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 W-bound L~Cage(C,n) = Gauge(C,n)*
  (1,i)`1
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: [1,i] in Indices G by A2,A1,MATRIX_0:30;
  thus W-bound L~f = w - (e - w)/(2|^n) by Th62
    .= |[w+((e-w)/(2|^n))*(1-2),s+((a-s)/(2|^n))*(i-2)]|`1 by EUCLID:52
    .= G*(1,i)`1 by A3,JORDAN8:def 1;
end;
