let n be Element of NAT ; for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-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 <= len (Gauge (C,n)) )
and
A3:
(Cage (C,n)) /. p = (Gauge (C,n)) * (q,1)
by Th77;
(Cage (C,n)) /. p in S-most (L~ (Cage (C,n)))
by A1, A2, A3, Th81;
then A4:
((Cage (C,n)) /. p) `2 = (S-min (L~ (Cage (C,n)))) `2
by PSCOMP_1:55;
4 <= len (Gauge (C,n))
by JORDAN8:10;
then
( len (Gauge (C,n)) = width (Gauge (C,n)) & 1 <= len (Gauge (C,n)) )
by JORDAN8:def 1, XXREAL_0:2;
then A5:
[q,1] in Indices (Gauge (C,n))
by A2, MATRIX_1:36;
thus S-bound (L~ (Cage (C,n))) =
(S-min (L~ (Cage (C,n)))) `2
by EUCLID:52
.=
|[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (q - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (1 - 2)))]| `2
by A3, A4, A5, JORDAN8:def 1
.=
(S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))
by EUCLID:52
; verum