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 & i in Seg width G & rng f misses rng Col(G,i) &
  width G > 1 implies f is_sequence_on DelCol(G,i)
proof
  set D = DelCol(G,i);
  assume that
A1: f is_sequence_on G and
A2: i in Seg width G and
A3: rng f misses rng Col(G,i) and
A4: width G > 1;
A5: len G = len D by MATRIX_0:def 13;
A6: Indices D = [:dom D,Seg width D:] by MATRIX_0:def 4;
A7: Indices G = [:dom G,Seg width G:] by MATRIX_0:def 4;
A8: dom G = Seg len G & dom D = Seg len D by FINSEQ_1:def 3;
  consider M be Nat such that
A9: width G = M+1 and
A10: M>0 by A4,SEQM_3:43;
A11: width D = M by A2,A9,MATRIX_0:63;
A12: now
    let n such that
A13: n in dom f & n+1 in dom f;
    let i1,i2,j1,j2 be Nat;
    assume that
A14: [i1,i2] in Indices D and
A15: [j1,j2] in Indices D and
A16: f/.n = D*(i1,i2) & f/.(n+1) = D*(j1,j2);
A17: i1 in dom D by A6,A14,ZFMISC_1:87;
A18: i2 in Seg width D by A6,A14,ZFMISC_1:87;
    then
A19: 1<=i2 by FINSEQ_1:1;
A20: i2<=M by A11,A18,FINSEQ_1:1;
    then 1<=i2+1 & i2+1<=M+1 by NAT_1:11,XREAL_1:6;
    then i2+1 in Seg(M+1) by FINSEQ_1:1;
    then
A21: [i1,i2+1] in Indices G by A5,A9,A8,A7,A17,ZFMISC_1:87;
A22: j1 in dom D by A6,A15,ZFMISC_1:87;
A23: j2 in Seg width D by A6,A15,ZFMISC_1:87;
    then
A24: 1<=j2 by FINSEQ_1:1;
    M<=M+1 by NAT_1:11;
    then
A25: Seg width D c= Seg width G by A9,A11,FINSEQ_1:5;
    then
A26: [j1,j2] in Indices G by A5,A8,A7,A22,A23,ZFMISC_1:87;
A27: j2<=M by A11,A23,FINSEQ_1:1;
    then 1<=j2+1 & j2+1<=M+1 by NAT_1:11,XREAL_1:6;
    then j2+1 in Seg(M+1) by FINSEQ_1:1;
    then
A28: [j1,j2+1] in Indices G by A5,A9,A8,A7,A22,ZFMISC_1:87;
A29: [i1,i2] in Indices G by A5,A8,A7,A17,A18,A25,ZFMISC_1:87;
    now
      per cases;
      case
        i2<i & j2<i;
        then f/.n=G*(i1,i2) & f/.(n+1)=G* (j1,j2) by A2,A5,A9,A10,A8,A16,A17
,A22,A19,A24,Th8;
        hence |.i1-j1.|+|.i2-j2.| = 1 by A1,A13,A29,A26;
      end;
      case
A30:    i<=i2 & j2<i;
        i2<=i2+1 by NAT_1:11;
        then i<=i2+1 by A30,XXREAL_0:2;
        then
A31:    j2<i2+1 by A30,XXREAL_0:2;
        then j2+1<=i2+1 by NAT_1:13;
        then
A32:    1<=i2+1-j2 by XREAL_1:19;
        f/.n=G*(i1,i2+1) & f/.(n+1)=G*(j1,j2) by A2,A5,A9,A8,A16,A17,A22,A20
,A24,A30,Th8,Th9;
        then
A33:    1=|.i1-j1.|+|.i2+1-j2.| by A1,A13,A26,A21;
        0<i2+1-j2 by A31,XREAL_1:50;
        then
A34:    |.i2+1-j2.| = i2+1-j2 by ABSVALUE:def 1;
        0<=|.i1-j1.| by COMPLEX1:46;
        then 0+(i2+1-j2)<=1 by A33,A34,XREAL_1:7;
        then i2+1-j2 = 1 by A32,XXREAL_0:1;
        hence contradiction by A30;
      end;
      case
A35:    i2<i & i<=j2;
        j2<=j2+1 by NAT_1:11;
        then i<=j2+1 by A35,XXREAL_0:2;
        then
A36:    i2<j2+1 by A35,XXREAL_0:2;
        then i2+1<=j2+1 by NAT_1:13;
        then
A37:    1<=j2+1-i2 by XREAL_1:19;
        f/.n=G*(i1,i2) & f/.(n+1)=G*(j1,j2+1) by A2,A5,A9,A8,A16,A17,A22,A19
,A27,A35,Th8,Th9;
        then
A38:    1=|.i1-j1.|+|.i2-(j2+1).| by A1,A13,A29,A28
          .=|.i1-j1.|+|.-(j2+1 -i2).|
          .=|.i1-j1.|+|.j2+1 -i2.| by COMPLEX1:52;
        0<j2+1-i2 by A36,XREAL_1:50;
        then
A39:    |.j2+1-i2.| = j2+1-i2 by ABSVALUE:def 1;
        0<=|.i1-j1.| by COMPLEX1:46;
        then 0+(j2+1-i2)<=1 by A38,A39,XREAL_1:7;
        then j2+1-i2 = 1 by A37,XXREAL_0:1;
        hence contradiction by A35;
      end;
      case
        i<=i2 & i<=j2;
        then f/.n=G*(i1,i2+1) & f/.(n+1)=G*(j1,j2+1) by A2,A5,A9,A10,A8,A16,A17
,A22,A20,A27,Th9;
        hence 1 = |.i1-j1.|+|.(i2+1)-(j2+1).| by A1,A13,A21,A28
          .= |.i1-j1.|+|.i2-j2.|;
      end;
    end;
    hence |.i1-j1.|+|.i2-j2.| = 1;
  end;
A40: 1<=i by A2,FINSEQ_1:1;
A41: i<=width G by A2,FINSEQ_1:1;
  now
    let n;
    assume
A42: n in dom f;
    then consider m,k such that
A43: [m,k] in Indices G and
A44: f/.n=G*(m,k) by A1;
    take m;
A45: m in dom G by A7,A43,ZFMISC_1:87;
A46: k in Seg width G by A7,A43,ZFMISC_1:87;
    then
A47: 1<=k by FINSEQ_1:1;
A48: k<=M+1 by A9,A46,FINSEQ_1:1;
    now
      per cases;
      suppose
A49:    k<i;
        take k;
        k<width G by A41,A49,XXREAL_0:2;
        then k<=M by A9,NAT_1:13;
        then k in Seg M by A47,FINSEQ_1:1;
        hence
        [m,k] in Indices D & f/.n=D*(m,k) by A2,A5,A9,A10,A11,A8,A6,A44,A45,A47
,A49,Th8,ZFMISC_1:87;
      end;
      suppose
        i<=k;
        then
A50:    i<k by A3,A42,A44,A45,MATRIX_0:43,XXREAL_0:1;
        then k-1 in NAT by A40,INT_1:5,XXREAL_0:2;
        then reconsider l = k-1 as Nat;

        take l;
A51:    l<=M by A48,XREAL_1:20;
        i+1<=k by A50,NAT_1:13;
        then
A52:    i<=k-1 by XREAL_1:19;
        then 1<=l by A40,XXREAL_0:2;
        then
A53:    l in Seg M by A51,FINSEQ_1:1;
        D*(m,l)=G*(m,l+1) by A2,A9,A40,A45,A52,A51,Th9;
        hence [m,l] in Indices D & f/.n=D*(m,l) by A5,A11,A8,A6,A44,A45,A53,
ZFMISC_1:87;
      end;
    end;
    hence ex k st [m,k] in Indices D & f/.n=D*(m,k);
  end;
  hence thesis by A12;
end;
