reserve n for Nat;

theorem Th33:
  for C be Simple_closed_curve for i,j,k be Nat st 1 <
  j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & n > 0 &
  Gauge(C,n)*(k,i) in Upper_Arc L~Cage(C,n) & Gauge(C,n)*(j,i) in Lower_Arc L~
  Cage(C,n) holds LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Upper_Arc C
proof
  let C be Simple_closed_curve;
  let i,j,k be Nat;
  assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: n > 0 and
A7: Gauge(C,n)*(k,i) in Upper_Arc L~Cage(C,n) and
A8: Gauge(C,n)*(j,i) in Lower_Arc L~Cage(C,n);
A9: L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n) by A6,JORDAN1G:56;
  L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) by A6,JORDAN1G:55;
  hence thesis by A1,A2,A3,A4,A5,A7,A8,A9,Th31;
end;
