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
  i < j implies S-bound L~Cage(C,i) < S-bound L~Cage(C,j)
proof
  assume
A1: i < j;
  defpred P[Nat] means
i < $1 implies S-bound L~Cage(C,i) < S-bound
  L~Cage(C,$1);
A2: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume
A3: P[n];
    set j = n + 1, a = N-bound C, s = S-bound C;
A4: s-(a-s)/(2|^j) - (s-(a-s)/(2|^n)) = s +- (a-s)/(2|^j) - s + (a-s)/(2 |^n)
      .= - (a-s)/(2|^n*2) + (a-s)/(2|^n) by NEWTON:6
      .= (-(a-s))/(2|^n*2) + (a-s)/(2|^n) by XCMPLX_1:187
      .= (-(a-s))/(2|^n*2) + (a-s)*2/(2|^n*2) by XCMPLX_1:91
      .= (-(a-s) + (a-s)*2) / (2|^n*2) by XCMPLX_1:62
      .= (a-s)/(2|^n*2);
    2|^n > 0 by NEWTON:83;
    then
A5: 2|^n*2 > 0 * 2 by XREAL_1:68;
A6: S-bound L~Cage(C,n) = s-(a-s)/(2|^n) & S-bound L~Cage(C,j) = s-(a-s)/(
    2|^j) by Th63;
    a - s > 0 by SPRECT_1:32,XREAL_1:50;
    then 0 < s-(a-s)/(2|^j) - (s-(a-s)/(2|^n)) by A5,A4,XREAL_1:139;
    then
A7: S-bound L~Cage(C,n) < S-bound L~Cage(C,j) by A6,XREAL_1:47;
    assume i < n+1;
    then i <= n by NAT_1:13;
    hence thesis by A3,A7,XXREAL_0:1,2;
  end;
A8: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A8,A2);
  hence thesis by A1;
end;
