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
  1 <= i & i <= len Gauge(C,n) implies N-bound L~Cage(C,n) = Gauge(C,n)*
  (i,len Gauge(C,n))`2
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);
A1: len G = width G by JORDAN8:def 1;
  assume
A2: 1 <= i & i <= len G;
  then 1 <= len G by XXREAL_0:2;
  then
A3: [i,len G] in Indices G by A2,A1,MATRIX_0:30;
  thus N-bound L~f = a + (a - s)/(2|^n) by JORDAN10:6
    .= s+((a-s)/(2|^n))*(len G-2) by Lm10
    .= |[w+((e-w)/(2|^n))*(i-2),s+((a-s)/(2|^n))*(len G-2)]|`2 by EUCLID:52
    .= G*(i,len G)`2 by A3,JORDAN8:def 1;
end;
