reserve a, b, i, k, m, n for Nat,
  r for Real,
  D for non empty Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2;

theorem
  ex j being Nat st 1 <= j & j <= width Gauge(C,n) & Gauge(C,
  n)*(len Gauge(C,n),j) in rng Cage(C,n)
proof
  consider j be Nat such that
A1: 1 <= j and
A2: j <= width Gauge(C,n) and
A3: E-min L~Cage(C,n) = Gauge(C,n)*(len Gauge(C,n),j) by Th24;
  take j;
  thus thesis by A1,A2,A3,SPRECT_2:45;
end;
