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/^n) & n <= len f & f is_sequence_on G implies
left_cell(f,k+n,G) = left_cell(f/^n,k,G) & right_cell(f,k+n,G) = right_cell(f/^
  n,k,G)
proof
  set g = (f/^n);
  assume that
A1: 1 <= k and
A2: k+1 <= len g and
A3: n <= len f and
A4: f is_sequence_on G;
A5: len g = len f - n & k+1+n <= (len g)+n by A2,A3,RFINSEQ:def 1,XREAL_1:6;
  k in dom g by A1,A2,SEQ_4:134;
  then
A6: g/.k = f/.(k+n) by FINSEQ_5:27;
  set lf = left_cell(f,k+n,G), lfn = left_cell(g,k,G), rf = right_cell(f,k+n,G
  ), rfn = right_cell(g,k,G);
A7: k+1+n = k+n+1 & 1 <= k+n by A1,NAT_1:12;
  k+1 in dom g by A1,A2,SEQ_4:134;
  then
A8: g/.(k+1) = f/.(k+1+n) by FINSEQ_5:27;
A9: g is_sequence_on G by A4,JORDAN8:2;
  then consider i1,j1,i2,j2 being Nat such that
A10: [i1,j1] in Indices G & g/.k = G*(i1,j1) & [i2,j2] in Indices G & g/.
  ( k+1) = G*(i2,j2) and
A11: 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,JORDAN8:3;
A12: j1+1 > j1 & j2+1 > j2 by NAT_1:13;
A13: i1+1 > i1 & i2+1 > i2 by NAT_1:13;
  now
    per cases by A11;
    suppose
A14:  i1 = i2 & j1+1 = j2;
      hence lf = cell(G,i1-'1,j1) by A4,A10,A12,A6,A8,A5,A7,Def2
        .= lfn by A1,A2,A9,A10,A12,A14,Def2;
      thus rf = cell(G,i1,j1) by A4,A10,A12,A6,A8,A5,A7,A14,Def1
        .= rfn by A1,A2,A9,A10,A12,A14,Def1;
    end;
    suppose
A15:  i1+1 = i2 & j1 = j2;
      hence lf = cell(G,i1,j1) by A4,A10,A13,A6,A8,A5,A7,Def2
        .= lfn by A1,A2,A9,A10,A13,A15,Def2;
      thus rf = cell(G,i1,j1-'1) by A4,A10,A13,A6,A8,A5,A7,A15,Def1
        .= rfn by A1,A2,A9,A10,A13,A15,Def1;
    end;
    suppose
A16:  i1 = i2+1 & j1 = j2;
      hence lf = cell(G,i2,j2-'1) by A4,A10,A13,A6,A8,A5,A7,Def2
        .= lfn by A1,A2,A9,A10,A13,A16,Def2;
      thus rf = cell(G,i2,j2) by A4,A10,A13,A6,A8,A5,A7,A16,Def1
        .= rfn by A1,A2,A9,A10,A13,A16,Def1;
    end;
    suppose
A17:  i1 = i2 & j1 = j2+1;
      hence lf = cell(G,i1,j2) by A4,A10,A12,A6,A8,A5,A7,Def2
        .= lfn by A1,A2,A9,A10,A12,A17,Def2;
      thus rf = cell(G,i1-'1,j2) by A4,A10,A12,A6,A8,A5,A7,A17,Def1
        .= rfn by A1,A2,A9,A10,A12,A17,Def1;
    end;
  end;
  hence thesis;
end;
