reserve n for Nat;

theorem Th4:
  for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
  for n be Nat st n > 0
  for i,j be Nat st 1 <= i & i <= len Gauge(C,n) &
  1 <= j & j <= width Gauge(C,n) & Gauge(C,n)*(i,j) in L~Cage(C,n) holds
  LSeg(Gauge(C,n)*(i,width Gauge(C,n)),Gauge(C,n)*(i,j)) meets
  Upper_Arc L~Cage(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
  assume
A1: n > 0;
  let i,j be Nat;
  assume that
A2: 1 <= i and
A3: i <= len Gauge(C,n) and
A4: 1 <= j and
A5: j <= width Gauge(C,n) and
A6: Gauge(C,n)*(i,j) in L~Cage(C,n);
  L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) by A1,JORDAN1G:55;
  hence thesis by A2,A3,A4,A5,A6,Th3;
end;
