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 Th63:
  S-bound L~Cage(C,n) = S-bound C - (N-bound C - S-bound C)/(2|^n)
proof
  set a = N-bound C, s = S-bound C, w = W-bound C, e = E-bound C, f = Cage(C,n
  ), G = Gauge(C,n);
  consider p, q being Nat such that
A1: 1 <= p & p < len f and
A2: 1 <= q & q <= len G and
A3: f/.p = G*(q,1) by Th56;
  f/.p in S-most L~f by A1,A2,A3,Th60;
  then
A4: (f/.p)`2 = (S-min L~f)`2 by PSCOMP_1:55;
  4 <= len G by JORDAN8:10;
  then len G = width G & 1 <= len G by JORDAN8:def 1,XXREAL_0:2;
  then
A5: [q,1] in Indices G by A2,MATRIX_0:30;
  thus S-bound L~f = (S-min L~f)`2 by EUCLID:52
    .= |[w+((e-w)/(2|^n))*(q-2), s+((a-s)/(2|^n))*(1-2)]|`2 by A3,A4,A5,
JORDAN8:def 1
    .= s-(a-s)/(2|^n) by EUCLID:52;
end;
