let n be Nat; :: thesis: for C being Simple_closed_curve
for j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
let C be Simple_closed_curve; :: thesis: for j, k being Nat st 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) holds
LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
let j, k be Nat; :: thesis: ( 1 < j & j <= k & k < len (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) implies LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C )
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len (Gauge (C,(n + 1))) and
A4: (Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1))))) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A5: (Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))) in Lower_Arc (L~ (Cage (C,(n + 1)))) ; :: thesis: LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C
A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 3 by XXREAL_0:2;
then Center (Gauge (C,(n + 1))) < len (Gauge (C,(n + 1))) by JORDAN1B:15;
then A7: Center (Gauge (C,(n + 1))) < width (Gauge (C,(n + 1))) by JORDAN8:def 1;
len (Gauge (C,(n + 1))) >= 2 by A6, XXREAL_0:2;
then 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
hence LSeg (((Gauge (C,(n + 1))) * (j,(Center (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * (k,(Center (Gauge (C,(n + 1))))))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, Th40; :: thesis: verum