let n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex j being Element of NAT st
( 1 <= j & j <= width (Gauge C,n) & (Gauge C,n) * 1,j in rng (Cage C,n) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ex j being Element of NAT st
( 1 <= j & j <= width (Gauge C,n) & (Gauge C,n) * 1,j in rng (Cage C,n) )
consider j being Element of NAT such that
A1: ( 1 <= j & j <= width (Gauge C,n) & W-min (L~ (Cage C,n)) = (Gauge C,n) * 1,j ) by Th34;
take j ; :: thesis: ( 1 <= j & j <= width (Gauge C,n) & (Gauge C,n) * 1,j in rng (Cage C,n) )
thus ( 1 <= j & j <= width (Gauge C,n) & (Gauge C,n) * 1,j in rng (Cage C,n) ) by A1, SPRECT_2:47; :: thesis: verum