reserve n for Nat;

theorem Th24:
  for C be Simple_closed_curve for i,j,k be Nat st 1 <
  i & i < len Gauge(C,n) & 1 <= j & j <= k & k <= width Gauge(C,n) & n > 0 &
  Gauge(C,n)*(i,k) in Upper_Arc L~Cage(C,n) & Gauge(C,n)*(i,j) in Lower_Arc L~
  Cage(C,n) holds LSeg(Gauge(C,n)*(i,j),Gauge(C,n)*(i,k)) meets Upper_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) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: n > 0 and
A7: Gauge(C,n)*(i,k) in Upper_Arc L~Cage(C,n) and
A8: Gauge(C,n)*(i,j) 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,Th22;
end;
