reserve i,i1,i2,i9,i19,j,j1,j2,j9,j19,k,k1,k2,l,m,n for Nat;
reserve r,s,r9,s9 for Real;
reserve D for non empty set, f for FinSequence of D;
reserve f for FinSequence of TOP-REAL 2, G for Go-board;

theorem
  1 <= k & k+1 <= len f & f is_sequence_on G implies left_cell(f,k,G) /\
  right_cell(f,k,G) = LSeg(f,k)
proof
  assume that
A1: 1 <= k and
A2: k+1 <= len f and
A3: f is_sequence_on G;
  k+1 >= 1 by NAT_1:11;
  then
A4: k+1 in dom f by A2,FINSEQ_3:25;
  then consider i2,j2 being Nat such that
A5: [i2,j2] in Indices G and
A6: f/.(k+1) = G*(i2,j2) by A3,GOBOARD1:def 9;
A7: 1 <= j2 by A5,MATRIX_0:32;
A8: i2 <= len G by A5,MATRIX_0:32;
A9: 1 <= i2 by A5,MATRIX_0:32;
A10: j2 <= width G by A5,MATRIX_0:32;
  k <= k+1 by NAT_1:11;
  then k <= len f by A2,XXREAL_0:2;
  then
A11: k in dom f by A1,FINSEQ_3:25;
  then consider i1,j1 being Nat such that
A12: [i1,j1] in Indices G and
A13: f/.k = G*(i1,j1) by A3,GOBOARD1:def 9;
A14: 0+1 <= j1 by A12,MATRIX_0:32;
  then j1 > 0;
  then consider j being Nat such that
A15: j+1 = j1 by NAT_1:6;
A16: |.i1-i2.|+|.j1-j2.| = 1 by A3,A11,A12,A13,A4,A5,A6,GOBOARD1:def 9;
A17: now
    per cases by A16,SEQM_3:42;
    case that
A18:  |.i1-i2.| = 1 and
A19:  j1 = j2;
      i1-i2 = 1 or -(i1-i2) = 1 by A18,ABSVALUE:def 1;
      hence i1 = i2+1 or i1+1 = i2;
      thus j1 = j2 by A19;
    end;
    case that
A20:  i1 = i2 and
A21:  |.j1-j2.| = 1;
      j1-j2 = 1 or -(j1-j2) = 1 by A21,ABSVALUE:def 1;
      hence j1 = j2+1 or j1+1 = j2;
      thus i1 = i2 by A20;
    end;
  end;
A22: j1-'1=j by A15,NAT_D:34;
A23: j1 <= width G by A12,MATRIX_0:32;
  then
A24: j < width G by A15,NAT_1:13;
A25: 0+1 <= i1 by A12,MATRIX_0:32;
  then i1 > 0;
  then consider i being Nat such that
A26: i+1 = i1 by NAT_1:6;
A27: i1 <= len G by A12,MATRIX_0:32;
  then
A28: i < len G by A26,NAT_1:13;
A29: i1-'1 = i by A26,NAT_D:34;
  reconsider i,j as Nat;
  per cases by A17;
  suppose
A30: i1 = i2 & j1+1 = j2;
    then
A31: right_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A12,A13,A5,A6,Th15;
    j1 < width G & left_cell(f,k,G) = cell(G,i,j1) by A1,A2,A3,A12,A13,A5,A6
,A10,A29,A30,Th14,NAT_1:13;
    hence left_cell(f,k,G) /\ right_cell(f,k,G) = LSeg(G*(i1,j1),G*(i1,j1+1))
    by A14,A26,A28,A31,GOBOARD5:25
      .= LSeg(f,k) by A1,A2,A13,A6,A30,TOPREAL1:def 3;
  end;
  suppose
A32: i1+1 = i2 & j1 = j2;
    then
A33: right_cell(f,k,G) = cell(G,i1,j) by A1,A2,A3,A12,A13,A5,A6,A22,Th17;
    i1 < len G & left_cell(f,k,G) = cell(G,i1,j1) by A1,A2,A3,A12,A13,A5,A6,A8
,A32,Th16,NAT_1:13;
    hence left_cell(f,k,G) /\ right_cell(f,k,G) = LSeg(G*(i1,j1),G*(i1+1,j1))
    by A25,A15,A24,A33,GOBOARD5:26
      .= LSeg(f,k) by A1,A2,A13,A6,A32,TOPREAL1:def 3;
  end;
  suppose
A34: i1 = i2+1 & j1 = j2;
    then
A35: right_cell(f,k,G) = cell(G,i2,j1) by A1,A2,A3,A12,A13,A5,A6,Th19;
    i2 < len G & left_cell(f,k,G) = cell(G,i2,j) by A1,A2,A3,A12,A13,A5,A6,A27
,A22,A34,Th18,NAT_1:13;
    hence left_cell(f,k,G) /\ right_cell(f,k,G) = LSeg(G*(i2+1,j1),G*(i2,j1))
    by A9,A15,A24,A35,GOBOARD5:26
      .= LSeg(f,k) by A1,A2,A13,A6,A34,TOPREAL1:def 3;
  end;
  suppose
A36: i1 = i2 & j1 = j2+1;
    then
A37: right_cell(f,k,G) = cell(G,i,j2) by A1,A2,A3,A12,A13,A5,A6,A29,Th21;
    j2 < width G & left_cell(f,k,G) = cell(G,i1,j2) by A1,A2,A3,A12,A13,A5,A6
,A23,A36,Th20,NAT_1:13;
    hence left_cell(f,k,G) /\ right_cell(f,k,G) = LSeg(G*(i1,j2+1),G*(i1,j2))
    by A7,A26,A28,A37,GOBOARD5:25
      .= LSeg(f,k) by A1,A2,A13,A6,A36,TOPREAL1:def 3;
  end;
end;
