let i, n be 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
S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( 1 <= i & i <= len (Gauge (C,n)) implies S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2 )
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: S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2
then 1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A3: [i,1] in Indices (Gauge (C,n)) by A2, A1, MATRIX_0:30;
thus S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n)) by Th63
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (1 - 2)))]| `2 by EUCLID:52
.= ((Gauge (C,n)) * (i,1)) `2 by A3, JORDAN8:def 1 ; :: thesis: verum