reserve n for Nat;

theorem
  for C be Simple_closed_curve for j,k be Nat holds 1 <= j &
  j <= k & k <= width Gauge(C,n+1) & Gauge(C,n+1)*(Center Gauge(C,n+1),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)) meets Lower_Arc C
proof
  let C be Simple_closed_curve;
  let j,k be Nat;
  assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= width Gauge(C,n+1) and
A4: Gauge(C,n+1)*(Center Gauge(C,n+1),k) in Upper_Arc L~Cage(C,n+1) and
A5: Gauge(C,n+1)*(Center Gauge(C,n+1),j) in Lower_Arc L~Cage(C,n+1);
A6: len Gauge(C,n+1) >= 4 by JORDAN8:10;
  then len Gauge(C,n+1) >= 2 by XXREAL_0:2;
  then
A7: 1 < Center Gauge(C,n+1) by JORDAN1B:14;
  len Gauge(C,n+1) >= 3 by A6,XXREAL_0:2;
  hence thesis by A1,A2,A3,A4,A5,A7,Th23,JORDAN1B:15;
end;
