dom g
              & m+1 in dom g holds LSeg(g,n) misses LSeg(g,m)
              proof
A563:           1<=len g1 by A24,A14,A47,XXREAL_0:2;
                then len g1 in dom g1 by FINSEQ_3:25;
                then
A564:           g/.len g1=g1/.len g1 by FINSEQ_4:68
                  .= ppi by A27,A14,A51,A46,A29,FINSEQ_4:71;
                reconsider qq=g2/.1 as Point of TOP-REAL 2;
                set l1 = {LSeg(g1,i): 1<=i & i+1 <= len g1}, l2 = {LSeg(g2,j):
                1<=j & j+1 <= len g2};
                let n,m;
                assume that
A565:           m>n+1 and
A566:           n in dom g and
A567:           n+1 in dom g and
A568:           m in dom g and
A569:           m+1 in dom g;
A570:           1<=n by A566,FINSEQ_3:25;
                i2+1<=j2 by A390,NAT_1:13;
                then
A571:           1<=l by XREAL_1:19;
                then
A572:           1 in dom g2 by A396,FINSEQ_3:25;
                then
A573:           qq`1=ppi`1 & qq`2>ppi`2 by A406;
A574:           g/.(len g1+1)=qq by A396,A571,SEQ_4:136;
A575:           qq`2<=pj`2 by A406,A572;
A576:           m+1<=len g by A569,FINSEQ_3:25;
A577:           1<=m+1 by A569,FINSEQ_3:25;
A578:           1<=n+1 by A567,FINSEQ_3:25;
A579:           n+1<=len g by A567,FINSEQ_3:25;
A580:           qq=|[qq`1,qq`2]| by EUCLID:53;
A581:           1<=m by A568,FINSEQ_3:25;
                set ql={z where z is Point of TOP-REAL 2: z`1=ppi`1 & ppi`2<=z
                `2 & z`2<=qq`2};
A582:           n<=n+1 by NAT_1:11;
A583:           len g=len g1+len g2 by FINSEQ_1:22;
                then len g1+1<=len g by A396,A571,XREAL_1:7;
                then
A584:           LSeg (g,len g1)=LSeg(ppi,qq) by A563,A564,A574,TOPREAL1:def 3
                  .= ql by A423,A573,A580,TOPREAL3:9;
A585:           m<=m+1 by NAT_1:11;
                then
A586:           n+1<=m+1 by A565,XXREAL_0:2;
                now
                  per cases;
                  suppose
A587:               m+1<=len g1;
                    then m<=len g1 by A585,XXREAL_0:2;
                    then
A588:               m in dom g1 by A581,FINSEQ_3:25;
                    m+1 in dom g1 by A577,A587,FINSEQ_3:25;
                    then
A589:               LSeg(g,m)=LSeg(g1,m) by A588,TOPREAL3:18;
A590:               n+1<=len g1 by A586,A587,XXREAL_0:2;
                    then n<=len g1 by A582,XXREAL_0:2;
                    then
A591:               n in dom g1 by A570,FINSEQ_3:25;
                    n+1 in dom g1 by A578,A590,FINSEQ_3:25;
                    then LSeg(g,n)=LSeg(g1,n) by A591,TOPREAL3:18;
                    hence thesis by A42,A565,A589;
                  end;
                  suppose
                    len g1<m+1;
                    then
A592:               len g1<=m by NAT_1:13;
                    then reconsider m1=m-len g1 as Element of NAT by INT_1:5;
                    now
                      per cases;
                      suppose
A593:                   m=len g1;
A594:                   LSeg(g,m) c= LSeg(f,k)
                        proof
                          let x be object;
                          assume x in LSeg(g,m);
                          then consider px be Point of TOP-REAL 2 such that
A595:                     px =x & px`1=ppi`1 & ppi`2<=px`2 and
A596:                     px`2<= qq`2 by A584,A593;
                          pj`2>=px`2 by A575,A596,XXREAL_0:2;
                          hence thesis by A478,A595;
                        end;
                        n<=len g1 by A565,A582,A593,XXREAL_0:2;
                        then
A597:                   n in dom g1 by A570,FINSEQ_3:25;
                        now
                          1<len g1 by A565,A578,A593,XXREAL_0:2;
                          then
A598:                     1+1<=len g1 by NAT_1:13;
                          assume k=1;
                          hence contradiction by A52,A598,TOPREAL1:23;
                        end;
                        then 1<k by A24,XXREAL_0:1;
                        then
A599:                   L~f1 /\ LSeg(f,k) ={f/.k} by A3,A6,A7,GOBOARD2:4;
A600:                   n+1 in dom g1 by A565,A578,A593,FINSEQ_3:25;
                        then
A601:                   LSeg(g,n)=LSeg(g1,n) by A597,TOPREAL3:18;
                        then LSeg(g,n) in l1 by A565,A570,A593;
                        then LSeg(g,n) c= L~f1 by A44,ZFMISC_1:74;
                        then
A602:                   LSeg(g,n) /\ LSeg(g,m) c= {f/.k} by A599,A594,
XBOOLE_1:27;
                        now
                          set x = the Element of LSeg(g,n)/\ LSeg(g,m);
                          assume
A603:                     LSeg(g,n)/\ LSeg(g,m)<>{};
                          then
A604:                     x in LSeg(g,n) by XBOOLE_0:def 4;
                          x in {f/.k} by A602,A603;
                          then
A605:                     x=f/.k by TARSKI:def 1;
                          f/.k=g1/.len g1 by A27,A14,A51,A46,FINSEQ_4:71;
                          hence contradiction by A40,A41,A42,A565,A593,A597
,A600,A601,A604,A605,GOBOARD2:2;
                        end;
                        hence thesis;
                      end;
                      suppose
                        m<>len g1;
                        then
A606:                   len g1<m by A592,XXREAL_0:1;
                        then len g1+1<= m by NAT_1:13;
                        then
A607:                   1<=m1 by XREAL_1:19;
                        m+1=m1+1+len g1;
                        then
A608:                   m1+1 <= len g2 by A576,A583,XREAL_1:6;
                        m = m1+len g1;
                        then
A609:                   LSeg(g,m)=LSeg(g2,m1) by A576,A606,GOBOARD2:5;
                        then LSeg(g,m) in l2 by A607,A608;
                        then
A610:                   LSeg(g,m) c= L~g2 by ZFMISC_1:74;
                        now
                          per cases;
                          suppose
A611:                       n+1<=len g1;
                            then n<=len g1 by A582,XXREAL_0:2;
                            then
A612:                       n in dom g1 by A570,FINSEQ_3:25;
                            n+1 in dom g1 by A578,A611,FINSEQ_3:25;
                            then LSeg(g,n)=LSeg(g1,n) by A612,TOPREAL3:18;
                            then LSeg(g,n) in l1 by A570,A611;
                            then LSeg(g,n) c= L~g1 by ZFMISC_1:74;
                            then LSeg(g,n) /\ LSeg(g,m) = {} by A561,A610,
XBOOLE_1:3,27;
                            hence thesis;
                          end;
                          suppose
                            len g1<n+1;
                            then
A613:                       len g1<=n by NAT_1:13;
                            then reconsider n1=n-len g1 as Element of NAT by
INT_1:5;
A614:                       n - len g1 + 1 = n + 1 - len g1;
A615:                       n = n1 + len g1;
                            now
                              per cases;
                              suppose
A616:                           len g1=n;
                                now
                                  reconsider q1=g2/.m1, q2=g2/.(m1+1) as Point
                                  of TOP-REAL 2;

set x = the Element of LSeg(g,n) /\ LSeg(g,m);
                                  set q1l={v where v is Point of TOP-REAL 2: v
                                  `1=ppi`1 & q1`2<=v`2 & v`2<=q2`2};
A617:                             q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]|
                                  by EUCLID:53;
                                  assume
A618:                             LSeg(g,n) /\ LSeg(g,m)<>{};
                                  then
A619:                             x in LSeg(g,m) by XBOOLE_0:def 4;
                                  x in LSeg(g,n) by A618,XBOOLE_0:def 4;
                                  then
A620:                             ex qx be Point of TOP-REAL 2 st qx=x
& qx`1=ppi`1 & ppi`2<=qx`2 & qx`2<=qq`2 by A584,A616;
A621:                             m1 in dom g2 by A607,A608,SEQ_4:134;
                                  then
A622:                             q1 `1=ppi`1 by A406;
A623:                             m1+1 in dom g2 by A607,A608,SEQ_4:134;
                                  then
A624:                             q2`1=ppi`1 by A406;
                                  m1<m1+1 by NAT_1:13;
                                  then
A625:                             q1`2<q2`2 by A424,A621,A623;
                                  LSeg(g2,m1)=LSeg(q1,q2) by A607,A608,
TOPREAL1:def 3
                                    .=q1l by A622,A624,A625,A617,TOPREAL3:9;
                                  then
A626:                             ex qy be Point of TOP-REAL 2 st qy=x
& qy`1=ppi`1 & q1`2<=qy`2 & qy`2<=q2`2 by A609,A619;
                                  m1 > n1 + 1 & n1 + 1 >= 1 by A565,A614,
NAT_1:11,XREAL_1:9;
                                  then m1 > 1 by XXREAL_0:2;
                                  then qq`2 < q1`2 by A424,A572,A621;
                                  hence contradiction by A620,A626,XXREAL_0:2;
                                end;
                                hence thesis;
                              end;
                              suppose
                                n<>len g1;
                                then len g1<n by A613,XXREAL_0:1;
                                then
A627:                           LSeg(g,n)=LSeg(g2,n1) by A579,A615,GOBOARD2:5;
                                m1>n1+1 by A565,A614,XREAL_1:9;
                                hence thesis by A450,A609,A627;
                              end;
                            end;
                            hence thesis;
                          end;
                        end;
                        hence thesis;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                hence thesis;
              end;
              hence g is s.n.c. by GOBOARD2:1;
              now
                set p=g1/.len g1, q=g2/.1;
                i2+1<=j2 by A390,NAT_1:13;
                then 1<=l by XREAL_1:19;
                then 1 in dom g2 by A396,FINSEQ_3:25;
                then q`1=ppi`1 by A406;
                hence p`1=q`1 or p`2=q`2 by A27,A14,A51,A46,A29,FINSEQ_4:71;
              end;
              hence g is special by A43,A415,GOBOARD2:8;
              thus L~g=L~f
              proof
                set lg = {LSeg(g,i): 1<=i & i+1 <= len g}, lf = {LSeg(f,j): 1
                <=j & j+1 <= len f};
A628:           len g = len g1 + len g2 by FINSEQ_1:22;
A629:           now
                  let j;
                  assume that
A630:             len g1<=j and
A631:             j<=len g;
                  reconsider w = j-len g1 as Element of NAT by A630,INT_1:5;
                  let p such that
A632:             p=g/.j;
                  per cases;
                  suppose
A633:               j=len g1;
                    1<=len g1 by A24,A14,A47,XXREAL_0:2;
                    then len g1 in dom g1 by FINSEQ_3:25;
                    then
A634:               g/.len g1 = f1/.len f1 by A46,FINSEQ_4:68
                      .= G*(i1,i2) by A27,A14,A51,A29,FINSEQ_4:71;
                    hence p`1=G*(i1,i2)`1 by A632,A633;
                    thus G*(i1,i2)`2<=p`2 & p`2<=G* (i1,j2)`2 by A68,A74,A71
,A30,A72,A390,A392,A394,A632,A633,A634,SEQM_3:def 1;
                    dom l1 = Seg len l1 by FINSEQ_1:def 3;
                    hence p in rng l1 by A68,A72,A391,A632,A633,A634,PARTFUN2:2
;
                  end;
                  suppose
                    j<>len g1;
                    then len g1 < j by A630,XXREAL_0:1;
                    then len g1 + 1<=j by NAT_1:13;
                    then
A635:               1<=w by XREAL_1:19;
A636:               w<=len g2 by A628,A631,XREAL_1:20;
                    then
A637:               w in dom g2 by A635,FINSEQ_3:25;
                    j = w + len g1;
                    then g/.j=g2/.w by A635,A636,SEQ_4:136;
                    hence p`1=ppi`1 & ppi`2<=p`2 & p`2<= pj`2 & p in rng l1 by
A406,A632,A637;
                  end;
                end;
                thus L~g c= L~f
                proof
                  let x be object;
                  assume x in L~g;
                  then consider X be set such that
A638:             x in X and
A639:             X in lg by TARSKI:def 4;
                  consider i such that
A640:             X=LSeg(g,i) and
A641:             1<=i and
A642:             i+1 <= len g by A639;
                  per cases;
                  suppose
A643:               i+1 <= len g1;
                    i<=i+1 by NAT_1:11;
                    then i<=len g1 by A643,XXREAL_0:2;
                    then
A644:               i in dom g1 by A641,FINSEQ_3:25;
                    1<=i+1 by NAT_1:11;
                    then i+1 in dom g1 by A643,FINSEQ_3:25;
                    then X=LSeg(g1,i) by A640,A644,TOPREAL3:18;
                    then X in {LSeg(g1,j): 1<=j & j+1 <= len g1} by A641,A643;
                    then
A645:               x in L~f1 by A44,A638,TARSKI:def 4;
                    L~f1 c= L~f by TOPREAL3:20;
                    hence thesis by A645;
                  end;
                  suppose
A646:               i+1 > len g1;
                    reconsider q1=g/.i, q2=g/.(i+1) as Point of TOP-REAL 2;
A647:               i<=len g by A642,NAT_1:13;
A648:               len g1<=i by A646,NAT_1:13;
                    then
A649:               q1 `1=ppi`1 by A629,A647;
A650:               q1`2<=pj`2 by A629,A648,A647;
A651:               ppi`2<=q1`2 by A629,A648,A647;
                    q2`1=ppi`1 by A629,A642,A646;
                    then
A652:               q2=|[q1`1,q2`2 ]| by A649,EUCLID:53;
A653:               q2`2<=pj`2 by A629,A642,A646;
A654:               q1=|[q1`1,q1`2]| & LSeg(g,i)=LSeg(q2,q1) by A641,A642,
EUCLID:53,TOPREAL1:def 3;
A655:               ppi`2<= q2`2 by A629,A642,A646;
                    now
                      per cases by XXREAL_0:1;
                      suppose
                        q1`2>q2`2;
                        then LSeg(g,i)={p2: p2`1=q1`1 & q2`2<=p2`2 & p2`2<=
                        q1`2} by A652,A654,TOPREAL3:9;
                        then consider p2 such that
A656:                   p2 =x & p2`1=q1`1 and
A657:                   q2`2<=p2`2 & p2`2<=q1`2 by A638,A640;
                        ppi`2<=p2`2 & p2`2<=pj`2 by A650,A655,A657,XXREAL_0:2;
                        then
A658:                   x in LSeg(f,k) by A478,A649,A656;
                        LSeg(f,k) in lf by A3,A24;
                        hence thesis by A658,TARSKI:def 4;
                      end;
                      suppose
                        q1`2=q2`2;
                        then LSeg(g,i)={q1} by A652,A654,RLTOPSP1:70;
                        then x=q1 by A638,A640,TARSKI:def 1;
                        then
A659:                   x in LSeg(f,k) by A478,A649,A651,A650;
                        LSeg(f,k) in lf by A3,A24;
                        hence thesis by A659,TARSKI:def 4;
                      end;
                      suppose
                        q1`2<q2`2;
                        then LSeg(g,i)= {p1: p1`1=q1`1 & q1`2<=p1`2 & p1`2<=
                        q2`2} by A652,A654,TOPREAL3:9;
                        then consider p2 such that
A660:                   p2 =x & p2`1=q1`1 and
A661:                   q1`2<=p2`2 & p2`2<=q2`2 by A638,A640;
                        ppi`2<=p2`2 & p2`2<=pj`2 by A651,A653,A661,XXREAL_0:2;
                        then
A662:                   x in LSeg(f,k) by A478,A649,A660;
                        LSeg(f,k) in lf by A3,A24;
                        hence thesis by A662,TARSKI:def 4;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                let x be object;
                assume x in L~f;
                then
A663:           x in L~f1 \/ LSeg(f,k) by A3,A13,GOBOARD2:3;
                now
                  per cases by A663,XBOOLE_0:def 3;
                  suppose
A664:               x in L~f1;
                    L~g1 c= L~g by GOBOARD2:6;
                    hence thesis by A44,A664;
                  end;
                  suppose
                    x in LSeg(f,k);
                    then consider p1 such that
A665:               p1=x and
A666:               p1`1=ppi`1 and
A667:               ppi`2<=p1`2 and
A668:               p1`2<=pj`2 by A478;
                    defpred P2[Nat] means len g1<=$1 & $1<=len g & for q st q=
                    g/.$1 holds q`2<=p1`2;
A669:               now
                      reconsider n=len g1 as Nat;
                      take n;
                      thus P2[n]
                      proof
                        thus len g1<=n & n<=len g by A628,XREAL_1:31;
                        1<=len g1 by A24,A14,A47,XXREAL_0:2;
                        then
A670:                   len g1 in dom g1 by FINSEQ_3:25;
                        let q;
                        assume q=g/.n;
                        then q=f1/.len f1 by A46,A670,FINSEQ_4:68
                          .=G*(i1,i2) by A27,A14,A51,A29,FINSEQ_4:71;
                        hence thesis by A667;
                      end;
                    end;
A671:               for n be Nat holds P2[n] implies n<=len g;
                    consider ma be Nat such that
A672:               P2[ma] & for n be Nat st P2[n] holds n<=ma from
                    NAT_1:sch 6 (A671,A669);
                    reconsider ma as Nat;
                    now
                      per cases;
                      suppose
A673:                   ma=len g;
                        i2+1<=j2 by A390,NAT_1:13;
                        then
A674:                   1<=l by XREAL_1:19;
                        then len g1+1<=ma by A396,A628,A673,XREAL_1:7;
                        then
A675:                   len g1<=ma-1 by XREAL_1:19;
                        then 0+1<=ma by XREAL_1:19;
                        then reconsider m1=ma-1 as Element of NAT by INT_1:5;
                        reconsider q=g/.m1 as Point of TOP-REAL 2;
A676:                   ma-1<=len g by A673,XREAL_1:43;
                        then
A677:                   q`1=ppi`1 by A629,A675;
A678:                   q`2<=pj`2 by A629,A676,A675;
                        set lq={e where e is Point of TOP-REAL 2: e`1=ppi`1 &
                        q`2<=e`2 & e`2<=pj`2};
A679:                   i2+l=j2;
A680:                   l in dom g2 by A396,A674,FINSEQ_3:25;
                        then
A681:                   g/.ma=g2/.l by A396,A628,A673,FINSEQ_4:69
                          .= pj by A396,A680,A679;
                        then pj`2<=p1`2 by A672;
                        then
A682:                   p1`2=pj`2 by A668,XXREAL_0:1;
                        1<=len g1 by A24,A14,A47,XXREAL_0:2;
                        then
A683:                   1<=m1 by A675,XXREAL_0:2;
A684:                   m1+1=ma;
                        then q=|[q`1,q`2]| & LSeg(g,m1)=LSeg(q,pj) by A673,A681
,A683,EUCLID:53,TOPREAL1:def 3;
                        then LSeg(g,m1)=lq by A462,A477,A677,A678,TOPREAL3:9;
                        then
A685:                   p1 in LSeg(g,m1) by A666,A682,A678;
                        LSeg(g,m1) in lg by A673,A683,A684;
                        hence thesis by A665,A685,TARSKI:def 4;
                      end;
                      suppose
                        ma<>len g;
                        then ma<len g by A672,XXREAL_0:1;
                        then
A686:                   ma+1<=len g by NAT_1:13;
                        reconsider qa=g/.ma, qa1=g/.(ma+1) as Point of
                        TOP-REAL 2;
                        set lma = {p2: p2`1=ppi`1 & qa`2<=p2`2 & p2`2<=qa1`2};
A687:                   qa1=|[qa1 `1, qa1 `2]| by EUCLID:53;
A688:                   qa`2<=p1`2 by A672;
A689:                   len g1<=ma+1 by A672,NAT_1:13;
                        then
A690:                   qa1 `1 = ppi`1 by A629,A686;
A691:                   now
                          assume qa1`2<=p1`2;
                          then for q holds q=g/.(ma+1) implies q`2<=p1`2;
                          then ma+1<=ma by A672,A686,A689;
                          hence contradiction by XREAL_1:29;
                        end;
A692:                   qa`1=ppi`1 & qa=|[qa`1,qa`2]| by A629,A672,EUCLID:53;
A693:                   1<=ma by A24,A14,A47,A672,NAT_1:13;
                        then LSeg(g,ma)=LSeg(qa,qa1) by A686,TOPREAL1:def 3
                          .= lma by A688,A691,A690,A692,A687,TOPREAL3:9
,XXREAL_0:2;
                        then
A694:                   x in LSeg(g,ma) by A665,A666,A688,A691;
                        LSeg(g,ma) in lg by A686,A693;
                        hence thesis by A694,TARSKI:def 4;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                hence thesis;
              end;
              1<=len g1 by A24,A14,A47,XXREAL_0:2;
              then 1 in dom g1 by FINSEQ_3:25;
              hence g/.1=f1/.1 by A45,FINSEQ_4:68
                .=f/.1 by A27,A25,FINSEQ_4:71;
A695:         len g=len g1 + l by A396,FINSEQ_1:22;
              i2+1<=j2 by A390,NAT_1:13;
              then
A696:         1<=l by XREAL_1:19;
              then
A697:         l in dom g2 by A396,FINSEQ_3:25;
              hence g/.len g=g2/.l by A695,FINSEQ_4:69
                .=G*(i1,i2+l) by A396,A697
                .=f/.len f by A3,A21,A76;
              thus len f<=len g by A3,A14,A47,A696,A695,XREAL_1:7;
            end;
          end;
          hence thesis;
        end;
        suppose
A698:     i2=j2;
          set ppi = G*(i1,i2), pj = G*(j1,i2);
          now
            per cases by XXREAL_0:1;
            case
A699:         i1>j1;
              c1/.i1=c1.i1 by A66,A60,PARTFUN1:def 6;
              then
A700:         c1/.i1=ppi by A66,MATRIX_0:def 8;
              then
A701:         x2.i1=ppi`1 by A66,A18,A63,A64,A59,GOBOARD1:def 1;
              c1/.j1=c1.j1 by A23,A60,PARTFUN1:def 6;
              then
A702:         c1/.j1=pj by A23,MATRIX_0:def 8;
              then
A703:         x2.j1 =pj`1 by A23,A18,A63,A64,A59,GOBOARD1:def 1;
              then
A704:         pj`1<ppi`1 by A66,A23,A18,A69,A63,A64,A59,A699,A701,SEQM_3:def 1;
              reconsider l=i1-j1 as Element of NAT by A699,INT_1:5;
              defpred P1[Nat,set] means for m st m=i1-$1 holds $2=G*(m,i2);
              set lk={w where w is Point of TOP-REAL 2: w`2=ppi`2 & pj`1<=w`1
              & w`1<= ppi`1};
A705:         ppi=|[ppi`1,ppi`2]| by EUCLID:53;
A706:         now
                let n;
                assume n in Seg l;
                then
A707:           n<=l by FINSEQ_1:1;
                l<=i1 by XREAL_1:43;
                then reconsider w=i1-n as Element of NAT by A707,INT_1:5
,XXREAL_0:2;
                i1-n<=i1 & i1<=len G by A66,FINSEQ_3:25,XREAL_1:43;
                then
A708:           w<=len G by XXREAL_0:2;
A709:           1<=j1 by A23,FINSEQ_3:25;
                i1-l<=i1-n by A707,XREAL_1:13;
                then 1<=w by A709,XXREAL_0:2;
                then w in dom G by A708,FINSEQ_3:25;
                hence
                i1-n is Nat & [i1-n,i2] in Indices G & i1-n in
                dom G by A22,A68,ZFMISC_1:87;
              end;
A710:         now
                let n be Nat;
                assume n in Seg l;
                then reconsider m=i1-n as Nat by A706;
                take p=G*(m,i2);
                thus P1[n,p];
              end;
              consider g2 such that
A711:         len g2 = l & for n being Nat st n in Seg l holds P1[n
              ,g2/.n] from FINSEQ_4:sch 1(A710);
              take g=g1^g2;
A712:         dom g2=Seg l by A711,FINSEQ_1:def 3;
              now
                let n;
                assume
A713:           n in dom g2;
                then reconsider m=i1-n as Nat by A706,A712;
                take m,k=i2;
                thus [m,k] in Indices G & g2/.n=G*(m,k) by A706,A711,A712,A713;
              end;
              then
A714:         for n st n in dom g ex i,j st [i,j] in Indices G & g/.n=G
              *(i,j) by A75,GOBOARD1:23;
A715:         Seg len g2=dom g2 by FINSEQ_1:def 3;
A716:         y2.i1=ppi`2 by A66,A18,A61,A59,A700,GOBOARD1:def 2;
A717:         now
                let n,p;
                assume that
A718:           n in dom g2 and
A719:           g2/.n=p;
                reconsider n1=i1-n as Nat by A706,A712,A718;
                n<=len g2 by A715,A718,FINSEQ_1:1;
                then
A720:           i1-len g2<= n1 by XREAL_1:13;
                set pn = G*(n1,i2);
A721:           g2/.n=pn by A711,A715,A718;
A722:           i1-n in dom G by A706,A711,A715,A718;
                then
A723:           y2.n1=y2.i1 by A66,A18,A70,A61,A59,SEQM_3:def 10;
                c1/.n1=c1.n1 by A60,A722,PARTFUN1:def 6;
                then
A724:           c1/.n1=pn by A722,MATRIX_0:def 8;
                then
A725:           x2.n1=pn`1 by A18,A63,A64,A59,A722,GOBOARD1:def 1;
                y2.n1=pn`2 by A18,A61,A59,A722,A724,GOBOARD1:def 2;
                hence
                p`2=ppi`2 & pj`1<=p`1 & p`1<=ppi`1 by A66,A23,A18,A69,A63,A64
,A59,A711,A716,A701,A703,A719,A722,A721,A720,A723,A725,SEQ_4:137,XREAL_1:43;
                thus p in rng c1 by A60,A719,A722,A721,A724,PARTFUN2:2;
                1<=n by A715,A718,FINSEQ_1:1;
                then n1<i1 by XREAL_1:44;
                hence p`1<ppi`1 by A66,A18,A69,A63,A64,A59,A701,A719,A722,A721
,A725,SEQM_3:def 1;
              end;
A726:         g2 is special
              proof
                let n be Nat;
                set p = g2/.n;
                assume
A727:           1<=n & n+1 <= len g2;
                then n in dom g2 by SEQ_4:134;
                then
A728:           p`2=ppi`2 by A717;
                n+1 in dom g2 by A727,SEQ_4:134;
                hence thesis by A717,A728;
              end;
A729:         now
                let n,m,p,q;
                assume that
A730:           n in dom g2 and
A731:           m in dom g2 and
A732:           n<m and
A733:           g2/.n=p & g2/.m=q;
A734:           i1-n in dom G by A706,A712,A730;
                reconsider n1=i1-n, m1=i1-m as Nat by A706,A712,A730
,A731;
                set pn = G*(n1,i2), pm = G*(m1,i2);
A735:           m1<n1 by A732,XREAL_1:15;
                c1/.n1=c1.n1 by A60,A706,A712,A730,PARTFUN1:def 6;
                then c1/.n1=pn by A734,MATRIX_0:def 8;
                then
A736:           x2.n1=pn`1 by A65,A60,A734,GOBOARD1:def 1;
A737:           i1-m in dom G by A706,A712,A731;
                c1/.m1 = c1.m1 by A60,A706,A712,A731,PARTFUN1:def 6;
                then c1/.m1= pm by A737,MATRIX_0:def 8;
                then
A738:           x2.m1=pm`1 by A65,A60,A737,GOBOARD1:def 1;
                g2/.n=pn & g2/.m=pm by A711,A712,A730,A731;
                hence q`1<p`1 by A69,A65,A60,A733,A734,A737,A735,A736,A738,
SEQM_3:def 1;
              end;
              for n,m st m>n+1 & n in dom g2 & n+1 in dom g2 & m in dom
              g2 & m+1 in dom g2 holds LSeg(g2,n) misses LSeg(g2,m)
              proof
                let n,m;
                assume that
A739:           m>n+1 and
A740:           n in dom g2 and
A741:           n+1 in dom g2 and
A742:           m in dom g2 and
A743:           m+1 in dom g2 and
A744:           LSeg(g2,n) /\ LSeg(g2,m) <> {};
                reconsider p1=g2/.n, p2=g2/.(n+1), q1=g2/.m, q2=g2/.(m+1) as
                Point of TOP-REAL 2;
A745:           p1`2=ppi`2 & p2`2= ppi`2 by A717,A740,A741;
                n<n+1 by NAT_1:13;
                then
A746:           p2`1<p1`1 by A729,A740,A741;
                set lp = {w where w is Point of TOP-REAL 2: w`2=ppi`2 & p2`1<=
w`1 & w`1<=p1`1}, lq = {w where w is Point of TOP-REAL 2: w`2=ppi`2 & q2`1<=w`1
                & w`1<=q1`1};
A747:           p1=|[p1`1,p1`2]| & p2=|[p2`1,p2`2]| by EUCLID:53;
                m<m+1 by NAT_1:13;
                then
A748:           q2`1<q1`1 by A729,A742,A743;
A749:           q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]| by EUCLID:53;
                set x = the Element of LSeg(g2,n) /\ LSeg(g2,m);
A750:           x in LSeg(g2,n) by A744,XBOOLE_0:def 4;
A751:           q1`2= ppi`2 & q2`2=ppi`2 by A717,A742,A743;
A752:           x in LSeg(g2,m) by A744,XBOOLE_0:def 4;
                1 <= m & m+1<= len g2 by A742,A743,FINSEQ_3:25;
                then LSeg(g2,m) = LSeg(q2,q1) by TOPREAL1:def 3
                  .=lq by A748,A751,A749,TOPREAL3:10;
                then
A753:           ex tm be Point of TOP-REAL 2 st tm=x & tm`2=ppi`2 & q2
                `1<=tm`1 & tm`1<=q1`1 by A752;
                1 <= n & n+1 <= len g2 by A740,A741,FINSEQ_3:25;
                then LSeg(g2,n) = LSeg(p2,p1) by TOPREAL1:def 3
                  .=lp by A746,A745,A747,TOPREAL3:10;
                then
A754:           ex tn be Point of TOP-REAL 2 st tn=x & tn`2=ppi`2 & p2
                `1<=tn`1 & tn`1<=p1`1 by A750;
                q1`1<p2`1 by A729,A739,A741,A742;
                hence contradiction by A754,A753,XXREAL_0:2;
              end;
              then
A755:         g2 is s.n.c. by GOBOARD2:1;
A756:         not f/.k in L~g2
              proof
                set ls={LSeg(g2,m): 1<=m & m+1 <= len g2};
                assume f/.k in L~g2;
                then consider X be set such that
A757:           f/.k in X and
A758:           X in ls by TARSKI:def 4;
                consider m such that
A759:           X=LSeg(g2,m) and
A760:           1<=m & m+1 <= len g2 by A758;
                reconsider q1=g2/.m, q2=g2/.(m+1) as Point of TOP-REAL 2;
A761:           m in dom g2 by A760,SEQ_4:134;
                then
A762:           q1`2=ppi`2 by A717;
                set lq={w where w is Point of TOP-REAL 2: w`2=ppi`2 & q2`1<=w
                `1 & w`1<=q1`1};
A763:           q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]| by EUCLID:53;
A764:           m+1 in dom g2 by A760,SEQ_4:134;
                then
A765:           q2`2=ppi`2 by A717;
                m<m+1 by NAT_1:13;
                then
A766:           q2`1<q1`1 by A729,A761,A764;
                LSeg(g2,m)=LSeg(q2,q1) by A760,TOPREAL1:def 3
                  .=lq by A762,A765,A766,A763,TOPREAL3:10;
                then ex p st p=f/.k & p`2=ppi`2 & q2`1<=p`1 & p`1<=q1`1 by A757
,A759;
                hence contradiction by A29,A717,A761;
              end;
              y2.j1=pj`2 by A23,A18,A61,A59,A702,GOBOARD1:def 2;
              then
A767:         ppi`2=pj`2 by A66,A23,A18,A70,A61,A59,A716,SEQM_3:def 10;
              now
                let n,m be Element of NAT;
                assume that
A768:           n in dom g2 & m in dom g2 and
A769:           n<>m;
                reconsider n1=i1-n, m1=i1-m as Nat by A706,A712,A768;
A770:           g2/.n=G*(n1,i2) & g2/.m=G*(m1,i2) by A711,A712,A768;
                assume
A771:           g2/.n=g2/.m;
                [i1-n,i2] in Indices G & [i1-m,i2] in Indices G by A706,A712
,A768;
                then n1=m1 by A770,A771,GOBOARD1:5;
                hence contradiction by A769;
              end;
              then for n,m being Element of NAT
st n in dom g2 & m in dom g2 & g2/.n = g2/.m
              holds n = m;
              then
A772:         g2 is one-to-one by PARTFUN2:9;
              reconsider m1=i1-l as Nat;
A773:         pj=|[pj`1,pj `2]| by EUCLID:53;
A774:         LSeg(f,k)=LSeg(pj,ppi) by A3,A24,A29,A21,A698,TOPREAL1:def 3
                .= lk by A704,A767,A705,A773,TOPREAL3:10;
A775:         rng g2 c= LSeg(f,k)
              proof
                let x be object;
                assume x in rng g2;
                then consider n being Element of NAT such that
A776:           n in dom g2 and
A777:           g2/.n=x by PARTFUN2:2;
                reconsider n1=i1-n as Nat by A706,A711,A715,A776;
                set pn = G*(n1,i2);
A778:           g2/.n=pn by A711,A715,A776;
                then
A779:           pn`1<=ppi`1 by A717,A776;
                pn`2=ppi`2 & pj`1<=pn`1 by A717,A776,A778;
                hence thesis by A774,A777,A778,A779;
              end;
A780:         now
                let n;
                assume that
A781:           n in dom g2 and
A782:           n+1 in dom g2;
                reconsider m1=i1-n,m2=i1-(n+1) as Nat by A706,A712
,A781,A782;
                let l1,l2,l3,l4 be Nat;
                assume that
A783:           [l1,l2] in Indices G and
A784:           [l3,l4] in Indices G and
A785:           g2/.n=G*(l1,l2) and
A786:           g2/.(n+1)=G*(l3,l4);
                [i1-(n+1),i2] in Indices G & g2/.(n+1)=G*(m2,i2) by A706,A711
,A712,A782;
                then
A787:           l3=m2 & l4=i2 by A784,A786,GOBOARD1:5;
                [i1-n,i2] in Indices G & g2/.n=G*(m1,i2) by A706,A711,A712,A781
;
                then l1=m1 & l2=i2 by A783,A785,GOBOARD1:5;
                hence |.l1-l3.|+|.l2-l4.|= |.i1-n-(i1-(n+1)).|+0 by A787,
ABSVALUE:2
                  .= 1 by ABSVALUE:def 1;
              end;
              now
                let l1,l2,l3,l4 be Nat;
                assume that
A788:           [ l1,l2] in Indices G and
A789:           [l3,l4] in Indices G and
A790:           g1/.len g1=G*(l1,l2) and
A791:           g2/.1=G*(l3,l4) and
                len g1 in dom g1 and
A792:           1 in dom g2;
                reconsider m1=i1-1 as Nat by A706,A712,A792;
                [i1-1,i2] in Indices G & g2/.1=G*(m1,i2) by A706,A711,A712,A792
;
                then
A793:           l3=m1 & l4=i2 by A789,A791,GOBOARD1:5;
                f1/.len f1=f/.k by A27,A14,A51,FINSEQ_4:71;
                then l1=i1 & l2=i2 by A46,A28,A29,A788,A790,GOBOARD1:5;
                hence |.l1-l3.|+|.l2-l4.|=|.i1-(i1-1).|+0 by A793,ABSVALUE:2
                  .=1 by ABSVALUE:def 1;
              end;
              then for n st n in dom g & n+1 in dom g holds for m,k,i,j st [
m,k] in Indices G & [i,j] in Indices G & g/.n=G*(m,k) & g/.(n+1)=G*(i,j) holds
              |.m-i.|+|.k-j.|=1 by A48,A780,GOBOARD1:24;
              hence g is_sequence_on G by A714,GOBOARD1:def 9;
A794:         LSeg(f,k)=LSeg(ppi,pj) by A3,A24,A29,A21,A698,TOPREAL1:def 3;
A795:         not f/.k in rng g2
              proof
                assume f/.k in rng g2;
                then consider n being Element of NAT such that
A796:           n in dom g2 and
A797:           g2/.n=f/.k by PARTFUN2:2;
                reconsider n1=i1-n as Nat by A706,A711,A715,A796;
                [i1-n,i2] in Indices G & g2/.n=G*(n1,i2) by A706,A711,A715,A796
;
                then
A798:           n1=i1 by A28,A29,A797,GOBOARD1:5;
                0<n by A715,A796,FINSEQ_1:1;
                hence contradiction by A798;
              end;
              rng g1 /\ rng g2 = {}
              proof
                set x = the Element of rng g1 /\ rng g2;
                assume
A799:           not thesis;
                then
A800:           x in rng g2 by XBOOLE_0:def 4;
A801:           x in rng g1 by A799,XBOOLE_0:def 4;
                now
                  per cases by A24,XXREAL_0:1;
                  suppose
                    k=1;
                    hence contradiction by A52,A795,A801,A800,TARSKI:def 1;
                  end;
                  suppose
                    1<k;
                    then x in L~f1 /\ LSeg(f,k) & L~f1 /\ LSeg(f,k)={f/.k}
                    by A3,A6,A7,A49,A775,A801,A800,GOBOARD2:4,XBOOLE_0:def 4;
                    hence contradiction by A795,A800,TARSKI:def 1;
                  end;
                end;
                hence contradiction;
              end;
              then rng g1 misses rng g2;
              hence g is one-to-one by A40,A772,FINSEQ_3:91;
A802:         for n st 1<=n & n+2 <= len g2 holds LSeg(g2,n) /\ LSeg(g2
              ,n+1) = {g2/.(n+1)}
              proof
                let n;
                assume that
A803:           1<=n and
A804:           n+2 <= len g2;
A805:           n+1 in dom g2 by A803,A804,SEQ_4:135;
                then g2/.(n+1) in rng g2 by PARTFUN2:2;
                then g2/.(n+1) in lk by A774,A775;
                then consider u1 be Point of TOP-REAL 2 such that
A806:           g2/.(n+1)=u1 and
A807:           u1`2=ppi`2 and
                pj`1<=u1`1 and
                u1`1<=ppi`1;
A808:           n+2 in dom g2 by A803,A804,SEQ_4:135;
                then g2/.(n+2) in rng g2 by PARTFUN2:2;
                then g2/.(n+2) in lk by A774,A775;
                then consider u2 be Point of TOP-REAL 2 such that
A809:           g2/.(n+2)=u2 and
A810:           u2`2=ppi`2 and
                pj`1<=u2`1 and
                u2`1<=ppi`1;
                1<= n+1 & n+1+1 = n+(1+1) by NAT_1:11;
                then
A811:           LSeg(g2,n+1)=LSeg( u1,u2) by A804,A806,A809,TOPREAL1:def 3;
                n+1<n+1+1 by NAT_1:13;
                then
A812:           u2`1<u1`1 by A729,A805,A808,A806,A809;
A813:           n in dom g2 by A803,A804,SEQ_4:135;
                then g2/.n in rng g2 by PARTFUN2:2;
                then g2/.n in lk by A774,A775;
                then consider u be Point of TOP-REAL 2 such that
A814:           g2/.n=u and
A815:           u`2=ppi`2 and
                pj`1<=u`1 and
                u`1<=ppi`1;
                n+1 <= n+2 by XREAL_1:6;
                then n+1 <= len g2 by A804,XXREAL_0:2;
                then
A816:           LSeg(g2,n)=LSeg(u,u1) by A803,A814,A806,TOPREAL1:def 3;
                set lg = {w where w is Point of TOP-REAL 2: w`2=ppi`2 & u2`1<=
                w`1 & w`1<=u`1};
                n<n+1 by NAT_1:13;
                then
A817:           u1`1<u`1 by A729,A813,A805,A814,A806;
                then
A818:           u1 in lg by A807,A812;
                u=|[u`1,u`2]| & u2=|[u2`1,u2`2 ]| by EUCLID:53;
                then LSeg(g2/.n,g2/.(n+2))=lg by A814,A815,A809,A810,A812,A817,
TOPREAL3:10,XXREAL_0:2;
                hence thesis by A814,A806,A809,A816,A811,A818,TOPREAL1:8;
              end;
              thus g is unfolded
              proof
                let n be Nat;
                assume that
A819:           1<=n and
A820:           n+2 <= len g;
A821:           n+1+1<=len g by A820;
                n+1<=n+1+1 by NAT_1:11;
                then
A822:           n+1 <= len g by A820,XXREAL_0:2;
A823:           len g=len g1+len g2 by FINSEQ_1:22;
                n+2-len g1 = n-len g1 +2;
                then
A824:           n-len g1 + 2 <= len g2 by A820,A823,XREAL_1:20;
A825:           1 <= n+1 by NAT_1:11;
A826:           n<=n+1 by NAT_1:11;
A827:           n+(1+1)=n+1+1;
                per cases;
                suppose
A828:             n+2 <= len g1;
A829:             n+(1+1)=n+1+1;
A830:             n+1 in dom g1 by A819,A828,SEQ_4:135;
                  then
A831:             g/.(n+1)=g1/.(n+1) by FINSEQ_4:68;
                  n in dom g1 by A819,A828,SEQ_4:135;
                  then
A832:             LSeg(g1,n)=LSeg(g,n) by A830,TOPREAL3:18;
                  n+2 in dom g1 by A819,A828,SEQ_4:135;
                  then LSeg(g1,n+1)=LSeg(g,n+1) by A830,A829,TOPREAL3:18;
                  hence thesis by A41,A819,A828,A832,A831;
                end;
                suppose
                  len g1 < n+2;
                  then len g1+1<=n+2 by NAT_1:13;
                  then
A833:             len g1<=n+2-1 by XREAL_1:19;
                  thus thesis
                  proof
                    per cases;
                    suppose
A834:                 len g1=n+1;
                      then 1<=len g-len g1 by A821,XREAL_1:19;
                      then 1 in dom g2 by A823,FINSEQ_3:25;
                      then
A835:                 g2/.1 in rng g2 by PARTFUN2:2;
                      then g2/.1 in lk by A774,A775;
                      then consider u1 be Point of TOP-REAL 2 such that
A836:                 g2/.1=u1 and
                      u1`2=ppi`2 and
                      pj`1<=u1`1 and
                      u1`1<=ppi`1;
                      ppi in LSeg(ppi,pj) by RLTOPSP1:68;
                      then
A837:                 LSeg(ppi,u1) c= LSeg(f,k) by A794,A775,A835,A836,
TOPREAL1:6;
                      1<=n+1 by NAT_1:11;
                      then
A838:                 n+1 in dom g1 by A834,FINSEQ_3:25;
                      then
A839:                 g/.(n+1)=f1/.len f1 by A46,A834,FINSEQ_4:68
                        .= ppi by A27,A14,A51,A29,FINSEQ_4:71;
                      now
                        1<len g1 by A819,A834,NAT_1:13;
                        then
A840:                   1+1<=len g1 by NAT_1:13;
                        assume k=1;
                        hence contradiction by A52,A840,TOPREAL1:23;
                      end;
                      then 1<k by A24,XXREAL_0:1;
                      then
A841:                 L~f1 /\ LSeg(f,k)={f/.k} by A3,A6,A7,GOBOARD2:4;
A842:                 LSeg(g1,n) c= L~f1 by A44,TOPREAL3:19;
                      n in dom g1 by A819,A826,A834,FINSEQ_3:25;
                      then
A843:                 LSeg(g,n)=LSeg(g1,n) by A838,TOPREAL3:18;
                      g/.(n+1) in LSeg(g,n) & g/.(n+1) in LSeg(g,n+1)
                      by A819,A820,A825,A822,A827,TOPREAL1:21;
                      then g/.(n+1) in LSeg(g,n) /\ LSeg(g,n+1) by
XBOOLE_0:def 4;
                      then
A844:                 {g/.(n+1)} c= LSeg(g,n) /\ LSeg(g,n+1) by ZFMISC_1:31;
                      1 <= len g2 by A820,A827,A823,A834,XREAL_1:6;
                      then g/.(n+2)=g2/.1 by A827,A834,SEQ_4:136;
                      then LSeg(g,n+1)=LSeg(ppi,u1 ) by A820,A825,A827,A839
,A836,TOPREAL1:def 3;
                      then LSeg(g,n) /\ LSeg(g,n+1) c= {g /.(n+1)} by A29,A842
,A841,A843,A839,A837,XBOOLE_1:27;
                      hence thesis by A844;
                    end;
                    suppose
                      len g1<>n+1;
                      then len g1<n+1 by A833,XXREAL_0:1;
                      then
A845:                 len g1<=n by NAT_1:13;
                      then reconsider n1=n-len g1 as Element of NAT by INT_1:5;
                      thus thesis
                      proof
                        per cases;
                        suppose
A846:                     len g1=n;
                          then
A847:                     2 <= len g2 by A820,A823,XREAL_1:6;
                          then 1<= len g2 by XXREAL_0:2;
                          then
A848:                     g/.(n+1)=g2/.1 by A846,SEQ_4:136;
                          1<=len g2 by A847,XXREAL_0:2;
                          then
A849:                     1 in dom g2 by FINSEQ_3:25;
                          then g2/.1 in rng g2 by PARTFUN2:2;
                          then g2/.1 in lk by A774,A775;
                          then consider u1 be Point of TOP-REAL 2 such that
A850:                     g2/.1=u1 and
A851:                     u1`2=ppi`2 and
                          pj`1<=u1`1 and
A852:                     u1`1<=ppi`1;
                          1<=len g1 by A24,A14,A47,XXREAL_0:2;
                          then len g1 in dom g1 by FINSEQ_3:25;
                          then g/.n=f1/.len f1 by A46,A846,FINSEQ_4:68
                            .= ppi by A27,A14,A51,A29,FINSEQ_4:71;
                          then
A853:                     LSeg(g,n)=LSeg(ppi,u1) by A819,A822,A848,A850,
TOPREAL1:def 3;
A854:                     2 in dom g2 by A847,FINSEQ_3:25;
                          then g2/.2 in rng g2 by PARTFUN2:2;
                          then g2/.2 in lk by A774,A775;
                          then consider u2 be Point of TOP-REAL 2 such that
A855:                     g2 /.2=u2 and
A856:                     u2`2=ppi`2 and
                          pj`1<=u2`1 and
A857:                     u2`1<=ppi`1;
                          set lg = {w where w is Point of TOP-REAL 2 : w`2=ppi
                          `2 & u2`1<=w`1 & w`1<=ppi`1};
                          u2=|[u2`1,u2`2]| by EUCLID:53;
                          then
A858:                     LSeg(ppi,g2/.2)=lg by A705,A855,A856,A857,TOPREAL3:10
;
                          u2`1<u1`1 by A729,A849,A854,A850,A855;
                          then
A859:                     u1 in lg by A851,A852;
                          g/.(n+2)=g2/.2 by A846,A847,SEQ_4:136;
                          then LSeg(g,n+1)=LSeg(u1,u2) by A820,A825,A827,A848
,A850,A855,TOPREAL1:def 3;
                          hence thesis by A848,A850,A855,A859,A853,A858,
TOPREAL1:8;
                        end;
                        suppose
                          len g1<>n;
                          then
A860:                     len g1<n by A845,XXREAL_0:1;
                          then len g1+1<=n by NAT_1:13;
                          then
A861:                     1<=n1 by XREAL_1:19;
                          n1 + len g1 = n;
                          then
A862:                     LSeg(g,n)=LSeg(g2,n1) by A822,A860,GOBOARD2:5;
A863:                     n+1 = n1+1+len g1;
                          n1 + 1 + len g1 = n + 1;
                          then n1+1 <= len g2 by A822,A823,XREAL_1:6;
                          then
A864:                     g/.(n+1)=g2/.(n1+1) by A863,NAT_1:11,SEQ_4:136;
                          len g1<n+1 by A826,A860,XXREAL_0:2;
                          then LSeg(g,n+1)=LSeg(g2,n1+1) by A821,A863,
GOBOARD2:5;
                          hence thesis by A802,A824,A862,A864,A861;
                        end;
                      end;
                    end;
                  end;
                end;
              end;
A865:         L~g2 c= LSeg(f,k)
              proof
                let x be object;
                set ls={LSeg(g2,m): 1<=m & m+1 <= len g2};
                assume x in L~g2;
                then consider X be set such that
A866:           x in X and
A867:           X in ls by TARSKI:def 4;
                consider m such that
A868:           X=LSeg(g2,m) and
A869:           1<=m & m+1 <= len g2 by A867;
                reconsider q1=g2/.m, q2=g2/.(m+1) as Point of TOP-REAL 2;
A870:           LSeg(g2,m)=LSeg(q1,q2) by A869,TOPREAL1:def 3;
                m+1 in dom g2 by A869,SEQ_4:134;
                then
A871:           g2/.(m+1) in rng g2 by PARTFUN2:2;
                m in dom g2 by A869,SEQ_4:134;
                then g2/.m in rng g2 by PARTFUN2:2;
                then LSeg(q1,q2) c= LSeg(ppi,pj) by A794,A775,A871,TOPREAL1:6;
                hence thesis by A794,A866,A868,A870;
              end;
A872:         L~g1 /\ L~g2 = {}
              proof
                per cases;
                suppose
                  k=1;
                  hence thesis by A52;
                end;
                suppose
                  k<>1;
                  then 1<k by A24,XXREAL_0:1;
                  then L~g1 /\ LSeg(f,k)={f/.k} by A3,A6,A7,A44,GOBOARD2:4;
                  then
A873:             L~g1 /\ L~g2 c= {f/.k} by A865,XBOOLE_1:26;
                  now
                    set x = the Element of L~g1 /\ L~g2;
                    assume L~g1 /\ L~g2 <> {};
                    then x in {f/.k} & x in L~g2 by A873,XBOOLE_0:def 4;
                    hence contradiction by A756,TARSKI:def 1;
                  end;
                  hence thesis;
                end;
              end;
              for n,m st m>n+1 & n in dom g & n+1 in dom g & m in dom g
              & m+1 in dom g holds LSeg(g,n) misses LSeg(g,m)
              proof
A874:           1<=len g1 by A24,A14,A47,XXREAL_0:2;
                then len g1 in dom g1 by FINSEQ_3:25;
                then
A875:           g/.len g1=g1/.len g1 by FINSEQ_4:68
                  .= ppi by A27,A14,A51,A46,A29,FINSEQ_4:71;
                reconsider qq=g2/.1 as Point of TOP-REAL 2;
                set l1 = {LSeg(g1,i): 1<=i & i+1 <= len g1}, l2 = {LSeg(g2,j):
                1<=j & j+1 <= len g2};
                let n,m;
                assume that
A876:           m>n+1 and
A877:           n in dom g and
A878:           n+1 in dom g and
A879:           m in dom g and
A880:           m+1 in dom g;
A881:           1<=n by A877,FINSEQ_3:25;
                j1+1<=i1 by A699,NAT_1:13;
                then
A882:           1<=l by XREAL_1:19;
                then
A883:           1 in dom g2 by A711,FINSEQ_3:25;
                then
A884:           qq`2=ppi`2 & qq`1<ppi`1 by A717;
A885:           g/.(len g1+1)=qq by A711,A882,SEQ_4:136;
A886:           pj`1<=qq`1 by A717,A883;
A887:           m+1<=len g by A880,FINSEQ_3:25;
A888:           1<=m+1 by A880,FINSEQ_3:25;
A889:           1<=n+1 by A878,FINSEQ_3:25;
A890:           n+1<=len g by A878,FINSEQ_3:25;
A891:           qq=|[qq`1,qq`2]| by EUCLID:53;
A892:           1<=m by A879,FINSEQ_3:25;
                set ql={z where z is Point of TOP-REAL 2: z`2=ppi`2 & qq`1<=z
                `1 & z`1<=ppi`1};
A893:           n<=n+1 by NAT_1:11;
A894:           len g=len g1+len g2 by FINSEQ_1:22;
                then len g1+1<=len g by A711,A882,XREAL_1:7;
                then
A895:           LSeg (g,len g1)=LSeg(qq,ppi) by A874,A875,A885,TOPREAL1:def 3
                  .= ql by A705,A884,A891,TOPREAL3:10;
A896:           m<=m+1 by NAT_1:11;
                then
A897:           n+1<=m+1 by A876,XXREAL_0:2;
                now
                  per cases;
                  suppose
A898:               m+1<=len g1;
                    then m<=len g1 by A896,XXREAL_0:2;
                    then
A899:               m in dom g1 by A892,FINSEQ_3:25;
                    m+1 in dom g1 by A888,A898,FINSEQ_3:25;
                    then
A900:               LSeg(g,m)=LSeg(g1,m) by A899,TOPREAL3:18;
A901:               n+1<=len g1 by A897,A898,XXREAL_0:2;
                    then n<=len g1 by A893,XXREAL_0:2;
                    then
A902:               n in dom g1 by A881,FINSEQ_3:25;
                    n+1 in dom g1 by A889,A901,FINSEQ_3:25;
                    then LSeg(g,n)=LSeg(g1,n) by A902,TOPREAL3:18;
                    hence thesis by A42,A876,A900;
                  end;
                  suppose
                    len g1<m+1;
                    then
A903:               len g1<=m by NAT_1:13;
                    then reconsider m1=m-len g1 as Element of NAT by INT_1:5;
                    now
                      per cases;
                      suppose
A904:                   m=len g1;
A905:                   LSeg(g,m) c= LSeg(f,k)
                        proof
                          let x be object;
                          assume x in LSeg(g,m);
                          then consider px be Point of TOP-REAL 2 such that
A906:                     px=x & px`2=ppi`2 and
A907:                     qq`1<=px`1 and
A908:                     px`1<= ppi`1 by A895,A904;
                          pj`1<=px`1 by A886,A907,XXREAL_0:2;
                          hence thesis by A774,A906,A908;
                        end;
                        n<=len g1 by A876,A893,A904,XXREAL_0:2;
                        then
A909:                   n in dom g1 by A881,FINSEQ_3:25;
                        now
                          1<len g1 by A876,A889,A904,XXREAL_0:2;
                          then
A910:                     1+1<=len g1 by NAT_1:13;
                          assume k=1;
                          hence contradiction by A52,A910,TOPREAL1:23;
                        end;
                        then 1<k by A24,XXREAL_0:1;
                        then
A911:                   L~f1 /\ LSeg(f,k) ={f/.k} by A3,A6,A7,GOBOARD2:4;
A912:                   n+1 in dom g1 by A876,A889,A904,FINSEQ_3:25;
                        then
A913:                   LSeg(g,n)=LSeg(g1,n) by A909,TOPREAL3:18;
                        then LSeg(g,n) in l1 by A876,A881,A904;
                        then LSeg(g,n) c= L~f1 by A44,ZFMISC_1:74;
                        then
A914:                   LSeg(g,n) /\ LSeg(g,m) c= {f/.k} by A911,A905,
XBOOLE_1:27;
                        now
                          set x = the Element of LSeg(g,n)/\ LSeg(g,m);
                          assume
A915:                     LSeg(g,n)/\ LSeg(g,m)<>{};
                          then
A916:                     x in LSeg(g,n) by XBOOLE_0:def 4;
                          x in {f/.k} by A914,A915;
                          then
A917:                     x=f/.k by TARSKI:def 1;
                          f/.k=g1/.len g1 by A27,A14,A51,A46,FINSEQ_4:71;
                          hence
                          contradiction by A40,A41,A42,A876,A904,A909,A912,A913
,A916,A917,GOBOARD2:2;
                        end;
                        hence thesis;
                      end;
                      suppose
                        m<>len g1;
                        then
A918:                   len g1<m by A903,XXREAL_0:1;
                        then len g1+1<= m by NAT_1:13;
                        then
A919:                   1<=m1 by XREAL_1:19;
                        m+1=m1+1+len g1;
                        then
A920:                   m1+1 <= len g2 by A887,A894,XREAL_1:6;
                        m = m1+len g1;
                        then
A921:                   LSeg(g,m)=LSeg(g2,m1) by A887,A918,GOBOARD2:5;
                        then LSeg(g,m) in l2 by A919,A920;
                        then
A922:                   LSeg(g,m) c= L~g2 by ZFMISC_1:74;
                        now
                          per cases;
                          suppose
A923:                       n+1<=len g1;
                            then n<=len g1 by A893,XXREAL_0:2;
                            then
A924:                       n in dom g1 by A881,FINSEQ_3:25;
                            n+1 in dom g1 by A889,A923,FINSEQ_3:25;
                            then LSeg(g,n)=LSeg(g1,n) by A924,TOPREAL3:18;
                            then LSeg(g,n) in l1 by A881,A923;
                            then LSeg(g,n) c= L~g1 by ZFMISC_1:74;
                            then LSeg(g,n) /\ LSeg(g,m) = {} by A872,A922,
XBOOLE_1:3,27;
                            hence thesis;
                          end;
                          suppose
                            len g1<n+1;
                            then
A925:                       len g1<=n by NAT_1:13;
                            then reconsider n1=n-len g1 as Element of NAT by
INT_1:5;
A926:                       n - len g1 + 1 = n + 1 - len g1;
A927:                       n = n1 + len g1;
                            now
                              per cases;
                              suppose
A928:                           len g1=n;
                                now
                                  reconsider q1=g2/.m1, q2=g2/.(m1+1) as Point
                                  of TOP-REAL 2;

set x = the Element of LSeg(g,n) /\ LSeg(g,m);
                                  set q1l={v where v is Point of TOP-REAL 2: v
                                  `2=ppi`2 & q2`1<=v`1 & v`1<=q1`1};
A929:                             q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]|
                                  by EUCLID:53;
                                  assume
A930:                             LSeg(g,n) /\ LSeg(g,m)<>{};
                                  then
A931:                             x in LSeg(g,m) by XBOOLE_0:def 4;
                                  x in LSeg(g,n) by A930,XBOOLE_0:def 4;
                                  then
A932:                             ex qx be Point of TOP-REAL 2 st qx=x
& qx`2=ppi`2 & qq`1<=qx`1 & qx`1<= ppi`1 by A895,A928;
A933:                             m1 in dom g2 by A919,A920,SEQ_4:134;
                                  then
A934:                             q1 `2=ppi`2 by A717;
A935:                             m1+1 in dom g2 by A919,A920,SEQ_4:134;
                                  then
A936:                             q2`2=ppi`2 by A717;
                                  m1<m1+1 by NAT_1:13;
                                  then
A937:                             q2`1<q1`1 by A729,A933,A935;
                                  LSeg(g2,m1)=LSeg(q2,q1) by A919,A920,
TOPREAL1:def 3
                                    .=q1l by A934,A936,A937,A929,TOPREAL3:10;
                                  then
A938:                             ex qy be Point of TOP-REAL 2 st qy=x
& qy`2=ppi`2 & q2`1<=qy`1 & qy`1<= q1`1 by A921,A931;
                                  m1 > n1 + 1 & n1 + 1 >= 1 by A876,A926,
NAT_1:11,XREAL_1:9;
                                  then m1 > 1 by XXREAL_0:2;
                                  then q1`1 < qq`1 by A729,A883,A933;
                                  hence contradiction by A932,A938,XXREAL_0:2;
                                end;
                                hence thesis;
                              end;
                              suppose
                                n<>len g1;
                                then len g1<n by A925,XXREAL_0:1;
                                then
A939:                           LSeg(g,n)=LSeg(g2,n1) by A890,A927,GOBOARD2:5;
                                m1>n1+1 by A876,A926,XREAL_1:9;
                                hence thesis by A755,A921,A939;
                              end;
                            end;
                            hence thesis;
                          end;
                        end;
                        hence thesis;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                hence thesis;
              end;
              hence g is s.n.c. by GOBOARD2:1;
              now
                set p=g1/.len g1, q=g2/.1;
                j1+1<=i1 by A699,NAT_1:13;
                then 1<=l by XREAL_1:19;
                then 1 in dom g2 by A712,FINSEQ_1:1;
                then q`2=ppi`2 by A717;
                hence p`1=q`1 or p`2=q`2 by A27,A14,A51,A46,A29,FINSEQ_4:71;
              end;
              hence g is special by A43,A726,GOBOARD2:8;
              thus L~g=L~f
              proof
                set lg = {LSeg(g,i): 1<=i & i+1 <= len g}, lf = {LSeg(f,j): 1
                <=j & j+1 <= len f};
A940:           len g = len g1 + len g2 by FINSEQ_1:22;
A941:           now
                  let j;
                  assume that
A942:             len g1<=j and
A943:             j<=len g;
                  reconsider w = j-len g1 as Element of NAT by A942,INT_1:5;
                  let p such that
A944:             p=g/.j;
                  per cases;
                  suppose
A945:               j=len g1;
                    1<=len g1 by A24,A14,A47,XXREAL_0:2;
                    then len g1 in dom g1 by FINSEQ_3:25;
                    then
A946:               g/.len g1 = f1/.len f1 by A46,FINSEQ_4:68
                      .= G*(i1,i2) by A27,A14,A51,A29,FINSEQ_4:71;
                    hence p`2=G*(i1,i2)`2 by A944,A945;
                    thus G*(j1,i2)`1<=p`1 & p`1<=G* (i1,i2)`1 by A66,A23,A18
,A69,A63,A64,A59,A699,A701,A703,A944,A945,A946,SEQM_3:def 1;
                    Seg len c1 = dom c1 by FINSEQ_1:def 3;
                    hence p in rng c1 by A66,A18,A59,A700,A944,A945,A946,
PARTFUN2:2;
                  end;
                  suppose
                    j<>len g1;
                    then len g1 < j by A942,XXREAL_0:1;
                    then len g1 + 1<=j by NAT_1:13;
                    then
A947:               1<=w by XREAL_1:19;
A948:               w<=len g2 by A940,A943,XREAL_1:20;
                    then
A949:               w in dom g2 by A947,FINSEQ_3:25;
                    j = w + len g1;
                    then g/.j=g2/.w by A947,A948,SEQ_4:136;
                    hence p`2=ppi`2 & pj`1<=p`1 & p`1<= ppi`1 & p in rng c1 by
A717,A944,A949;
                  end;
                end;
                thus L~g c= L~f
                proof
                  let x be object;
                  assume x in L~g;
                  then consider X be set such that
A950:             x in X and
A951:             X in lg by TARSKI:def 4;
                  consider i such that
A952:             X=LSeg(g,i) and
A953:             1<=i and
A954:             i+1 <= len g by A951;
                  per cases;
                  suppose
A955:               i+1 <= len g1;
                    i<=i+1 by NAT_1:11;
                    then i<=len g1 by A955,XXREAL_0:2;
                    then
A956:               i in dom g1 by A953,FINSEQ_3:25;
                    1<=i+1 by NAT_1:11;
                    then i+1 in dom g1 by A955,FINSEQ_3:25;
                    then X=LSeg(g1,i) by A952,A956,TOPREAL3:18;
                    then X in {LSeg(g1,j): 1<=j & j+1 <= len g1} by A953,A955;
                    then
A957:               x in L~f1 by A44,A950,TARSKI:def 4;
                    L~f1 c= L~f by TOPREAL3:20;
                    hence thesis by A957;
                  end;
                  suppose
A958:               i+1 > len g1;
                    reconsider q1=g/.i, q2=g/.(i+1) as Point of TOP-REAL 2;
A959:               i<=len g by A954,NAT_1:13;
A960:               len g1<=i by A958,NAT_1:13;
                    then
A961:               q1`2=ppi`2 by A941,A959;
A962:               q1`1<=ppi`1 by A941,A960,A959;
A963:               pj`1<=q1`1 by A941,A960,A959;
                    q2`2=ppi`2 by A941,A954,A958;
                    then
A964:               q2=|[q2`1,q1`2 ]| by A961,EUCLID:53;
A965:               q2`1<=ppi`1 by A941,A954,A958;
A966:               q1=|[q1`1,q1`2]| & LSeg(g,i)=LSeg(q2,q1) by A953,A954,
EUCLID:53,TOPREAL1:def 3;
A967:               pj`1<= q2`1 by A941,A954,A958;
                    now
                      per cases by XXREAL_0:1;
                      suppose
                        q1`1>q2`1;
                        then LSeg(g,i)={p2: p2`2=q1`2 & q2`1<=p2`1 & p2`1<=
                        q1`1} by A964,A966,TOPREAL3:10;
                        then consider p2 such that
A968:                   p2 =x & p2`2=q1`2 and
A969:                   q2`1<=p2`1 & p2`1<=q1`1 by A950,A952;
                        pj`1<=p2`1 & p2`1<=ppi`1 by A962,A967,A969,XXREAL_0:2;
                        then
A970:                   x in LSeg(f,k) by A774,A961,A968;
                        LSeg(f,k) in lf by A3,A24;
                        hence thesis by A970,TARSKI:def 4;
                      end;
                      suppose
                        q1`1=q2`1;
                        then LSeg(g,i)={q1} by A964,A966,RLTOPSP1:70;
                        then x=q1 by A950,A952,TARSKI:def 1;
                        then
A971:                   x in LSeg(f,k) by A774,A961,A963,A962;
                        LSeg(f,k) in lf by A3,A24;
                        hence thesis by A971,TARSKI:def 4;
                      end;
                      suppose
                        q1`1<q2`1;
                        then LSeg(g,i)= {p1: p1`2=q1`2 & q1`1<=p1`1 & p1`1<=
                        q2`1} by A964,A966,TOPREAL3:10;
                        then consider p2 such that
A972:                   p2 =x & p2`2=q1`2 and
A973:                   q1`1<=p2`1 & p2`1<=q2`1 by A950,A952;
                        pj`1<=p2`1 & p2`1<=ppi`1 by A963,A965,A973,XXREAL_0:2;
                        then
A974:                   x in LSeg(f,k) by A774,A961,A972;
                        LSeg(f,k) in lf by A3,A24;
                        hence thesis by A974,TARSKI:def 4;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                let x be object;
                assume x in L~f;
                then
A975:           x in L~f1 \/ LSeg(f,k) by A3,A13,GOBOARD2:3;
                per cases by A975,XBOOLE_0:def 3;
                suppose
A976:             x in L~f1;
                  L~g1 c= L~g by GOBOARD2:6;
                  hence thesis by A44,A976;
                end;
                suppose
                  x in LSeg(f,k);
                  then consider p1 such that
A977:             p1=x and
A978:             p1`2=ppi`2 and
A979:             pj`1<=p1`1 and
A980:             p1`1<=ppi`1 by A774;
                  defpred P2[Nat] means len g1<=$1 & $1<=len g & for q st q=g
                  /.$1 holds q`1>=p1`1;
A981:             now
                    reconsider n=len g1 as Nat;
                    take n;
                    thus P2[n]
                    proof
                      thus len g1<=n & n<=len g by A940,XREAL_1:31;
                      1<=len g1 by A24,A14,A47,XXREAL_0:2;
                      then
A982:                 len g1 in dom g1 by FINSEQ_3:25;
                      let q;
                      assume q=g/.n;
                      then q=f1/.len f1 by A46,A982,FINSEQ_4:68
                        .=G*(i1,i2) by A27,A14,A51,A29,FINSEQ_4:71;
                      hence thesis by A980;
                    end;
                  end;
A983:             for n be Nat holds P2[n] implies n<=len g;
                  consider ma be Nat such that
A984:             P2[ma] & for n be Nat st P2[n] holds n<=ma from
                  NAT_1:sch 6 (A983,A981);
                  reconsider ma as Nat;
                  now
                    per cases;
                    suppose
A985:                 ma=len g;
                      j1+1<=i1 by A699,NAT_1:13;
                      then
A986:                 1<=l by XREAL_1:19;
                      then len g1+1<=ma by A711,A940,A985,XREAL_1:7;
                      then
A987:                 len g1<=ma-1 by XREAL_1:19;
                      then 0+1<=ma by XREAL_1:19;
                      then reconsider m1=ma-1 as Element of NAT by INT_1:5;
                      reconsider q=g/.m1 as Point of TOP-REAL 2;
A988:                 ma-1<=len g by A985,XREAL_1:43;
                      then
A989:                 q`2=ppi`2 by A941,A987;
A990:                 pj`1<=q`1 by A941,A988,A987;
                      set lq={e where e is Point of TOP-REAL 2: e`2=ppi`2 & pj
                      `1<=e`1 & e`1<=q`1};
A991:                 i1-l=j1;
A992:                 l in dom g2 by A711,A986,FINSEQ_3:25;
                      then
A993:                 g/.ma=g2/.l by A711,A940,A985,FINSEQ_4:69
                        .= pj by A711,A712,A992,A991;
                      then p1`1<=pj`1 by A984;
                      then
A994:                 p1`1=pj`1 by A979,XXREAL_0:1;
                      1<=len g1 by A24,A14,A47,XXREAL_0:2;
                      then
A995:                 1<=m1 by A987,XXREAL_0:2;
A996:                 m1+1=ma;
                      then q=|[q`1,q`2]| & LSeg(g,m1)=LSeg(pj,q) by A985,A993
,A995,EUCLID:53,TOPREAL1:def 3;
                      then LSeg(g,m1)=lq by A767,A773,A989,A990,TOPREAL3:10;
                      then
A997:                 p1 in LSeg(g,m1) by A978,A994,A990;
                      LSeg(g,m1) in lg by A985,A995,A996;
                      hence thesis by A977,A997,TARSKI:def 4;
                    end;
                    suppose
                      ma<>len g;
                      then ma<len g by A984,XXREAL_0:1;
                      then
A998:                 ma+1<=len g by NAT_1:13;
                      reconsider qa=g/.ma, qa1=g/.(ma+1) as Point of TOP-REAL
                      2;
                      set lma = {p2: p2`2=ppi`2 & qa1`1<=p2`1 & p2`1<=qa`1};
A999:                 qa1=|[qa1 `1, qa1 `2]| by EUCLID:53;
A1000:                p1`1<=qa`1 by A984;
A1001:                len g1<=ma+1 by A984,NAT_1:13;
                      then
A1002:                qa1 `2 = ppi`2 by A941,A998;
A1003:                now
                        assume p1`1<=qa1`1;
                        then for q holds q=g/.(ma+1) implies p1`1<=q`1;
                        then ma+1<=ma by A984,A998,A1001;
                        hence contradiction by XREAL_1:29;
                      end;
A1004:                qa`2=ppi`2 & qa=|[qa`1,qa`2]| by A941,A984,EUCLID:53;
A1005:                1<=ma by A24,A14,A47,A984,NAT_1:13;
                      then LSeg(g,ma)=LSeg(qa1,qa) by A998,TOPREAL1:def 3
                        .= lma by A1000,A1003,A1002,A1004,A999,TOPREAL3:10
,XXREAL_0:2;
                      then
A1006:                x in LSeg(g,ma) by A977,A978,A1000,A1003;
                      LSeg(g,ma) in lg by A998,A1005;
                      hence thesis by A1006,TARSKI:def 4;
                    end;
                  end;
                  hence thesis;
                end;
              end;
A1007:        len g=len g1 + len g2 by FINSEQ_1:22;
              1<=len g1 by A24,A14,A47,XXREAL_0:2;
              then 1 in dom g1 by FINSEQ_3:25;
              hence g/.1=f1/.1 by A45,FINSEQ_4:68
                .=f/.1 by A27,A25,FINSEQ_4:71;
              j1+1<=i1 by A699,NAT_1:13;
              then
A1008:        1<=l by XREAL_1:19;
              then
A1009:        l in dom g2 by A712,FINSEQ_1:1;
              hence g/.len g=g2/.l by A711,A1007,FINSEQ_4:69
                .=G*(m1,i2) by A711,A712,A1009
                .=f/.len f by A3,A21,A698;
              thus len f<=len g by A3,A14,A47,A711,A1008,A1007,XREAL_1:7;
            end;
            case
A1010:        i1=j1;
              k<>k+1;
              hence contradiction by A5,A27,A29,A19,A21,A698,A1010,PARTFUN2:10;
            end;
            case
A1011:        i1<j1;
              c1/.i1=c1.i1 by A66,A60,PARTFUN1:def 6;
              then
A1012:        c1/.i1=ppi by A66,MATRIX_0:def 8;
              then
A1013:        x2.i1=ppi`1 by A66,A65,A60,GOBOARD1:def 1;
              c1/.j1=c1.j1 by A23,A60,PARTFUN1:def 6;
              then
A1014:        c1/.j1=pj by A23,MATRIX_0:def 8;
              then
A1015:        x2.j1 =pj`1 by A23,A65,A60,GOBOARD1:def 1;
              then
A1016:        ppi`1<pj`1 by A66,A23,A69,A65,A60,A1011,A1013,SEQM_3:def 1;
              reconsider l=j1-i1 as Element of NAT by A1011,INT_1:5;
              deffunc F(Nat) = G*(i1+$1,i2);
              consider g2 such that
A1017:        len g2 = l & for n being Nat st n in dom g2 holds g2
              /.n = F(n) from FINSEQ_4:sch 2;
              take g=g1^g2;
A1018:        now
                let n;
A1019:          n<=i1+n by NAT_1:11;
                assume
A1020:          n in Seg l;
                then n<=l by FINSEQ_1:1;
                then
A1021:          i1+n<=l+i1 by XREAL_1:7;
                j1<=len G by A23,FINSEQ_3:25;
                then
A1022:          i1+n<=len G by A1021,XXREAL_0:2;
                1<=n by A1020,FINSEQ_1:1;
                then 1<=i1+n by A1019,XXREAL_0:2;
                hence i1+n in dom G by A1022,FINSEQ_3:25;
                hence [i1+n,i2] in Indices G by A22,A68,ZFMISC_1:87;
              end;
A1023:        Seg len g2 = dom g2 by FINSEQ_1:def 3;
              now
                let n such that
A1024:          n in dom g2;
                take m=i1+n,k=i2;
                thus [m,k] in Indices G & g2/.n=G*(m,k) by A1017,A1018,A1023
,A1024;
              end;
              then
A1025:        for n st n in dom g ex i,j st [i,j] in Indices G & g/.n=G
              *(i,j) by A75,GOBOARD1:23;
A1026:        y2.i1=ppi`2 by A66,A63,A64,A65,A61,A60,A1012,GOBOARD1:def 2;
A1027:        now
                let n,p;
                assume that
A1028:          n in dom g2 and
A1029:          g2/.n=p;
A1030:          g2/.n=G*(i1+n,i2) by A1017,A1028;
                set n1=i1+n;
                set pn = G*(n1,i2);
A1031:          i1+ n in dom G by A1017,A1018,A1023,A1028;
                then
A1032:          y2.n1=y2.i1 by A66,A70,A63,A64,A65,A61,A60,SEQM_3:def 10;
                c1/.n1=c1.n1 by A60,A1017,A1018,A1023,A1028,PARTFUN1:def 6;
                then
A1033:          c1/.n1=pn by A1031,MATRIX_0:def 8;
                then
A1034:          x2.n1=pn`1 by A65,A60,A1031,GOBOARD1:def 1;
                n<=len g2 by A1028,FINSEQ_3:25;
                then
A1035:          n1<=i1+len g2 by XREAL_1:7;
                y2.n1=pn`2 by A63,A64,A65,A61,A60,A1031,A1033,GOBOARD1:def 2;
                hence p`2=ppi`2 & ppi`1<=p`1 & p`1<=pj`1 by A66,A23,A69,A65,A60
,A1017,A1026,A1013,A1015,A1029,A1030,A1031,A1035,A1032,A1034,SEQ_4:137
,XREAL_1:31;
                thus p in rng c1 by A60,A1029,A1030,A1031,A1033,PARTFUN2:2;
                1<=n by A1028,FINSEQ_3:25;
                then i1<n1 by XREAL_1:29;
                hence p`1>ppi`1 by A66,A69,A65,A60,A1013,A1029,A1030,A1031
,A1034,SEQM_3:def 1;
              end;
A1036:        g2 is special
              proof
                let n be Nat;
                set p = g2/.n;
                assume
A1037:          1<=n & n+1 <= len g2;
                then n in dom g2 by SEQ_4:134;
                then
A1038:          p`2=ppi`2 by A1027;
                n+1 in dom g2 by A1037,SEQ_4:134;
                hence thesis by A1027,A1038;
              end;
              now
                let n,m be Element of NAT;
                assume that
A1039:          n in dom g2 & m in dom g2 and
A1040:          n<>m;
A1041:          g2/.n=G*(i1+n,i2) & g2/.m=G*(i1+m,i2) by A1017,A1039;
                assume
A1042:          g2/.n=g2/.m;
                [ i1+n,i2] in Indices G & [i1+m,i2] in Indices G by A1017,A1018
,A1023,A1039;
                then i1+n=i1+m by A1041,A1042,GOBOARD1:5;
                hence contradiction by A1040;
              end;
              then for n,m being Element of NAT
st n in dom g2 & m in dom g2 & g2/.n = g2/.m
              holds n = m;
              then
A1043:        g2 is one-to-one by PARTFUN2:9;
              set lk={w where w is Point of TOP-REAL 2: w`2=ppi`2 & ppi`1<=w`1
              & w`1<= pj`1};
A1044:        ppi=|[ppi`1,ppi`2]| by EUCLID:53;
A1045:        now
                let n,m,p,q;
                assume that
A1046:          n in dom g2 and
A1047:          m in dom g2 and
A1048:          n<m and
A1049:          g2/.n=p & g2/.m=q;
A1050:          i1+n in dom G by A1017,A1018,A1023,A1046;
                set n1=i1+n, m1=i1+m;
                set pn = G*(n1,i2), pm = G*(m1,i2);
A1051:          n1<m1 by A1048,XREAL_1:8;
                c1/.n1=c1.n1 by A60,A1017,A1018,A1023,A1046,PARTFUN1:def 6;
                then c1/.n1=pn by A1050,MATRIX_0:def 8;
                then
A1052:          x2.n1=pn`1 by A65,A60,A1050,GOBOARD1:def 1;
A1053:          i1+m in dom G by A1017,A1018,A1023,A1047;
                c1/.m1 = c1.m1 by A60,A1017,A1018,A1023,A1047,PARTFUN1:def 6;
                then c1/.m1=pm by A1053,MATRIX_0:def 8;
                then
A1054:          x2.m1=pm`1 by A65,A60,A1053,GOBOARD1:def 1;
                g2/.n=G*(i1+n,i2) & g2/.m=G*(i1+m,i2) by A1017,A1046,A1047;
                hence p`1<q`1 by A69,A65,A60,A1049,A1050,A1053,A1051,A1052
,A1054,SEQM_3:def 1;
              end;
              for n,m st m>n+1 & n in dom g2 & n+1 in dom g2 & m in dom
              g2 & m+1 in dom g2 holds LSeg(g2,n) misses LSeg(g2,m)
              proof
                let n,m;
                assume that
A1055:          m>n+1 and
A1056:          n in dom g2 and
A1057:          n+1 in dom g2 and
A1058:          m in dom g2 and
A1059:          m+1 in dom g2 and
A1060:          LSeg(g2,n) /\ LSeg(g2,m) <> {};
                reconsider p1=g2/.n, p2=g2/.(n+1), q1=g2/.m, q2=g2/.(m+1) as
                Point of TOP-REAL 2;
A1061:          p1`2=ppi`2 & p2`2= ppi`2 by A1027,A1056,A1057;
                n<n+1 by NAT_1:13;
                then
A1062:          p1`1<p2`1 by A1045,A1056,A1057;
                set lp = {w where w is Point of TOP-REAL 2: w`2=ppi`2 & p1`1<=
w`1 & w`1<=p2`1}, lq = {w where w is Point of TOP-REAL 2: w`2=ppi`2 & q1`1<=w`1
                & w`1<=q2`1};
A1063:          p1=|[p1`1,p1`2]| & p2=|[p2`1,p2`2]| by EUCLID:53;
                m<m+1 by NAT_1:13;
                then
A1064:          q1`1<q2`1 by A1045,A1058,A1059;
A1065:          q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]| by EUCLID:53;
                set x = the Element of LSeg(g2,n) /\ LSeg(g2,m);
A1066:          x in LSeg(g2,n) by A1060,XBOOLE_0:def 4;
A1067:          q1`2= ppi`2 & q2`2=ppi`2 by A1027,A1058,A1059;
A1068:          x in LSeg(g2,m) by A1060,XBOOLE_0:def 4;
                1 <= m & m+1<= len g2 by A1058,A1059,FINSEQ_3:25;
                then LSeg(g2,m) = LSeg(q1,q2) by TOPREAL1:def 3
                  .=lq by A1064,A1067,A1065,TOPREAL3:10;
                then
A1069:          ex tm be Point of TOP-REAL 2 st tm=x & tm`2=ppi`2 & q1
                `1<=tm`1 & tm`1<=q2`1 by A1068;
                1 <= n & n+1 <= len g2 by A1056,A1057,FINSEQ_3:25;
                then LSeg(g2,n) = LSeg(p1,p2) by TOPREAL1:def 3
                  .=lp by A1062,A1061,A1063,TOPREAL3:10;
                then
A1070:          ex tn be Point of TOP-REAL 2 st tn=x & tn`2=ppi`2 & p1
                `1<=tn`1 & tn`1<=p2`1 by A1066;
                p2`1<q1`1 by A1045,A1055,A1057,A1058;
                hence contradiction by A1070,A1069,XXREAL_0:2;
              end;
              then
A1071:        g2 is s.n.c. by GOBOARD2:1;
A1072:        not f/.k in L~g2
              proof
                set ls={LSeg(g2,m): 1<=m & m+1 <= len g2};
                assume f/.k in L~g2;
                then consider X be set such that
A1073:          f/.k in X and
A1074:          X in ls by TARSKI:def 4;
                consider m such that
A1075:          X=LSeg(g2,m) and
A1076:          1<=m & m+1 <= len g2 by A1074;
                reconsider q1=g2/.m, q2=g2/.(m+1) as Point of TOP-REAL 2;
A1077:          m in dom g2 by A1076,SEQ_4:134;
                then
A1078:          q1`2=ppi`2 by A1027;
                set lq={w where w is Point of TOP-REAL 2: w`2=ppi`2 & q1`1<=w
                `1 & w`1<=q2`1};
A1079:          q1=|[q1`1,q1`2]| & q2=|[q2`1,q2`2]| by EUCLID:53;
A1080:          m+1 in dom g2 by A1076,SEQ_4:134;
                then
A1081:          q2`2=ppi`2 by A1027;
                m<m+1 by NAT_1:13;
                then
A1082:          q1`1<q2`1 by A1045,A1077,A1080;
                LSeg(g2,m)=LSeg(q1,q2) by A1076,TOPREAL1:def 3
                  .=lq by A1078,A1081,A1082,A1079,TOPREAL3:10;
                then ex p st p=f/.k & p`2=ppi`2 & q1`1<=p`1 & p`1<=q2`1 by
A1073,A1075;
                hence contradiction by A29,A1027,A1077;
              end;
              y2.j1=pj`2 by A23,A63,A64,A65,A61,A60,A1014,GOBOARD1:def 2;
              then
A1083:        ppi`2=pj`2 by A66,A23,A70,A63,A64,A65,A61,A60,A1026,SEQM_3:def 10
;
A1084:        now
                let n;
                assume that
A1085:          n in dom g2 and
A1086:          n+1 in dom g2;
                let l1,l2,l3,l4 be Nat;
                assume that
A1087:          [l1,l2] in Indices G and
A1088:          [l3,l4] in Indices G and
A1089:          g2/.n=G*(l1,l2) and
A1090:          g2/.(n+1)=G*(l3,l4);
                g2/.(n+1)=G*(i1+(n+1),i2) & [i1+(n+1),i2] in Indices G
                by A1017,A1018,A1023,A1086;
                then
A1091:          l3=i1+(n+1) & l4=i2 by A1088,A1090,GOBOARD1:5;
                g2/.n=G*(i1+n,i2) & [i1+n,i2] in Indices G by A1017,A1018,A1023
,A1085;
                then l1=i1+n & l2=i2 by A1087,A1089,GOBOARD1:5;
                hence |.l1-l3.|+|.l2-l4.|= |.i1+n-(i1+(n+1)).|+0 by A1091,
ABSVALUE:2
                  .= |.-1.|
                  .= |.1.| by COMPLEX1:52
                  .= 1 by ABSVALUE:def 1;
              end;
              now
                let l1,l2,l3,l4 be Nat;
                assume that
A1092:          [l1,l2] in Indices G and
A1093:          [l3,l4] in Indices G and
A1094:          g1/.len g1=G*(l1, l2) and
A1095:          g2/.1=G*(l3,l4) and
                len g1 in dom g1 and
A1096:          1 in dom g2;
                g2/.1=G* (i1+1,i2) & [i1+1,i2] in Indices G by A1017,A1018
,A1023,A1096;
                then
A1097:          l3=i1+1 & l4=i2 by A1093,A1095,GOBOARD1:5;
                f1/.len f1=f/.k by A27,A14,A51,FINSEQ_4:71;
                then l1=i1 & l2=i2 by A46,A28,A29,A1092,A1094,GOBOARD1:5;
                hence |.l1-l3.|+|.l2-l4.|=|.i1-(i1+1).|+0 by A1097,
ABSVALUE:2
                  .=|.i1-i1+-1.|
                  .=|.1.| by COMPLEX1:52
                  .=1 by ABSVALUE:def 1;
              end;
              then for n st n in dom g & n+1 in dom g holds for m,k,i,j st [
m,k] in Indices G & [i,j] in Indices G & g/.n=G*(m,k) & g/.(n+1)=G*(i,j) holds
              |.m-i.|+|.k-j.|=1 by A48,A1084,GOBOARD1:24;
              hence g is_sequence_on G by A1025,GOBOARD1:def 9;
A1098:        pj=|[pj`1,pj`2]| by EUCLID:53;
A1099:        LSeg(f,k)=LSeg(ppi,pj) by A3,A24,A29,A21,A698,TOPREAL1:def 3
                .= lk by A1016,A1083,A1044,A1098,TOPREAL3:10;
A1100:        rng g2 c= LSeg(f,k)
              proof
                let x be object;
                assume x in rng g2;
                then consider n being Element of NAT such that
A1101:          n in dom g2 and
A1102:          g2/.n=x by PARTFUN2:2;
                set pn = G*((i1+n),i2);
A1103:          g2/.n=G*(i1+n,i2) by A1017,A1101;
                then
A1104:          pn`1<=pj`1 by A1027,A1101;
                pn`2=ppi`2 & ppi`1<=pn`1 by A1027,A1101,A1103;
                hence thesis by A1099,A1102,A1103,A1104;
              end;
A1105:        Seg l = dom g2 by A1017,FINSEQ_1:def 3;
A1106:        not f/.k in rng g2
              proof
                assume f/.k in rng g2;
                then consider n being Element of NAT such that
A1107:          n in dom g2 and
A1108:          g2/.n=f/.k by PARTFUN2:2;
                g2/.n=G*(i1+n,i2) & [i1+n,i2] in Indices G by A1017,A1105,A1018
,A1107;
                then
A1109:          i1+n=i1 by A28,A29,A1108,GOBOARD1:5;
                0<n by A1107,FINSEQ_3:25;
                hence contradiction by A1109;
              end;
              rng g1 /\ rng g2 = {}
              proof
                set x = the Element of rng g1 /\ rng g2;
                assume
A1110:          not thesis;
                then
A1111:          x in rng g2 by XBOOLE_0:def 4;
A1112:          x in rng g1 by A1110,XBOOLE_0:def 4;
                now
                  per cases by A24,XXREAL_0:1;
                  suppose
                    k=1;
                    hence contradiction by A52,A1106,A1112,A1111,TARSKI:def 1;
                  end;
                  suppose
                    1<k;
                    then x in L~f1 /\ LSeg(f,k) & L~f1 /\ LSeg(f,k)={f/.k}
by A3,A6,A7,A49,A1100,A1112,A1111,GOBOARD2:4,XBOOLE_0:def 4;
                    hence contradiction by A1106,A1111,TARSKI:def 1;
                  end;
                end;
                hence contradiction;
              end;
              then rng g1 misses rng g2;
              hence g is one-to-one by A40,A1043,FINSEQ_3:91;
A1113:        LSeg(f,k)=LSeg(ppi,pj) by A3,A24,A29,A21,A698,TOPREAL1:def 3;
A1114:        for n st 1<=n & n+2 <= len g2 holds LSeg(g2,n) /\ LSeg(g2
              ,n+1) = {g2/.(n+1)}
              proof
                let n;
                assume that
A1115:          1<=n and
A1116:          n+2 <= len g2;
A1117:          n+1 in dom g2 by A1115,A1116,SEQ_4:135;
                then g2/.(n+1) in rng g2 by PARTFUN2:2;
                then g2/.(n+1) in lk by A1099,A1100;
                then consider u1 be Point of TOP-REAL 2 such that
A1118:          g2/.(n+1)=u1 and
A1119:          u1`2=ppi`2 and
                ppi`1<=u1`1 and
                u1`1<=pj`1;
A1120:          n+2 in dom g2 by A1115,A1116,SEQ_4:135;
                then g2/.(n+2) in rng g2 by PARTFUN2:2;
                then g2/.(n+2) in lk by A1099,A1100;
                then consider u2 be Point of TOP-REAL 2 such that
A1121:          g2/.(n+2)=u2 and
A1122:          u2`2=ppi`2 and
                ppi`1<=u2`1 and
                u2`1<=pj`1;
                1<= n+1 & n+1+1 = n+(1+1) by NAT_1:11;
                then
A1123:          LSeg(g2,n+1)=LSeg( u1,u2) by A1116,A1118,A1121,TOPREAL1:def 3;
                n+1<n+1+1 by NAT_1:13;
                then
A1124:          u1`1<u2`1 by A1045,A1117,A1120,A1118,A1121;
A1125:          n in dom g2 by A1115,A1116,SEQ_4:135;
                then g2/.n in rng g2 by PARTFUN2:2;
                then g2/.n in lk by A1099,A1100;
                then consider u be Point of TOP-REAL 2 such that
A1126:          g2/.n=u and
A1127:          u`2=ppi`2 and
                ppi`1<=u`1 and
                u`1<=pj`1;
                n+1 <= n+2 by XREAL_1:6;
                then n+1 <= len g2 by A1116,XXREAL_0:2;
                then
A1128:          LSeg(g2,n)=LSeg(u,u1) by A1115,A1126,A1118,TOPREAL1:def 3;
                set lg = {w where w is Point of TOP-REAL 2: w`2=ppi`2 & u`1<=w
                `1 & w`1<=u2`1};
                n<n+1 by NAT_1:13;
                then
A1129:          u`1<u1`1 by A1045,A1125,A1117,A1126,A1118;
                then
A1130:          u1 in lg by A1119,A1124;
                u=|[u`1,u`2]| & u2=|[u2`1,u2`2 ]| by EUCLID:53;
                then LSeg(g2/.n,g2/.(n+2))=lg by A1126,A1127,A1121,A1122,A1124
,A1129,TOPREAL3:10,XXREAL_0:2;
                hence thesis by A1126,A1118,A1121,A1128,A1123,A1130,TOPREAL1:8;
              end;
              thus g is unfolded
              proof
                let n be Nat;
                assume that
A1131:          1<=n and
A1132:          n+2 <= len g;
A1133:          n+1+1<=len g by A1132;
                n+1<=n+1+1 by NAT_1:11;
                then
A1134:          n+1 <= len g by A1132,XXREAL_0:2;
A1135:          len g=len g1+len g2 by FINSEQ_1:22;
                n+2-len g1 = n-len g1 +2;
                then
A1136:          n-len g1 + 2 <= len g2 by A1132,A1135,XREAL_1:20;
A1137:          1<= n+1 by NAT_1:11;
A1138:          n<=n+1 by NAT_1:11;
A1139:          n+(1+1)=n+1+1;
                per cases;
                suppose
A1140:            n+2 <= len g1;
A1141:            n+(1+1)=n+1+1;
A1142:            n+1 in dom g1 by A1131,A1140,SEQ_4:135;
                  then
A1143:            g/.(n+1)=g1/.(n+1) by FINSEQ_4:68;
                  n in dom g1 by A1131,A1140,SEQ_4:135;
                  then
A1144:            LSeg(g1,n)=LSeg(g,n) by A1142,TOPREAL3:18;
                  n+2 in dom g1 by A1131,A1140,SEQ_4:135;
                  then LSeg(g1,n+1)=LSeg(g,n+1) by A1142,A1141,TOPREAL3:18;
                  hence thesis by A41,A1131,A1140,A1144,A1143;
                end;
                suppose
                  len g1 < n+2;
                  then len g1+1<=n+2 by NAT_1:13;
                  then
A1145:            len g1<=n+2-1 by XREAL_1:19;
                  now
                    per cases;
                    suppose
A1146:                len g1=n+1;
                      then 1<=len g-len g1 by A1133,XREAL_1:19;
                      then 1 in dom g2 by A1135,FINSEQ_3:25;
                      then
A1147:                g2/.1 in rng g2 by PARTFUN2:2;
                      then g2/.1 in lk by A1099,A1100;
                      then consider u1 be Point of TOP-REAL 2 such that
A1148:                g2/.1=u1 and
                      u1`2=ppi`2 and
                      ppi`1<=u1`1 and
                      u1`1<=pj`1;
                      ppi in LSeg(ppi,pj) by RLTOPSP1:68;
                      then
A1149:                LSeg(ppi,u1) c= LSeg(f,k) by A1113,A1100,A1147,A1148,
TOPREAL1:6;
                      1<=n+1 by NAT_1:11;
                      then
A1150:                n+1 in dom g1 by A1146,FINSEQ_3:25;
                      then
A1151:                g/.(n+1)=f1/.len f1 by A46,A1146,FINSEQ_4:68
                        .= ppi by A27,A14,A51,A29,FINSEQ_4:71;
                      now
                        1<len g1 by A1131,A1146,NAT_1:13;
                        then
A1152:                  1+1<=len g1 by NAT_1:13;
                        assume k=1;
                        hence contradiction by A52,A1152,TOPREAL1:23;
                      end;
                      then 1<k by A24,XXREAL_0:1;
                      then
A1153:                L~f1 /\ LSeg(f,k)={f/.k} by A3,A6,A7,GOBOARD2:4;
A1154:                LSeg(g1,n) c= L~f1 by A44,TOPREAL3:19;
                      n in dom g1 by A1131,A1138,A1146,FINSEQ_3:25;
                      then
A1155:                LSeg(g,n)=LSeg(g1,n) by A1150,TOPREAL3:18;
                      g/.(n+1) in LSeg(g,n) & g/.(n+1) in LSeg(g,n+1)
                      by A1131,A1132,A1137,A1134,A1139,TOPREAL1:21;
                      then g/.(n+1) in LSeg(g,n) /\ LSeg(g,n+1) by
XBOOLE_0:def 4;
                      then
A1156:                {g/.(n+1)} c= LSeg(g,n) /\ LSeg(g,n+1) by ZFMISC_1:31;
                      n+2 = 1+len g1 & 1 <= len g2 by A1132,A1139,A1135,A1146,
XREAL_1:6;
                      then g/.(n+2)=g2/.1 by SEQ_4:136;
                      then LSeg(g,n+1)=LSeg(ppi,u1 ) by A1132,A1137,A1139,A1151
,A1148,TOPREAL1:def 3;
                      then LSeg(g,n) /\ LSeg(g,n+1) c= {g /.(n+1)} by A29,A1154
,A1153,A1155,A1151,A1149,XBOOLE_1:27;
                      hence thesis by A1156;
                    end;
                    suppose
                      len g1<>n+1;
                      then len g1<n+1 by A1145,XXREAL_0:1;
                      then
A1157:                len g1<=n by NAT_1:13;
                      then reconsider n1=n-len g1 as Element of NAT by INT_1:5;
                      now
                        per cases;
                        suppose
A1158:                    len g1=n;
                          then
A1159:                    2 <= len g2 by A1132,A1135,XREAL_1:6;
                          then 1 <= len g2 by XXREAL_0:2;
                          then
A1160:                    g/.(n+1)=g2/.1 by A1158,SEQ_4:136;
                          1<=len g2 by A1159,XXREAL_0:2;
                          then
A1161:                    1 in dom g2 by FINSEQ_3:25;
                          then g2/.1 in rng g2 by PARTFUN2:2;
                          then g2/.1 in lk by A1099,A1100;
                          then consider u1 be Point of TOP-REAL 2 such that
A1162:                    g2/.1=u1 and
A1163:                    u1`2=ppi`2 & ppi`1<=u1`1 and
                          u1`1<=pj`1;
                          1<=len g1 by A24,A14,A47,XXREAL_0:2;
                          then len g1 in dom g1 by FINSEQ_3:25;
                          then g/.n=f1/.len f1 by A46,A1158,FINSEQ_4:68
                            .= ppi by A27,A14,A51,A29,FINSEQ_4:71;
                          then
A1164:                    LSeg(g,n)=LSeg(ppi,u1) by A1131,A1134,A1160,A1162,
TOPREAL1:def 3;
A1165:                    2 in dom g2 by A1159,FINSEQ_3:25;
                          then g2/.2 in rng g2 by PARTFUN2:2;
                          then g2/.2 in lk by A1099,A1100;
                          then consider u2 be Point of TOP-REAL 2 such that
A1166:                    g2 /.2=u2 and
A1167:                    u2`2=ppi`2 & ppi`1<=u2`1 and
                          u2`1<=pj`1;
                          set lg = {w where w is Point of TOP-REAL 2 : w`2=ppi
                          `2 & ppi`1<=w`1 & w`1<=u2`1};
                          u1`1<u2`1 by A1045,A1161,A1165,A1162,A1166;
                          then u2=|[u2`1,u2`2]| & u1 in lg by A1163,EUCLID:53;
                          then
A1168:                    u1 in LSeg(ppi,u2) by A1044,A1167,TOPREAL3:10;
                          g/.(n+2)=g2/.2 by A1158,A1159,SEQ_4:136;
                          then LSeg(g,n+1)=LSeg(u1,u2) by A1132,A1137,A1139
,A1160,A1162,A1166,TOPREAL1:def 3;
                          hence thesis by A1160,A1162,A1164,A1168,TOPREAL1:8;
                        end;
                        suppose
                          len g1<>n;
                          then
A1169:                    len g1<n by A1157,XXREAL_0:1;
                          then len g1+1<=n by NAT_1:13;
                          then
A1170:                    1<=n1 by XREAL_1:19;
                          n1 + len g1 = n;
                          then
A1171:                    LSeg(g,n)=LSeg(g2,n1) by A1134,A1169,GOBOARD2:5;
A1172:                    n+1 = n1+1+len g1;
                          then n1+1<=len g2 by A1134,A1135,XREAL_1:6;
                          then
A1173:                    g/.(n+1)=g2/.(n1+1) by A1172,NAT_1:11,SEQ_4:136;
                          len g1<n+1 by A1138,A1169,XXREAL_0:2;
                          then LSeg(g,n+1)=LSeg(g2,n1+1) by A1133,A1172,
GOBOARD2:5;
                          hence thesis by A1114,A1136,A1171,A1173,A1170;
                        end;
                      end;
                      hence thesis;
                    end;
                  end;
                  hence thesis;
                end;
              end;
A1174:        L~g2 c= LSeg(f,k)
              proof
                let x be object;
                set ls={LSeg(g2,m): 1<=m & m+1 <= len g2};
                assume x in L~g2;
                then consider X be set such that
A1175:          x in X and
A1176:          X in ls by TARSKI:def 4;
                consider m such that
A1177:          X=LSeg(g2,m) and
A1178:          1<=m & m+1 <= len g2 by A1176;
                reconsider q1=g2/.m, q2=g2/.(m+1) as Point of TOP-REAL 2;
A1179:          LSeg(g2,m)=LSeg(q1,q2) by A1178,TOPREAL1:def 3;
                m+1 in dom g2 by A1178,SEQ_4:134;
                then
A1180:          g2/.(m+1) in rng g2 by PARTFUN2:2;
                m in dom g2 by A1178,SEQ_4:134;
                then g2/.m in rng g2 by PARTFUN2:2;
                then LSeg(q1,q2) c= LSeg(ppi,pj) by A1113,A1100,A1180,
TOPREAL1:6;
                hence thesis by A1113,A1175,A1177,A1179;
              end;
A1181:        L~g1 /\ L~g2 = {}
              proof
                per cases;
                suppose
                  k=1;
                  hence thesis by A52;
                end;
                suppose
                  k<>1;
                  then 1<k by A24,XXREAL_0:1;
                  then L~g1 /\ LSeg(f,k)={f/.k} by A3,A6,A7,A44,GOBOARD2:4;
                  then
A1182:            L~g1 /\ L~g2 c= {f/.k} by A1174,XBOOLE_1:26;
                  now
                    set x = the Element of L~g1 /\ L~g2;
                    assume L~g1 /\ L~g2 <> {};
                    then x in {f/.k} & x in L~g2 by A1182,XBOOLE_0:def 4;
                    hence contradiction by A1072,TARSKI:def 1;
                  end;
                  hence thesis;
                end;
              end;
              for n,m st m>n+1 & n in dom g & n+1 in dom g & m in dom g
              & m+1 in dom g holds LSeg(g,n) misses LSeg(g,m)
              proof
A1183:          1<=len g1 by A24,A14,A47,XXREAL_0:2;
                then len g1 in dom g1 by FINSEQ_3:25;
                then
A1184:          g/.len g1=g1/.len g1 by FINSEQ_4:68
                  .= ppi by A27,A14,A51,A46,A29,FINSEQ_4:71;
                reconsider qq=g2/.1 as Point of TOP-REAL 2;
                set l1 = {LSeg(g1,i): 1<=i & i+1 <= len g1}, l2 = {LSeg(g2,j):
                1<=j & j+1 <= len g2};
                let n,m;
                assume that
A1185:          m>n+1 and
A1186:          n in dom g and
A1187:          n+1 in dom g and
A1188:          m in dom g and
A1189:          m+1 in dom g;
A1190:          1<=n by A1186,FINSEQ_3:25;
                i1+1<=j1 by A1011,NAT_1:13;
                then
A1191:          1<=l by XREAL_1:19;
                then
A1192:          1 in dom g2 by A1017,FINSEQ_3:25;
                then
A1193:          qq`2=ppi`2 & qq`1>ppi`1 by A1027;
A1194:          g/.(len g1+1)=qq by A1017,A1191,SEQ_4:136;
A1195:          qq`1<=pj`1 by A1027,A1192;
A1196:          m+1<=len g by A1189,FINSEQ_3:25;
A1197:          1<=m+1 by A1189,FINSEQ_3:25;
A1198:          1<=n+1 by A1187,FINSEQ_3:25;
A1199:          n+1<=len g by A1187,FINSEQ_3:25;
A1200:          qq=|[qq`1,qq`2]| by EUCLID:53;
A1201:          1<=m by A1188,FINSEQ_3:25;
                set ql={z where z is Point of TOP-REAL 2: z`2=ppi`2 & ppi`1<=z
                `1 & z`1<=qq`1};
A1202:          n<=n+1 by NAT_1:11;
A1203:          len g=len g1+len g2 by FINSEQ_1:22;
                then len g1+1<=len g by A1017,A1191,XREAL_1:7;
                then
A1204:          LSeg(g,len g1)=LSeg(ppi,qq) by A1183,A1184,A1194,TOPREAL1:def 3
                  .= ql by A1044,A1193,A1200,TOPREAL3:10;
A1205:          m<=m+1 by NAT_1:11;
                then
A1206:          n+1<=m+1 by A1185,XXREAL_0:2;
                now
                  per cases;
                  suppose
A1207:              m+1<=len g1;
                    then m<=len g1 by A1205,XXREAL_0:2;
                    then
A1208:              m in dom g1 by A1201,FINSEQ_3:25;
                    m+1 in dom g1 by A1197,A1207,FINSEQ_3:25;
                    then
A1209:              LSeg(g,m)=LSeg(g1,m) by A1208,TOPREAL3:18;
A1210:              n+1<=len g1 by A1206,A1207,XXREAL_0:2;
                    then n<=len g1 by A1202,XXREAL_0:2;
                    then
A1211:              n in dom g1 by A1190,FINSEQ_3:25;
                    n+1 in dom g1 by A1198,A1210,FINSEQ_3:25;
                    then LSeg(g,n)=LSeg(g1,n) by A1211,TOPREAL3:18;
                    hence thesis by A42,A1185,A1209;
                  end;
                  suppose
                    len g1<m+1;
                    then
A1212:              len g1<=m by NAT_1:13;
                    then reconsider m1=m-len g1 as Element of NAT by INT_1:5;
                    now
                      per cases;
                      suppose
A1213:                  m=len g1;
A1214:                  LSeg(g,m) c= LSeg(f,k)
                        proof
                          let x be object;
                          assume x in LSeg(g,m);
                          then consider px be Point of TOP-REAL 2 such that
A1215:                    px =x & px`2=ppi`2 & ppi`1<=px`1 and
A1216:                    px`1<=qq`1 by A1204,A1213;
                          pj`1>=px`1 by A1195,A1216,XXREAL_0:2;
                          hence thesis by A1099,A1215;
                        end;
                        n<=len g1 by A1185,A1202,A1213,XXREAL_0:2;
                        then
A1217:                  n in dom g1 by A1190,FINSEQ_3:25;
                        now
                          1<len g1 by A1185,A1198,A1213,XXREAL_0:2;
                          then
A1218:                    1+1<=len g1 by NAT_1:13;
                          assume k=1;
                          hence contradiction by A52,A1218,TOPREAL1:23;
                        end;
                        then 1<k by A24,XXREAL_0:1;
                        then
A1219:                  L~f1 /\ LSeg(f,k) ={f/.k} by A3,A6,A7,GOBOARD2:4;
A1220:                  n+1 in dom g1 by A1185,A1198,A1213,FINSEQ_3:25;
                        then
A1221:                  LSeg(g,n)=LSeg(g1,n) by A1217,TOPREAL3:18;
                        then LSeg(g,n) in l1 by A1185,A1190,A1213;
                        then LSeg(g,n) c= L~f1 by A44,ZFMISC_1:74;
                        then
A1222:                  LSeg(g,n) /\ LSeg(g,m) c= {f/.k} by A1219,A1214,
XBOOLE_1:27;
                        now
                          set x = the Element of LSeg(g,n)/\ LSeg(g,m);
                          assume
A1223:                    LSeg(g,n)/\ LSeg(g,m)<>{};
                          then
A1224:                    x in LSeg(g,n) by XBOOLE_0:def 4;
                          x in {f/.k} by A1222,A1223;
                          then
A1225:                    x=f/.k by TARSKI:def 1;
                          f/.k=g1/.len g1 by A27,A14,A51,A46,FINSEQ_4:71;
                          hence
                          contradiction by A40,A41,A42,A1185,A1213,A1217,A1220
,A1221,A1224,A1225,GOBOARD2:2;
                        end;
                        hence thesis;
                      end;
                      suppose
                        m<>len g1;
                        then
A1226:                  len g1<m by A1212,XXREAL_0:1;
                        then len g1+1<= m by NAT_1:13;
                        then
A1227:                  1<=m1 by XREAL_1:19;
                        m+1=m1+1+len g1;
                        then
A1228:                  m1+1 <= len g2 by A1196,A1203,XREAL_1:6;
                        m = m1+len g1;
                        then
A1229:                  LSeg(g,m)=LSeg(g2,m1) by A1196,A1226,GOBOARD2:5;
                        then LSeg(g,m) in l2 by A1227,A1228;
                        then
A1230:                  LSeg(g,m) c= L~g2 by ZFMISC_1:74;
                        now
                          per cases;
                          suppose
A1231:                      n+1<=len g1;
                            then n<=len g1 by A1202,XXREAL_0:2;
                            then
A1232:                      n in dom g1 by A1190,FINSEQ_3:25;
                            n+1 in dom g1 by A1198,A1231,FINSEQ_3:25;
                            then LSeg(g,n)=LSeg(g1,n) by A1232,TOPREAL3:18;
                            then LSeg(g,n) in l1 by A1190,A1231;
                            then LSeg(g,n) c= L~g1 by ZFMISC_1:74;
                            then LSeg(g,n) /\ LSeg(g,m) = {} by A1181,A1230,
XBOOLE_1:3,27;
                            hence thesis;
                          end;
                          suppose
                            len g1<n+1;
                            then
A1233:                      len g1<=n by NAT_1:13;
                            then reconsider n1=n-len g1 as Element of NAT by
INT_1:5;
A1234:                      n - len g1 + 1 = n + 1 - len g1;
A1235:                      n = n1 + len g1;
                            now
                              per cases;
                              suppose
A1236:                          len g1=n;
                                now
                                  reconsider q1=g2/.m1, q2=g2/.(m1+1) as Point
                                  of TOP-REAL 2;

set x = the Element of LSeg(g,n) /\ LSeg(g,m);
                                  set q1l={v where v is Point of TOP-REAL 2: v
                                  `2=ppi`2 & q1`1<=v`1 & v`1<=q2`1};
A1237:                            q1=|[q1`1,q1 `2]| & q2=|[q2`1,q2`2]|
                                  by EUCLID:53;
                                  assume
A1238:                            LSeg(g,n) /\ LSeg(g,m)<>{};
                                  then
A1239:                            x in LSeg(g,m) by XBOOLE_0:def 4;
                                  x in LSeg(g,n) by A1238,XBOOLE_0:def 4;
                                  then
A1240:                            ex qx be Point of TOP-REAL 2 st qx=x
& qx`2=ppi`2 & ppi`1<=qx`1 & qx`1<=qq`1 by A1204,A1236;
A1241:                            m1 in dom g2 by A1227,A1228,SEQ_4:134;
                                  then
A1242:                            q1 `2=ppi`2 by A1027;
A1243:                            m1+1 in dom g2 by A1227,A1228,SEQ_4:134;
                                  then
A1244:                            q2`2=ppi`2 by A1027;
                                  m1<m1+1 by NAT_1:13;
                                  then
A1245:                            q1`1<q2`1 by A1045,A1241,A1243;
                                  LSeg(g2,m1)=LSeg(q1,q2) by A1227,A1228,
TOPREAL1:def 3
                                    .=q1l by A1242,A1244,A1245,A1237,
TOPREAL3:10;
                                  then
A1246:                            ex qy be Point of TOP-REAL 2 st qy=x
& qy`2=ppi`2 & q1`1<=qy`1 & qy`1<=q2`1 by A1229,A1239;
                                  m1 > n1 + 1 & n1 + 1 >= 1 by A1185,A1234,
NAT_1:11,XREAL_1:9;
                                  then m1 > 1 by XXREAL_0:2;
                                  then qq`1<q1`1 by A1045,A1192,A1241;
                                  hence contradiction by A1240,A1246,XXREAL_0:2
;
                                end;
                                hence thesis;
                              end;
                              suppose
                                n<>len g1;
                                then len g1<n by A1233,XXREAL_0:1;
                                then
A1247:                          LSeg(g,n)=LSeg(g2,n1) by A1199,A1235,GOBOARD2:5
;
                                m1>n1+1 by A1185,A1234,XREAL_1:9;
                                hence thesis by A1071,A1229,A1247;
                              end;
                            end;
                            hence thesis;
                          end;
                        end;
                        hence thesis;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                hence thesis;
              end;
              hence g is s.n.c. by GOBOARD2:1;
              now
                set p=g1/.len g1, q=g2/.1;
                i1+1<=j1 by A1011,NAT_1:13;
                then 1<=l by XREAL_1:19;
                then 1 in dom g2 by A1017,FINSEQ_3:25;
                then q`2=ppi`2 by A1027;
                hence p`1=q`1 or p`2=q`2 by A27,A14,A51,A46,A29,FINSEQ_4:71;
              end;
              hence g is special by A43,A1036,GOBOARD2:8;
              thus L~g=L~f
              proof
                set lg = {LSeg(g,i): 1<=i & i+1 <= len g}, lf = {LSeg(f,j): 1
                <=j & j+1 <= len f};
A1248:          len g = len g1 + len g2 by FINSEQ_1:22;
A1249:          now
                  let j;
                  assume that
A1250:            len g1<=j and
A1251:            j<=len g;
                  reconsider w = j-len g1 as Element of NAT by A1250,INT_1:5;
                  let p such that
A1252:            p=g/.j;
                  now
                    per cases;
                    suppose
A1253:                j=len g1;
                      1<=len g1 by A24,A14,A47,XXREAL_0:2;
                      then len g1 in dom g1 by FINSEQ_3:25;
                      then
A1254:                g/.len g1 = f1/.len f1 by A46,FINSEQ_4:68
                        .= G*(i1,i2) by A27,A14,A51,A29,FINSEQ_4:71;
                      hence p`2=G*(i1,i2)`2 by A1252,A1253;
                      thus G*(i1,i2)`1<=p`1 & p`1<=G* (j1,i2)`1 by A66,A23,A69
,A65,A60,A1011,A1013,A1015,A1252,A1253,A1254,SEQM_3:def 1;
                      thus p in rng c1 by A66,A60,A1012,A1252,A1253,A1254,
PARTFUN2:2;
                    end;
                    suppose
                      j<>len g1;
                      then len g1 < j by A1250,XXREAL_0:1;
                      then len g1 + 1 <= j by NAT_1:13;
                      then
A1255:                1<=w by XREAL_1:19;
A1256:                w<=len g2 by A1248,A1251,XREAL_1:20;
                      then
A1257:                w in dom g2 by A1255,FINSEQ_3:25;
                      j = w + len g1;
                      then g/.j=g2/.w by A1255,A1256,SEQ_4:136;
                      hence p`2=ppi`2 & ppi`1<=p`1 & p`1<= pj`1 & p in rng c1
                      by A1027,A1252,A1257;
                    end;
                  end;
                  hence p`2=ppi`2 & ppi`1<=p`1 & p`1<=pj`1 & p in rng c1;
                end;
                thus L~g c= L~f
                proof
                  let x be object;
                  assume x in L~g;
                  then consider X be set such that
A1258:            x in X and
A1259:            X in lg by TARSKI:def 4;
                  consider i such that
A1260:            X=LSeg(g,i) and
A1261:            1<=i and
A1262:            i+1 <= len g by A1259;
                  now
                    per cases;
                    suppose
A1263:                i+1 <= len g1;
                      i<=i+1 by NAT_1:11;
                      then i<=len g1 by A1263,XXREAL_0:2;
                      then
A1264:                i in dom g1 by A1261,FINSEQ_3:25;
                      1<=i+1 by NAT_1:11;
                      then i+1 in dom g1 by A1263,FINSEQ_3:25;
                      then X=LSeg(g1,i) by A1260,A1264,TOPREAL3:18;
                      then X in {LSeg(g1,j): 1<=j & j+1 <= len g1} by A1261
,A1263;
                      then
A1265:                x in L~f1 by A44,A1258,TARSKI:def 4;
                      L~f1 c= L~f by TOPREAL3:20;
                      hence thesis by A1265;
                    end;
                    suppose
A1266:                i+1 > len g1;
                      reconsider q1=g/.i, q2=g/.(i+1) as Point of TOP-REAL 2;
A1267:                i<=len g by A1262,NAT_1:13;
A1268:                len g1<=i by A1266,NAT_1:13;
                      then
A1269:                q1`2=ppi`2 by A1249,A1267;
A1270:                q1`1<=pj`1 by A1249,A1268,A1267;
A1271:                ppi`1<=q1`1 by A1249,A1268,A1267;
                      q2`2=ppi`2 by A1249,A1262,A1266;
                      then
A1272:                q2=|[q2`1, q1`2]| by A1269,EUCLID:53;
A1273:                q2`1<=pj`1 by A1249,A1262,A1266;
A1274:                q1=|[q1`1,q1`2]| & LSeg(g,i)=LSeg(q2,q1) by A1261,A1262,
EUCLID:53,TOPREAL1:def 3;
A1275:                ppi`1<= q2`1 by A1249,A1262,A1266;
                      now
                        per cases by XXREAL_0:1;
                        suppose
                          q1`1>q2`1;
                          then LSeg(g,i)={p2: p2`2=q1`2 & q2`1<=p2`1 & p2`1
                          <=q1`1} by A1272,A1274,TOPREAL3:10;
                          then consider p2 such that
A1276:                    p2 =x & p2`2=q1`2 and
A1277:                    q2`1<=p2`1 & p2`1<=q1`1 by A1258,A1260;
                          ppi`1<=p2`1 & p2`1<=pj`1 by A1270,A1275,A1277,
XXREAL_0:2;
                          then
A1278:                    x in LSeg(f,k) by A1099,A1269,A1276;
                          LSeg(f,k) in lf by A3,A24;
                          hence thesis by A1278,TARSKI:def 4;
                        end;
                        suppose
                          q1`1=q2`1;
                          then LSeg(g,i)={q1} by A1272,A1274,RLTOPSP1:70;
                          then x=q1 by A1258,A1260,TARSKI:def 1;
                          then
A1279:                    x in LSeg(f,k) by A1099,A1269,A1271,A1270;
                          LSeg(f,k) in lf by A3,A24;
                          hence thesis by A1279,TARSKI:def 4;
                        end;
                        suppose
                          q1`1<q2`1;
                          then LSeg(g,i)= {p1: p1`2=q1`2 & q1`1<=p1`1 & p1`1
                          <=q2`1} by A1272,A1274,TOPREAL3:10;
                          then consider p2 such that
A1280:                    p2 =x & p2`2=q1`2 and
A1281:                    q1`1<=p2`1 & p2`1<=q2`1 by A1258,A1260;
                          ppi`1<=p2`1 & p2`1<=pj`1 by A1271,A1273,A1281,
XXREAL_0:2;
                          then
A1282:                    x in LSeg(f,k) by A1099,A1269,A1280;
                          LSeg(f,k) in lf by A3,A24;
                          hence thesis by A1282,TARSKI:def 4;
                        end;
                      end;
                      hence thesis;
                    end;
                  end;
                  hence thesis;
                end;
                let x be object;
                assume x in L~f;
                then
A1283:          x in L~f1 \/ LSeg(f,k) by A3,A13,GOBOARD2:3;
                now
                  per cases by A1283,XBOOLE_0:def 3;
                  suppose
A1284:              x in L~f1;
                    L~g1 c= L~g by GOBOARD2:6;
                    hence thesis by A44,A1284;
                  end;
                  suppose
                    x in LSeg(f,k);
                    then consider p1 such that
A1285:              p1=x and
A1286:              p1`2=ppi`2 and
A1287:              ppi`1<=p1`1 and
A1288:              p1`1<=pj`1 by A1099;
                    defpred P2[Nat] means len g1<=$1 & $1<=len g & for q st q=
                    g/.$1 holds q`1<=p1`1;
A1289:              now
                      reconsider n=len g1 as Nat;
                      take n;
                      thus P2[n]
                      proof
                        thus len g1<=n & n<=len g by A1248,XREAL_1:31;
                        1<=len g1 by A24,A14,A47,XXREAL_0:2;
                        then
A1290:                  len g1 in dom g1 by FINSEQ_3:25;
                        let q;
                        assume q=g/.n;
                        then q=f1/.len f1 by A46,A1290,FINSEQ_4:68
                          .=G*(i1,i2) by A27,A14,A51,A29,FINSEQ_4:71;
                        hence thesis by A1287;
                      end;
                    end;
A1291:              for n be Nat holds P2[n] implies n<=len g;
                    consider ma be Nat such that
A1292:              P2[ma] & for n be Nat st P2[n] holds n<=ma from
                    NAT_1:sch 6 (A1291,A1289);
                    reconsider ma as Nat;
                    now
                      per cases;
                      suppose
A1293:                  ma=len g;
                        i1+1<=j1 by A1011,NAT_1:13;
                        then
A1294:                  1<=l by XREAL_1:19;
                        then len g1+1<=ma by A1017,A1248,A1293,XREAL_1:7;
                        then
A1295:                  len g1<=ma-1 by XREAL_1:19;
                        then 0+1<=ma by XREAL_1:19;
                        then reconsider m1=ma-1 as Element of NAT by INT_1:5;
                        reconsider q=g/.m1 as Point of TOP-REAL 2;
A1296:                  ma-1<=len g by A1293,XREAL_1:43;
                        then
A1297:                  q`2=ppi`2 by A1249,A1295;
A1298:                  q`1<=pj`1 by A1249,A1296,A1295;
                        set lq={e where e is Point of TOP-REAL 2: e`2=ppi`2 &
                        q`1<=e`1 & e`1<=pj`1};
A1299:                  i1+l=j1;
A1300:                  l in dom g2 by A1017,A1294,FINSEQ_3:25;
                        then
A1301:                  g/.ma=g2/.l by A1017,A1248,A1293,FINSEQ_4:69
                          .= pj by A1017,A1300,A1299;
                        then pj`1<=p1`1 by A1292;
                        then
A1302:                  p1`1=pj`1 by A1288,XXREAL_0:1;
                        1<=len g1 by A24,A14,A47,XXREAL_0:2;
                        then
A1303:                  1<=m1 by A1295,XXREAL_0:2;
A1304:                  m1+1=ma;
                        then q=|[q`1,q`2]| & LSeg(g,m1)=LSeg(q,pj) by A1293
,A1301,A1303,EUCLID:53,TOPREAL1:def 3;
                        then LSeg(g,m1)=lq by A1083,A1098,A1297,A1298,
TOPREAL3:10;
                        then
A1305:                  p1 in LSeg(g,m1) by A1286,A1302,A1298;
                        LSeg(g,m1) in lg by A1293,A1303,A1304;
                        hence thesis by A1285,A1305,TARSKI:def 4;
                      end;
                      suppose
                        ma<>len g;
                        then ma<len g by A1292,XXREAL_0:1;
                        then
A1306:                  ma+1<=len g by NAT_1:13;
                        reconsider qa=g/.ma, qa1=g/.(ma+1) as Point of
                        TOP-REAL 2;
                        set lma = {p2: p2`2=ppi`2 & qa`1<=p2`1 & p2`1<=qa1`1};
A1307:                  qa1=|[qa1 `1, qa1 `2]| by EUCLID:53;
A1308:                  qa`1<=p1`1 by A1292;
A1309:                  len g1<=ma+1 by A1292,NAT_1:13;
                        then
A1310:                  qa1 `2 = ppi`2 by A1249,A1306;
A1311:                  now
                          assume qa1`1<=p1`1;
                          then for q holds q=g/.(ma+1) implies q`1<=p1`1;
                          then ma+1<=ma by A1292,A1306,A1309;
                          hence contradiction by XREAL_1:29;
                        end;
A1312:                  qa`2=ppi`2 & qa=|[qa`1,qa`2]| by A1249,A1292,EUCLID:53;
A1313:                  1<=ma by A24,A14,A47,A1292,NAT_1:13;
                        then LSeg(g,ma)=LSeg(qa,qa1) by A1306,TOPREAL1:def 3
                          .= lma by A1308,A1311,A1310,A1312,A1307,TOPREAL3:10
,XXREAL_0:2;
                        then
A1314:                  x in LSeg(g,ma) by A1285,A1286,A1308,A1311;
                        LSeg(g,ma) in lg by A1306,A1313;
                        hence thesis by A1314,TARSKI:def 4;
                      end;
                    end;
                    hence thesis;
                  end;
                end;
                hence thesis;
              end;
              1<=len g1 by A24,A14,A47,XXREAL_0:2;
              then 1 in dom g1 by FINSEQ_3:25;
              hence g/.1=f1/.1 by A45,FINSEQ_4:68
                .=f/.1 by A27,A25,FINSEQ_4:71;
A1315:        len g=len g1 + l by A1017,FINSEQ_1:22;
              i1+1<=j1 by A1011,NAT_1:13;
              then
A1316:        1<=l by XREAL_1:19;
              then
A1317:        l in dom g2 by A1017,FINSEQ_3:25;
              hence g/.len g=g2/.l by A1315,FINSEQ_4:69
                .=G*(i1+l,i2) by A1017,A1317
                .=f/.len f by A3,A21,A698;
              thus len f<=len g by A3,A14,A47,A1316,A1315,XREAL_1:7;
            end;
          end;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
  end;
A1318: P[0]
  proof
    let f such that
A1319: len f=0 and
A1320: ( for n st n in dom f ex i,j st [i,j] in Indices G & f/.n=G*(i,j))
    & f is one-to-one unfolded s.n.c. special;
    take g=f;
    f= {} by A1319;
    then
    for n holds n in dom g & n+1 in dom g implies for m,k,i,j st [m,k] in
Indices G & [i,j] in Indices G & g/.n=G*(m,k) & g/.(n+1)=G*(i,j) holds |.m-i.|
    +|.k-j.|=1;
    hence thesis by A1320,GOBOARD1:def 9;
  end;
  for k holds P[k] from NAT_1:sch 2(A1318,A1);
  hence thesis;
end;
