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 & k+1 <= n implies
left_cell(f,k,G) = left_cell(f|n,k,G) & right_cell(f,k,G) = right_cell(f|n,k,G)
proof
  assume that
A1: 1 <= k and
A2: k+1 <= len f and
A3: f is_sequence_on G and
A4: k+1 <= n;
  per cases;
  suppose
    len f <= n;
    hence thesis by FINSEQ_1:58;
  end;
  suppose
    n < len f;
    then
A5: len(f|n) = n by FINSEQ_1:59;
    then k in dom(f|n) by A1,A4,SEQ_4:134;
    then
A6: (f|n)/.k = f/.k by FINSEQ_4:70;
    k+1 in dom(f|n) by A1,A4,A5,SEQ_4:134;
    then
A7: (f|n)/.(k+1) = f/.(k+1) by FINSEQ_4:70;
    set lf = left_cell(f,k,G), lfn = left_cell(f|n,k,G), rf = right_cell(f,k,G
    ), rfn = right_cell(f|n,k,G);
A8: f|n is_sequence_on G by A3,GOBOARD1:22;
    consider i1,j1,i2,j2 being Nat such that
A9: [i1,j1] in Indices G & f/.k = G*(i1,j1) & [i2,j2] in Indices G & f
    /.( k+1) = G*(i2,j2) and
A10: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2
    or i1 = i2 & j1 = j2+1 by A1,A2,A3,JORDAN8:3;
A11: j1+1 > j1 & j2+1 > j2 by NAT_1:13;
A12: i1+1 > i1 & i2+1 > i2 by NAT_1:13;
    now
      per cases by A10;
      suppose
A13:    i1 = i2 & j1+1 = j2;
        hence lf = cell(G,i1-'1,j1) by A1,A2,A3,A9,A11,Def2
          .= lfn by A1,A4,A9,A11,A8,A5,A6,A7,A13,Def2;
        thus rf = cell(G,i1,j1) by A1,A2,A3,A9,A11,A13,Def1
          .= rfn by A1,A4,A9,A11,A8,A5,A6,A7,A13,Def1;
      end;
      suppose
A14:    i1+1 = i2 & j1 = j2;
        hence lf = cell(G,i1,j1) by A1,A2,A3,A9,A12,Def2
          .= lfn by A1,A4,A9,A12,A8,A5,A6,A7,A14,Def2;
        thus rf = cell(G,i1,j1-'1) by A1,A2,A3,A9,A12,A14,Def1
          .= rfn by A1,A4,A9,A12,A8,A5,A6,A7,A14,Def1;
      end;
      suppose
A15:    i1 = i2+1 & j1 = j2;
        hence lf = cell(G,i2,j2-'1) by A1,A2,A3,A9,A12,Def2
          .= lfn by A1,A4,A9,A12,A8,A5,A6,A7,A15,Def2;
        thus rf = cell(G,i2,j2) by A1,A2,A3,A9,A12,A15,Def1
          .= rfn by A1,A4,A9,A12,A8,A5,A6,A7,A15,Def1;
      end;
      suppose
A16:    i1 = i2 & j1 = j2+1;
        hence lf = cell(G,i1,j2) by A1,A2,A3,A9,A11,Def2
          .= lfn by A1,A4,A9,A11,A8,A5,A6,A7,A16,Def2;
        thus rf = cell(G,i1-'1,j2) by A1,A2,A3,A9,A11,A16,Def1
          .= rfn by A1,A4,A9,A11,A8,A5,A6,A7,A16,Def1;
      end;
    end;
    hence thesis;
  end;
end;
