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 Simple_closed_curve holds LSeg(Gauge(C,n)*(Center Gauge(C,n),
  1), Gauge(C,n)*(Center Gauge(C,n),len Gauge(C,n))) meets Lower_Arc C
proof
  let C be Simple_closed_curve;
A1: 4 <= len Gauge(C,n) by JORDAN8:10;
  then len Gauge(C,n) >= 2 by XXREAL_0:2;
  then
A2: 1 < Center Gauge(C,n) by Th14;
  len Gauge(C,n) >= 3 by A1,XXREAL_0:2;
  hence thesis by A2,Th15,Th26;
end;
