reserve p for Point of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  v, v1,v2 for FinSequence of REAL,
  r,s for Real,
  n,m,i,j,k for Nat,
  x for set;
reserve G for Go-board;
reserve D for set,
  f for FinSequence of D,
  M for Matrix of D;
reserve f for FinSequence of TOP-REAL 2;

theorem
  f is_sequence_on G & f/.1 in rng Col(G,1) & f/.len f in rng Col(G,
  width G) & width G > 1 & 1<=len f implies ex g st g/.1 in rng Col(DelCol(G,
width G),1) & g/.len g in rng Col(DelCol(G,width G),width DelCol(G,width G)) &
  1<=len g & g is_sequence_on DelCol(G,width G) & rng g c= rng f
proof
  set D = DelCol(G,width G);
  assume that
A1: f is_sequence_on G and
A2: f/.1 in rng Col(G,1) and
A3: f/.len f in rng Col(G,width G) and
A4: width G > 1 and
A5: 1<=len f;
  consider k such that
A6: width G=k+1 and
A7: k>0 by A4,SEQM_3:43;
A8: width G in Seg width G by A4,FINSEQ_1:1;
A9: len D=len G by MATRIX_0:def 13;
A10: 0+1<=k by A7,NAT_1:13;
  then 1<width G by A6,NAT_1:13;
  then
A11: Col(D,1)=Col(G,1) by A6,A7,A8,Th6;
A12: dom G = Seg len G by FINSEQ_1:def 3;
  defpred P[Nat] means $1 in dom f & f/.$1 in rng Col(G,k);
  k<=k+1 by NAT_1:11;
  then ex m st P[m] by A1,A2,A3,A5,A6,A10,Th26;
  then
A13: ex m be Nat st P[m];
  consider m be Nat such that
A14: P[m] & for i be Nat st P[i] holds m<=i from NAT_1:sch 5(A13);
A15: width D = k by A6,A8,MATRIX_0:63;
  then width D<width G by A6,NAT_1:13;
  then
A16: Col(D,width D)=Col(G,width D) by A6,A10,A8,A15,Th6;
A17: dom D = Seg len D by FINSEQ_1:def 3;
A18: for i st P[i] holds m<=i by A14;
  reconsider m as Element of NAT by ORDINAL1:def 12;
A19: 1<=m by A14,FINSEQ_3:25;
  then
A20: 1 in Seg m by FINSEQ_1:1;
A21: Indices G = [:dom G,Seg width G:] by MATRIX_0:def 4;
  take t = f|m;
  m<=len f by A14,FINSEQ_3:25;
  then
A22: len t = m by FINSEQ_1:59;
  then len t in Seg m by A19,FINSEQ_1:1;
  hence
  t/.1 in rng Col(D,1) & t/.len t in rng Col(D,width D) & 1<=len t by A2,A14
,A15,A11,A16,A22,A20,FINSEQ_1:1,FINSEQ_4:71;
A23: dom t=Seg len t by FINSEQ_1:def 3;
A24: Indices D = [:dom D,Seg width D:] by MATRIX_0:def 4;
A25: now
    k<=k+1 by NAT_1:11;
    then
A26: k in Seg width G by A6,A10,FINSEQ_1:1;
    let n;
    assume
A27: n in dom t;
    then
A28: n<=m by A22,FINSEQ_3:25;
A29: n in dom f by A14,A22,A23,A27,FINSEQ_4:71;
    then consider i,j such that
A30: [i,j] in Indices G and
A31: f/.n=G*(i,j) by A1;
A32: j in Seg width G by A21,A30,ZFMISC_1:87;
    then
A33: 1<=j by FINSEQ_1:1;
    take i,j;
A34: len Col(G,j) = len G & dom Col(G,j)=Seg len Col(G,j) by FINSEQ_1:def 3
,MATRIX_0:def 8;
A35: i in dom G by A21,A30,ZFMISC_1:87;
    then Col(G,j).i=G*(i,j) by MATRIX_0:def 8;
    then f/.n in rng Col(G,j) by A12,A31,A35,A34,FUNCT_1:def 3;
    then j<=k by A1,A2,A18,A29,A28,A32,A26,Th27;
    then
A36: j in Seg k by A33,FINSEQ_1:1;
    hence [i,j] in Indices D by A9,A12,A17,A15,A24,A35,ZFMISC_1:87;
    thus t/.n = G*(i,j) by A14,A22,A23,A27,A31,FINSEQ_4:71
      .= D*(i,j) by A6,A7,A35,A36,Th13;
  end;
  now
    let n;
    assume that
A37: n in dom t and
A38: n+1 in dom t;
A39: n in dom f & n+1 in dom f by A14,A22,A23,A37,A38,FINSEQ_4:71;
    let i1,i2,j1,j2 be Nat;
    assume that
A40: [i1,i2] in Indices D and
A41: [j1,j2] in Indices D and
A42: t/.n=D*(i1,i2) and
A43: t/.(n+1)=D*(j1,j2);
A44: i1 in dom D & i2 in Seg k by A15,A24,A40,ZFMISC_1:87;
A45: j1 in dom D & j2 in Seg k by A15,A24,A41,ZFMISC_1:87;
    t/.n=f/.n by A14,A22,A23,A37,FINSEQ_4:71;
    then
A46: f/.n=G*(i1,i2) by A6,A7,A9,A12,A17,A42,A44,Th13;
    k<=k+1 by NAT_1:11;
    then
A47: Seg k c= Seg width G by A6,FINSEQ_1:5;
    then
A48: [j1,j2] in Indices G by A9,A21,A12,A17,A45,ZFMISC_1:87;
    t/.(n+1)=f/.(n+1) by A14,A22,A23,A38,FINSEQ_4:71;
    then
A49: f/.(n+1)=G*(j1,j2) by A6,A7,A9,A12,A17,A43,A45,Th13;
    [i1,i2] in Indices G by A9,A21,A12,A17,A44,A47,ZFMISC_1:87;
    hence |.i1-j1.|+|.i2-j2.| = 1 by A1,A39,A46,A49,A48;
  end;
  hence t is_sequence_on D by A25;
  t = f|Seg m by FINSEQ_1:def 16;
  hence thesis by RELAT_1:70;
end;
