let n be Nat; for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n)))))) meets Lower_Arc (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n)))))) meets Lower_Arc (L~ (Cage (C,n)))
A1:
4 <= len (Gauge (C,n))
by JORDAN8:10;
then
len (Gauge (C,n)) >= 3
by XXREAL_0:2;
then A2:
Center (Gauge (C,n)) < len (Gauge (C,n))
by Th15;
len (Gauge (C,n)) >= 2
by A1, XXREAL_0:2;
then
1 < Center (Gauge (C,n))
by Th14;
hence
LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n)))))) meets Lower_Arc (L~ (Cage (C,n)))
by A2, Th30; verum