theorem Th31: :: JORDAN15:31

for n being Nat
for C being Simple_closed_curve
for i, j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,n)) & 1 <= i & i <= width (Gauge (C,n)) & (Gauge (C,n)) * (k,i) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (j,i) in L~ (Lower_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (j,i)),((Gauge (C,n)) * (k,i))) meets Upper_Arc C