let i, n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge C,n) holds
W-bound (L~ (Cage C,n)) = ((Gauge C,n) * 1,i) `1
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( 1 <= i & i <= len (Gauge C,n) implies W-bound (L~ (Cage C,n)) = ((Gauge C,n) * 1,i) `1 )
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;
A1: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
assume A2: ( 1 <= i & i <= len (Gauge C,n) ) ; :: thesis: W-bound (L~ (Cage C,n)) = ((Gauge C,n) * 1,i) `1
then 1 <= len (Gauge C,n) by XXREAL_0:2;
then A3: [1,i] in Indices (Gauge C,n) by A2, A1, MATRIX_1:37;
thus W-bound (L~ (Cage C,n)) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n)) by Th83
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (1 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (i - 2)))]| `1 by EUCLID:56
.= ((Gauge C,n) * 1,i) `1 by A3, JORDAN8:def 1 ; :: thesis: verum