reserve n for Nat;

theorem
  for C be Simple_closed_curve for j,k be Nat holds 1 < j & j
<= k & k < len Gauge(C,n+1) & Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Upper_Arc
L~Cage(C,n+1) & Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1)
implies LSeg(Gauge(C,n+1)*(j,Center Gauge(C,n+1)), Gauge(C,n+1)*(k,Center Gauge
  (C,n+1))) 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 < len Gauge(C,n+1) and
A4: Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Upper_Arc L~Cage(C,n+1) and
A5: Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1);
A6: len Gauge(C,n+1) >= 4 by JORDAN8:10;
  then len Gauge(C,n+1) >= 3 by XXREAL_0:2;
  then Center Gauge(C,n+1) < len Gauge(C,n+1) by JORDAN1B:15;
  then
A7: Center Gauge(C,n+1) < width Gauge(C,n+1) by JORDAN8:def 1;
  len Gauge(C,n+1) >= 2 by A6,XXREAL_0:2;
  then 1 < Center Gauge(C,n+1) by JORDAN1B:14;
  hence thesis by A1,A2,A3,A4,A5,A7,Th40;
end;
