let n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex i being Element of NAT st
( 1 <= i & i <= len (Gauge (C,n)) & (Gauge (C,n)) * (i,1) in rng (Cage (C,n)) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ex i being Element of NAT st
( 1 <= i & i <= len (Gauge (C,n)) & (Gauge (C,n)) * (i,1) in rng (Cage (C,n)) )
consider i being Element of NAT such that
A1: 1 <= i and
A2: i <= len (Gauge (C,n)) and
A3: S-min (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,1) by Th31;
take i ; :: thesis: ( 1 <= i & i <= len (Gauge (C,n)) & (Gauge (C,n)) * (i,1) in rng (Cage (C,n)) )
thus ( 1 <= i & i <= len (Gauge (C,n)) & (Gauge (C,n)) * (i,1) in rng (Cage (C,n)) ) by A1, A2, A3, SPRECT_2:41; :: thesis: verum