let n be Nat; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex i being Nat st
( 1 <= i & i <= len (Gauge (C,n)) & (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Cage (C,n)) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ex i being Nat st
( 1 <= i & i <= len (Gauge (C,n)) & (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Cage (C,n)) )
consider i being Nat such that
A1: 1 <= i and
A2: i <= len (Gauge (C,n)) and
A3: N-min (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by Th21;
take i ; :: thesis: ( 1 <= i & i <= len (Gauge (C,n)) & (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Cage (C,n)) )
thus ( 1 <= i & i <= len (Gauge (C,n)) & (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Cage (C,n)) ) by A1, A2, A3, SPRECT_2:39; :: thesis: verum