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)) * ((len (Gauge (C,n))),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)) * ((len (Gauge (C,n))),j) in rng (Cage (C,n)) )
consider j being Element of NAT such that
A1: 1 <= j and
A2: j <= width (Gauge (C,n)) and
A3: E-min (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),j) by Th28;
take j ; :: thesis: ( 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * ((len (Gauge (C,n))),j) in rng (Cage (C,n)) )
thus ( 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * ((len (Gauge (C,n))),j) in rng (Cage (C,n)) ) by A1, A2, A3, SPRECT_2:45; :: thesis: verum