let n be Element of NAT ; :: thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge C,(n + 1)) & 1 <= j & j <= k & k <= width (Gauge C,(n + 1)) & (Gauge C,(n + 1)) * i,k in Upper_Arc (L~ (Cage C,(n + 1))) & (Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),j in Lower_Arc (L~ (Cage C,(n + 1))) holds
(LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),j),((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k)) \/ (LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k),((Gauge C,(n + 1)) * i,k)) meets Lower_Arc C
let C be Simple_closed_curve; :: thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge C,(n + 1)) & 1 <= j & j <= k & k <= width (Gauge C,(n + 1)) & (Gauge C,(n + 1)) * i,k in Upper_Arc (L~ (Cage C,(n + 1))) & (Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),j in Lower_Arc (L~ (Cage C,(n + 1))) holds
(LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),j),((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k)) \/ (LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k),((Gauge C,(n + 1)) * i,k)) meets Lower_Arc C
let i, j, k be Element of NAT ; :: thesis: ( 1 < i & i < len (Gauge C,(n + 1)) & 1 <= j & j <= k & k <= width (Gauge C,(n + 1)) & (Gauge C,(n + 1)) * i,k in Upper_Arc (L~ (Cage C,(n + 1))) & (Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),j in Lower_Arc (L~ (Cage C,(n + 1))) implies (LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),j),((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k)) \/ (LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k),((Gauge C,(n + 1)) * i,k)) meets Lower_Arc C )
assume that
A1: ( 1 < i & i < len (Gauge C,(n + 1)) ) and
A2: ( 1 <= j & j <= k & k <= width (Gauge C,(n + 1)) ) and
A3: (Gauge C,(n + 1)) * i,k in Upper_Arc (L~ (Cage C,(n + 1))) and
A4: (Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),j in Lower_Arc (L~ (Cage C,(n + 1))) ; :: thesis: (LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),j),((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k)) \/ (LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k),((Gauge C,(n + 1)) * i,k)) meets Lower_Arc C
len (Gauge C,(n + 1)) >= 4 by JORDAN8:13;
then ( len (Gauge C,(n + 1)) >= 2 & len (Gauge C,(n + 1)) >= 3 ) by XXREAL_0:2;
then ( 1 < Center (Gauge C,(n + 1)) & Center (Gauge C,(n + 1)) < len (Gauge C,(n + 1)) ) by JORDAN1B:15, JORDAN1B:16;
hence (LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),j),((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k)) \/ (LSeg ((Gauge C,(n + 1)) * (Center (Gauge C,(n + 1))),k),((Gauge C,(n + 1)) * i,k)) meets Lower_Arc C by A1, A2, A3, A4, Th51; :: thesis: verum