reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem
  N-bound L~Cage(C,n) + S-bound L~Cage(C,n) = N-bound L~Cage(C,m) +
  S-bound L~Cage(C,m)
proof
  thus N-bound L~Cage(C,n) + S-bound L~Cage(C,n) = N-bound C + (N-bound C -
  S-bound C)/(2|^n) + S-bound L~Cage(C,n) by JORDAN10:6
    .= N-bound C + (N-bound C - S-bound C)/(2|^n) + (S-bound C - (N-bound C
  - S-bound C)/(2|^n)) by Th63
    .= N-bound C + (N-bound C - S-bound C)/(2|^m) + (S-bound C - (N-bound C
  - S-bound C)/(2|^m))
    .= N-bound C + (N-bound C - S-bound C)/(2|^m) + S-bound L~Cage(C,m) by Th63
    .= N-bound L~Cage(C,m) + S-bound L~Cage(C,m) by JORDAN10:6;
end;
