reserve p,q for Point of TOP-REAL 2,
  i,i1,i2,j,j1,j2,k for Nat,
  r,s for Real,
  G for Matrix of TOP-REAL 2;
reserve f for standard special_circular_sequence;

theorem Th29:
  1 <= k & k+1 <= len f & [i,j+1] in Indices GoB f & [i+1,j+1] in
  Indices GoB f &
  f/.k = (GoB f)*(i+1,j+1) & f/.(k+1) = (GoB f)*(i,j+1) implies
  left_cell(f,k) = cell(GoB f,i,j) & right_cell(f,k) = cell(GoB f,i,j+1)
proof
  assume that
A1: 1 <= k and
A2: k+1 <= len f and
A3: [i,j+1] in Indices GoB f and
A4: [i+1,j+1] in Indices GoB f and
A5: f/.k = (GoB f)*(i+1,j+1) and
A6: f/.(k+1) = (GoB f)*(i,j+1);
A7: i < i+1 by XREAL_1:29;
A8: i+1 <= i+1+1 by NAT_1:11;
  hence left_cell(f,k) = cell(GoB f,i,j+1-'1) by A1,A2,A3,A4,A5,A6,A7,Def7
    .= cell(GoB f,i,j) by NAT_D:34;
  thus thesis by A1,A2,A3,A4,A5,A6,A7,A8,Def6;
end;
