reserve n for Nat;

theorem
  for C be Simple_closed_curve for i,j,k be Nat holds 1 < i &
i < len Gauge(C,n+1) & 1 <= j & j <= k & k <= width Gauge(C,n+1) & Gauge(C,n+1)
  *(i,k) in Upper_Arc L~Cage(C,n+1) & Gauge(C,n+1)*(Center Gauge(C,n+1),j) in
  Lower_Arc L~Cage(C,n+1) implies LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j),
Gauge(C,n+1)*(Center Gauge(C,n+1),k)) \/ LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1)
  ,k),Gauge(C,n+1)*(i,k)) meets Lower_Arc C
proof
  let C be Simple_closed_curve;
  let i,j,k be Nat;
  assume that
A1: 1 < i and
A2: i < len Gauge(C,n+1) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n+1) and
A6: Gauge(C,n+1)*(i,k) in Upper_Arc L~Cage(C,n+1) and
A7: Gauge(C,n+1)*(Center Gauge(C,n+1),j) in Lower_Arc L~Cage(C,n+1);
A8: len Gauge(C,n+1) >= 4 by JORDAN8:10;
  then len Gauge(C,n+1) >= 3 by XXREAL_0:2;
  then
A9: Center Gauge(C,n+1) < len Gauge(C,n+1) by JORDAN1B:15;
  len Gauge(C,n+1) >= 2 by A8,XXREAL_0:2;
  then 1 < Center Gauge(C,n+1) by JORDAN1B:14;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,A9,Th49;
end;
