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 i being Nat st 1 <= i & i <= len Gauge(C,n) & Gauge(C,n)
  *(i,1) in rng Cage(C,n)
proof
  consider i be Nat such that
A1: 1 <= i and
A2: i <= len Gauge(C,n) and
A3: S-min L~Cage(C,n) = Gauge(C,n)*(i,1) by Th27;
  take i;
  thus thesis by A1,A2,A3,SPRECT_2:41;
end;
