let n, i, a, b be Element of NAT ; for D being non empty Subset of (TOP-REAL 2) st 2 <= n & n < len (Gauge D,i) & 1 <= a & a <= len (Gauge D,i) & 1 <= b & b <= len (Gauge D,(i + 1)) holds
((Gauge D,i) * a,n) `2 = ((Gauge D,(i + 1)) * b,((2 * n) -' 2)) `2
let D be non empty Subset of (TOP-REAL 2); ( 2 <= n & n < len (Gauge D,i) & 1 <= a & a <= len (Gauge D,i) & 1 <= b & b <= len (Gauge D,(i + 1)) implies ((Gauge D,i) * a,n) `2 = ((Gauge D,(i + 1)) * b,((2 * n) -' 2)) `2 )
set I = Gauge D,i;
set J = Gauge D,(i + 1);
set z = N-bound D;
set e = E-bound D;
set s = S-bound D;
set w = W-bound D;
assume that
A1:
2 <= n
and
A2:
n < len (Gauge D,i)
and
A3:
1 <= a
and
A4:
a <= len (Gauge D,i)
and
A5:
1 <= b
and
A6:
b <= len (Gauge D,(i + 1))
; ((Gauge D,i) * a,n) `2 = ((Gauge D,(i + 1)) * b,((2 * n) -' 2)) `2
n < (2 |^ i) + 3
by A2, JORDAN8:def 1;
then
(2 * n) -' 2 <= (2 |^ (i + 1)) + 3
by Lm13;
then A7:
(2 * n) -' 2 <= len (Gauge D,(i + 1))
by JORDAN8:def 1;
A8:
len (Gauge D,(i + 1)) = width (Gauge D,(i + 1))
by JORDAN8:def 1;
1 <= (2 * n) -' 2
by A1, Lm11;
then A9:
[b,((2 * n) -' 2)] in Indices (Gauge D,(i + 1))
by A5, A6, A8, A7, MATRIX_1:37;
A10:
len (Gauge D,i) = width (Gauge D,i)
by JORDAN8:def 1;
1 <= n
by A1, XXREAL_0:2;
then
[a,n] in Indices (Gauge D,i)
by A2, A3, A4, A10, MATRIX_1:37;
hence ((Gauge D,i) * a,n) `2 =
|[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ i)) * (a - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ i)) * (n - 2)))]| `2
by JORDAN8:def 1
.=
(S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ i)) * (n - 2))
by EUCLID:56
.=
(S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ (i + 1))) * (((2 * n) -' 2) - 2))
by A1, Lm10
.=
|[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ (i + 1))) * (b - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ (i + 1))) * (((2 * n) -' 2) - 2)))]| `2
by EUCLID:56
.=
((Gauge D,(i + 1)) * b,((2 * n) -' 2)) `2
by A9, JORDAN8:def 1
;
verum