let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); for j, n being Nat st j <= len (Gauge C,n) holds
cell (Gauge C,n),0 ,j misses C
let j, n be Nat; ( j <= len (Gauge C,n) implies cell (Gauge C,n),0 ,j misses C )
set G = Gauge C,n;
A1:
j in NAT
by ORDINAL1:def 13;
assume A2:
j <= len (Gauge C,n)
; cell (Gauge C,n),0 ,j misses C
A3:
len (Gauge C,n) = width (Gauge C,n)
by Def1;
assume
(cell (Gauge C,n),0 ,j) /\ C <> {}
; XBOOLE_0:def 7 contradiction
then consider p being Point of (TOP-REAL 2) such that
A4:
p in (cell (Gauge C,n),0 ,j) /\ C
by SUBSET_1:10;
A5:
p in cell (Gauge C,n),0 ,j
by A4, XBOOLE_0:def 4;
A6:
p in C
by A4, XBOOLE_0:def 4;
4 <= len (Gauge C,n)
by Th13;
then A7:
1 <= len (Gauge C,n)
by XXREAL_0:2;
set W = W-bound C;
set S = S-bound C;
set E = E-bound C;
set N = N-bound C;
set EW = ((E-bound C) - (W-bound C)) / (2 |^ n);
[1,1] in Indices (Gauge C,n)
by A3, A7, MATRIX_1:37;
then
(Gauge C,n) * 1,1 = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (1 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (1 - 2)))]|
by Def1;
then A8:
((Gauge C,n) * 1,1) `1 = (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (- 1))
by EUCLID:56;
A9:
2 |^ n > 0
by NEWTON:102;
E-bound C > W-bound C
by Th11;
then
(E-bound C) - (W-bound C) > 0
by XREAL_1:52;
then
((E-bound C) - (W-bound C)) / (2 |^ n) > 0
by A9, XREAL_1:141;
then
(((E-bound C) - (W-bound C)) / (2 |^ n)) * (- 1) < 0 * (- 1)
by XREAL_1:71;
then A10:
((Gauge C,n) * 1,1) `1 < (W-bound C) + 0
by A8, XREAL_1:8;
( j = 0 or j > 0 )
by NAT_1:3;
then A11:
( j = 0 or j >= 1 + 0 )
by NAT_1:9;
per cases
( j = 0 or j = len (Gauge C,n) or ( 1 <= j & j < len (Gauge C,n) ) )
by A2, A11, XXREAL_0:1;
suppose
j = 0
;
contradictionthen
cell (Gauge C,n),
0 ,
j = { |[r,s]| where r, s is Real : ( r <= ((Gauge C,n) * 1,1) `1 & s <= ((Gauge C,n) * 1,1) `2 ) }
by GOBRD11:24;
then consider r,
s being
Real such that A12:
p = |[r,s]|
and A13:
r <= ((Gauge C,n) * 1,1) `1
and
s <= ((Gauge C,n) * 1,1) `2
by A5;
p `1 = r
by A12, EUCLID:56;
then
W-bound C > p `1
by A10, A13, XXREAL_0:2;
hence
contradiction
by A6, PSCOMP_1:71;
verum end; suppose
j = len (Gauge C,n)
;
contradictionthen
cell (Gauge C,n),
0 ,
j = { |[r,s]| where r, s is Real : ( r <= ((Gauge C,n) * 1,1) `1 & ((Gauge C,n) * 1,(width (Gauge C,n))) `2 <= s ) }
by A3, GOBRD11:25;
then consider r,
s being
Real such that A14:
p = |[r,s]|
and A15:
r <= ((Gauge C,n) * 1,1) `1
and
((Gauge C,n) * 1,(width (Gauge C,n))) `2 <= s
by A5;
p `1 = r
by A14, EUCLID:56;
then
W-bound C > p `1
by A10, A15, XXREAL_0:2;
hence
contradiction
by A6, PSCOMP_1:71;
verum end; suppose
( 1
<= j &
j < len (Gauge C,n) )
;
contradictionthen
cell (Gauge C,n),
0 ,
j = { |[r,s]| where r, s is Real : ( r <= ((Gauge C,n) * 1,1) `1 & ((Gauge C,n) * 1,j) `2 <= s & s <= ((Gauge C,n) * 1,(j + 1)) `2 ) }
by A1, A3, GOBRD11:26;
then consider r,
s being
Real such that A16:
p = |[r,s]|
and A17:
r <= ((Gauge C,n) * 1,1) `1
and
((Gauge C,n) * 1,j) `2 <= s
and
s <= ((Gauge C,n) * 1,(j + 1)) `2
by A5;
p `1 = r
by A16, EUCLID:56;
then
W-bound C > p `1
by A10, A17, XXREAL_0:2;
hence
contradiction
by A6, PSCOMP_1:71;
verum end;