let n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds W-bound (L~ (Cage (C,n))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: W-bound (L~ (Cage (C,n))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set f = Cage (C,n);
set G = Gauge (C,n);
consider p, q being Element of NAT such that
A1: ( 1 <= p & p < len (Cage (C,n)) ) and
A2: ( 1 <= q & q <= width (Gauge (C,n)) ) and
A3: (Cage (C,n)) /. p = (Gauge (C,n)) * (1,q) by Th76;
(Cage (C,n)) /. p in W-most (L~ (Cage (C,n))) by A1, A2, A3, Th80;
then A4: ((Cage (C,n)) /. p) `1 = (W-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:31;
4 <= len (Gauge (C,n)) by JORDAN8:10;
then 1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A5: [1,q] in Indices (Gauge (C,n)) by A2, MATRIX_1:36;
thus W-bound (L~ (Cage (C,n))) = (W-min (L~ (Cage (C,n)))) `1 by EUCLID:52
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (1 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (q - 2)))]| `1 by A3, A4, A5, JORDAN8:def 1
.= (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n)) by EUCLID:52 ; :: thesis: verum