reserve n for Nat;

theorem Th38:
  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) & Gauge(C,n)*(
  j,i) in L~Upper_Seq(C,n) & Gauge(C,n)*(k,i) in L~Lower_Seq(C,n) holds LSeg(
  Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Lower_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: Gauge(C,n)*(j,i) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(k,i) 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)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Upper_Seq(C,n) = {
  Gauge(C,n)*(j1,i)} and
A12: LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Lower_Seq(C,n) = {
  Gauge(C,n)*(k1,i)} by A1,A2,A3,A4,A5,A6,A7,Th20;
A13: k1 < len Gauge(C,n) by A3,A10,XXREAL_0:2;
  1 < j1 by A1,A8,XXREAL_0:2;
  then
  LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) meets Lower_Arc C by A4,A5,A9,A11
,A12,A13,Th36;
  hence thesis by A1,A3,A4,A5,A8,A9,A10,Th6,XBOOLE_1:63;
end;
