theorem Th44: :: JORDAN15:44

for n being Nat
for C being Simple_closed_curve
for i1, i2, j, k being Nat st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C