let i, n be Nat; :: thesis: for T being non empty Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (T,n)) holds
((Gauge (T,n)) * (i,2)) `2 = S-bound T
let T be non empty Subset of (TOP-REAL 2); :: thesis: ( 1 <= i & i <= len (Gauge (T,n)) implies ((Gauge (T,n)) * (i,2)) `2 = S-bound T )
set G = Gauge (T,n);
set W = W-bound T;
set S = S-bound T;
set E = E-bound T;
set N = N-bound T;
assume that
A1: 1 <= i and
A2: i <= len (Gauge (T,n)) ; :: thesis: ((Gauge (T,n)) * (i,2)) `2 = S-bound T
A3: len (Gauge (T,n)) = width (Gauge (T,n)) by Def1;
len (Gauge (T,n)) >= 4 by Th10;
then 2 <= len (Gauge (T,n)) by XXREAL_0:2;
then [i,2] in Indices (Gauge (T,n)) by A1, A2, A3, MATRIX_0:30;
then (Gauge (T,n)) * (i,2) = |[((W-bound T) + ((((E-bound T) - (W-bound T)) / (2 |^ n)) * (i - 2))),((S-bound T) + ((((N-bound T) - (S-bound T)) / (2 |^ n)) * (2 - 2)))]| by Def1;
hence ((Gauge (T,n)) * (i,2)) `2 = S-bound T by EUCLID:52; :: thesis: verum