reserve n for Nat;

theorem Th22:
  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) & Gauge(C,n)*(
  i,k) in L~Upper_Seq(C,n) & Gauge(C,n)*(i,j) in L~Lower_Seq(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: Gauge(C,n)*(i,k) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(i,j) in L~Lower_Seq(C,n);
  consider j1,k1 be Nat such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Lower_Seq(C,n) = {
  Gauge(C,n)*(i,j1)} and
A12: LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Upper_Seq(C,n) = {
  Gauge(C,n)*(i,k1)} by A1,A2,A3,A4,A5,A6,A7,Th11;
A13: k1 <= width Gauge(C,n) by A5,A10,XXREAL_0:2;
  1 <= j1 by A3,A8,XXREAL_0:2;
  then
  LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) meets Upper_Arc C by A1,A2,A9,A11
,A12,A13,JORDAN1J:59;
  hence thesis by A1,A2,A3,A5,A8,A9,A10,Th5,XBOOLE_1:63;
end;
