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
  E-bound L~Cage(C,n) + W-bound L~Cage(C,n) = E-bound L~Cage(C,m) +
  W-bound L~Cage(C,m)
proof
  thus E-bound L~Cage(C,n) + W-bound L~Cage(C,n) = E-bound C + (E-bound C -
  W-bound C)/(2|^n) + W-bound L~Cage(C,n) by Th64
    .= E-bound C + (E-bound C - W-bound C)/(2|^n) + (W-bound C - (E-bound C
  - W-bound C)/(2|^n)) by Th62
    .= E-bound C + (E-bound C - W-bound C)/(2|^m) + (W-bound C - (E-bound C
  - W-bound C)/(2|^m))
    .= E-bound C + (E-bound C - W-bound C)/(2|^m) + W-bound L~Cage(C,m) by Th62
    .= E-bound L~Cage(C,m) + W-bound L~Cage(C,m) by Th64;
end;
