reserve 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,
  i, j, m, n for Nat,
  p for Point of TOP-REAL 2;

theorem
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 holds LSeg(Gauge(C,n)*(Center Gauge(C,n),1), Gauge(C,n)*(Center
  Gauge(C,n),len Gauge(C,n))) meets Lower_Arc L~Cage(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
A1: 4 <= len Gauge(C,n) by JORDAN8:10;
  then len Gauge(C,n) >= 3 by XXREAL_0:2;
  then
A2: Center Gauge(C,n) < len Gauge(C,n) by Th15;
  len Gauge(C,n) >= 2 by A1,XXREAL_0:2;
  then 1 < Center Gauge(C,n) by Th14;
  hence thesis by A2,Th30;
end;
