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) & LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j),
Gauge(C,n+1)*(Center Gauge(C,n+1),k)) /\ Upper_Arc L~Cage(C,n+1) = {Gauge(C,n+1
)*(Center Gauge(C,n+1),k)} & LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j), Gauge(C
  ,n+1)*(Center Gauge(C,n+1),k)) /\ Lower_Arc L~Cage(C,n+1) = {Gauge(C,n+1)*(
  Center Gauge(C,n+1),j)} implies LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j),
  Gauge(C,n+1)*(Center Gauge(C,n+1),k)) meets Upper_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: LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j), Gauge(C,n+1)*(Center
Gauge(C,n+1),k)) /\ Upper_Arc L~Cage(C,n+1) = {Gauge(C,n+1)*(Center Gauge(C,n+1
  ),k)} and
A5: LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j), Gauge(C,n+1)*(Center
Gauge(C,n+1),k)) /\ Lower_Arc L~Cage(C,n+1) = {Gauge(C,n+1)*(Center Gauge(C,n+1
  ),j)};
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,Th61,JORDAN1B:15;
end;
