let n be Nat; for C being Simple_closed_curve
for j, k being Nat st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
let C be Simple_closed_curve; for j, k being Nat st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
let j, k be Nat; ( 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} implies LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C )
assume that
A1:
1 <= j
and
A2:
j <= k
and
A3:
k <= width (Gauge (C,(n + 1)))
and
A4:
(LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))}
and
A5:
(LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))}
; LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
A6:
len (Gauge (C,(n + 1))) >= 4
by JORDAN8:10;
then
len (Gauge (C,(n + 1))) >= 2
by XXREAL_0:2;
then A7:
1 < Center (Gauge (C,(n + 1)))
by JORDAN1B:14;
len (Gauge (C,(n + 1))) >= 3
by A6, XXREAL_0:2;
hence
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
by A1, A2, A3, A4, A5, A7, Th61, JORDAN1B:15; verum