let n be Nat; :: thesis: for C being Simple_closed_curve holds LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n)))))) meets Lower_Arc C

let C be Simple_closed_curve; :: thesis: LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n)))))) meets Lower_Arc C

A1: 4 <= len (Gauge (C,n)) by JORDAN8:10;

then len (Gauge (C,n)) >= 2 by XXREAL_0:2;

then A2: 1 < Center (Gauge (C,n)) by Th14;

len (Gauge (C,n)) >= 3 by A1, XXREAL_0:2;

hence LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n)))))) meets Lower_Arc C by A2, Th15, Th26; :: thesis: verum

let C be Simple_closed_curve; :: thesis: LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n)))))) meets Lower_Arc C

A1: 4 <= len (Gauge (C,n)) by JORDAN8:10;

then len (Gauge (C,n)) >= 2 by XXREAL_0:2;

then A2: 1 < Center (Gauge (C,n)) by Th14;

len (Gauge (C,n)) >= 3 by A1, XXREAL_0:2;

hence LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n)))))) meets Lower_Arc C by A2, Th15, Th26; :: thesis: verum