theorem :: JORDAN19:15

for n being Nat
for C being Simple_closed_curve
for i, j, k being Nat st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C