let n be Nat; for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex j being 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); ex j being 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 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 Th24;
take
j
; ( 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; verum