:: Gauges and Cages
:: by Artur Korni{\l}owicz, Robert Milewski, Adam Naumowicz and
::
:: Received September 12, 2000
:: Copyright (c) 2000-2011 Association of Mizar Users

begin

3 = (2 * 1) + 1
;

then Lm1: 3 div 2 = 1
by NAT_D:def 1;

1 = (2 * 0) + 1
;

then Lm2: 1 div 2 = 0
by NAT_D:def 1;

definition

let f be FinSequence;
func Center f -> Element of NAT equals :: JORDAN1A:def 1
((len f) div 2) + 1;
coherence
((len f) div 2) + 1 is Element of NAT ;

end;

:: deftheorem defines Center JORDAN1A:def 1 :
for f being FinSequence holds Center f = ((len f) div 2) + 1;

theorem :: JORDAN1A:1

canceled;

theorem :: JORDAN1A:2

canceled;

theorem :: JORDAN1A:3

canceled;

theorem :: JORDAN1A:4

canceled;

theorem :: JORDAN1A:5

canceled;

theorem :: JORDAN1A:6

canceled;

theorem :: JORDAN1A:7

canceled;

theorem :: JORDAN1A:8

canceled;

theorem :: JORDAN1A:9

for f being FinSequence st not len f is even holds
len f = (2 * (Center f)) - 1

proof end;

theorem :: JORDAN1A:10

for f being FinSequence st len f is even holds
len f = (2 * (Center f)) - 2

proof end;

begin

registration

cluster non empty being_simple_closed_curve compact non horizontal non vertical Element of K6( the U1 of (TOP-REAL 2));
existence
ex b₁ being Subset of (TOP-REAL 2) st
( b₁ is compact & not b₁ is vertical & not b₁ is horizontal & b₁ is being_simple_closed_curve & not b₁ is empty )

proof end;

end;

theorem Th11: :: JORDAN1A:11

for D being non empty Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in N-most D holds
p `2 = N-bound D

proof end;

theorem Th12: :: JORDAN1A:12

for D being non empty Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in E-most D holds
p `1 = E-bound D

proof end;

theorem Th13: :: JORDAN1A:13

for D being non empty Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in S-most D holds
p `2 = S-bound D

proof end;

theorem Th14: :: JORDAN1A:14

for D being non empty Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in W-most D holds
p `1 = W-bound D

proof end;

theorem :: JORDAN1A:15

for D being Subset of (TOP-REAL 2) holds BDD D misses D

proof end;

theorem :: JORDAN1A:16

canceled;

theorem Th17: :: JORDAN1A:17

for p being Point of (TOP-REAL 2) holds p in Vertical_Line (p `1)

proof end;

theorem :: JORDAN1A:18

for r, s being real number holds |[r,s]| in Vertical_Line r

proof end;

theorem :: JORDAN1A:19

for s being real number
for A being Subset of (TOP-REAL 2) st A c= Vertical_Line s holds
A is vertical

proof end;

theorem :: JORDAN1A:20

canceled;

theorem :: JORDAN1A:21

for p, q being Point of (TOP-REAL 2)
for r being real number st p `1 = q `1 & r in [.(proj2 . p),(proj2 . q).] holds
|[(p `1),r]| in LSeg (p,q)

proof end;

theorem :: JORDAN1A:22

for p, q being Point of (TOP-REAL 2)
for r being real number st p `2 = q `2 & r in [.(proj1 . p),(proj1 . q).] holds
|[r,(p `2)]| in LSeg (p,q)

proof end;

theorem :: JORDAN1A:23

for p, q being Point of (TOP-REAL 2)
for s being real number st p in Vertical_Line s & q in Vertical_Line s holds
LSeg (p,q) c= Vertical_Line s

proof end;

theorem :: JORDAN1A:24

for A, B being Subset of (TOP-REAL 2) st A meets B holds
proj2 .: A meets proj2 .: B

proof end;

theorem :: JORDAN1A:25

for s being real number
for A, B being Subset of (TOP-REAL 2) st A misses B & A c= Vertical_Line s & B c= Vertical_Line s holds
proj2 .: A misses proj2 .: B

proof end;

begin

theorem :: JORDAN1A:26

canceled;

theorem :: JORDAN1A:27

canceled;

theorem :: JORDAN1A:28

canceled;

theorem :: JORDAN1A:29

canceled;

theorem :: JORDAN1A:30

canceled;

theorem :: JORDAN1A:31

canceled;

theorem :: JORDAN1A:32

canceled;

theorem :: JORDAN1A:33

canceled;

theorem :: JORDAN1A:34

canceled;

theorem :: JORDAN1A:35

canceled;

theorem :: JORDAN1A:36

canceled;

theorem :: JORDAN1A:37

for i, j being Element of NAT
for G being Go-board st 1 <= i & i <= len G & 1 <= j & j <= width G holds
G * (i,j) in LSeg ((G * (i,1)),(G * (i,(width G))))

proof end;

theorem :: JORDAN1A:38

for i, j being Element of NAT
for G being Go-board st 1 <= i & i <= len G & 1 <= j & j <= width G holds
G * (i,j) in LSeg ((G * (1,j)),(G * ((len G),j)))

proof end;

theorem Th39: :: JORDAN1A:39

for j1, j2, i1, i2 being Element of NAT
for G being Go-board st 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 <= i2 & i2 <= len G holds
(G * (i1,j1)) `1 <= (G * (i2,j2)) `1

proof end;

theorem Th40: :: JORDAN1A:40

for i1, i2, j1, j2 being Element of NAT
for G being Go-board st 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 <= j2 & j2 <= width G holds
(G * (i1,j1)) `2 <= (G * (i2,j2)) `2

proof end;

theorem Th41: :: JORDAN1A:41

for t being Element of NAT
for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= len G holds
(G * (t,(width G))) `2 >= N-bound (L~ f)

proof end;

theorem Th42: :: JORDAN1A:42

for t being Element of NAT
for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= width G holds
(G * (1,t)) `1 <= W-bound (L~ f)

proof end;

theorem Th43: :: JORDAN1A:43

for t being Element of NAT
for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= len G holds
(G * (t,1)) `2 <= S-bound (L~ f)

proof end;

theorem Th44: :: JORDAN1A:44

for t being Element of NAT
for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= width G holds
(G * ((len G),t)) `1 >= E-bound (L~ f)

proof end;

theorem Th45: :: JORDAN1A:45

for i, j being Element of NAT
for G being Go-board st i <= len G & j <= width G holds
not cell (G,i,j) is empty

proof end;

theorem Th46: :: JORDAN1A:46

for i, j being Element of NAT
for G being Go-board st i <= len G & j <= width G holds
cell (G,i,j) is connected

proof end;

theorem Th47: :: JORDAN1A:47

for i being Element of NAT
for G being Go-board st i <= len G holds
not cell (G,i,0) is Bounded

proof end;

theorem Th48: :: JORDAN1A:48

for i being Element of NAT
for G being Go-board st i <= len G holds
not cell (G,i,(width G)) is Bounded

proof end;

begin

theorem :: JORDAN1A:49

for n being Element of NAT
for D being non empty Subset of (TOP-REAL 2) holds width (Gauge (D,n)) = (2 |^ n) + 3

proof end;

theorem :: JORDAN1A:50

for i, j being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st i < j holds
len (Gauge (D,i)) < len (Gauge (D,j))

proof end;

theorem :: JORDAN1A:51

for i, j being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st i <= j holds
len (Gauge (D,i)) <= len (Gauge (D,j))

proof end;

theorem Th52: :: JORDAN1A:52

for m, n, i being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i < len (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < len (Gauge (D,n)) )

proof end;

theorem Th53: :: JORDAN1A:53

for m, n, i being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i < width (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < width (Gauge (D,n)) )

proof end;

theorem Th54: :: JORDAN1A:54

for m, n, i, j being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i < len (Gauge (D,m)) & 1 < j & j < width (Gauge (D,m)) holds
for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
(Gauge (D,m)) * (i,j) = (Gauge (D,n)) * (i1,j1)

proof end;

theorem Th55: :: JORDAN1A:55

for m, n, i being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i + 1 < len (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= len (Gauge (D,n)) )

proof end;

theorem Th56: :: JORDAN1A:56

for m, n, i being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i + 1 < width (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= width (Gauge (D,n)) )

proof end;

Lm3: now

let D be non empty Subset of (TOP-REAL 2); :: thesis: for n being Element of NAT holds
( 1 <= Center (Gauge (D,n)) & Center (Gauge (D,n)) <= len (Gauge (D,n)) )
let n be Element of NAT ; :: thesis: ( 1 <= Center (Gauge (D,n)) & Center (Gauge (D,n)) <= len (Gauge (D,n)) )
set G = Gauge (D,n);
0 + 1 <= ((len (Gauge (D,n))) div 2) + 1 by XREAL_1:8;
hence 1 <= Center (Gauge (D,n)) ; :: thesis: Center (Gauge (D,n)) <= len (Gauge (D,n))
0 < len (Gauge (D,n)) by JORDAN8:13;
then (len (Gauge (D,n))) div 2 < len (Gauge (D,n)) by INT_1:83;
hence Center (Gauge (D,n)) <= len (Gauge (D,n)) by NAT_1:13; :: thesis: verum

end;

Lm4: now

let D be non empty Subset of (TOP-REAL 2); :: thesis: for n, j being Element of NAT st 1 <= j & j <= len (Gauge (D,n)) holds
[(Center (Gauge (D,n))),j] in Indices (Gauge (D,n))
let n, j be Element of NAT ; :: thesis: ( 1 <= j & j <= len (Gauge (D,n)) implies [(Center (Gauge (D,n))),j] in Indices (Gauge (D,n)) )
set G = Gauge (D,n);
assume A1: ( 1 <= j & j <= len (Gauge (D,n)) ) ; :: thesis: [(Center (Gauge (D,n))),j] in Indices (Gauge (D,n))
A2: ( len (Gauge (D,n)) = width (Gauge (D,n)) & 0 + 1 <= ((len (Gauge (D,n))) div 2) + 1 ) by JORDAN8:def 1, XREAL_1:8;
Center (Gauge (D,n)) <= len (Gauge (D,n)) by Lm3;
hence [(Center (Gauge (D,n))),j] in Indices (Gauge (D,n)) by A1, A2, MATRIX_1:37; :: thesis: verum

end;

Lm5: now

let D be non empty Subset of (TOP-REAL 2); :: thesis: for n, j being Element of NAT st 1 <= j & j <= len (Gauge (D,n)) holds
[j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n))
let n, j be Element of NAT ; :: thesis: ( 1 <= j & j <= len (Gauge (D,n)) implies [j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n)) )
set G = Gauge (D,n);
assume A1: ( 1 <= j & j <= len (Gauge (D,n)) ) ; :: thesis: [j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n))
A2: ( len (Gauge (D,n)) = width (Gauge (D,n)) & 0 + 1 <= ((len (Gauge (D,n))) div 2) + 1 ) by JORDAN8:def 1, XREAL_1:8;
Center (Gauge (D,n)) <= len (Gauge (D,n)) by Lm3;
hence [j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n)) by A1, A2, MATRIX_1:37; :: thesis: verum

end;

Lm6: for n being Element of NAT
for D being non empty Subset of (TOP-REAL 2)
for w being real number st n > 0 holds
(w / (2 |^ n)) * ((Center (Gauge (D,n))) - 2) = w / 2

proof end;

Lm7: now

let m, n be Element of NAT ; :: thesis: for c, d being real number st 0 <= c & m <= n holds
c / (2 |^ n) <= c / (2 |^ m)
let c, d be real number ; :: thesis: ( 0 <= c & m <= n implies c / (2 |^ n) <= c / (2 |^ m) )
assume A1: 0 <= c ; :: thesis: ( m <= n implies c / (2 |^ n) <= c / (2 |^ m) )
assume m <= n ; :: thesis: c / (2 |^ n) <= c / (2 |^ m)
then ( 0 < 2 |^ m & 2 |^ m <= 2 |^ n ) by NEWTON:102, PREPOWER:107;
hence c / (2 |^ n) <= c / (2 |^ m) by A1, XREAL_1:120; :: thesis: verum

end;

Lm8: now

let m, n be Element of NAT ; :: thesis: for c, d being real number st 0 <= c & m <= n holds
d + (c / (2 |^ n)) <= d + (c / (2 |^ m))
let c, d be real number ; :: thesis: ( 0 <= c & m <= n implies d + (c / (2 |^ n)) <= d + (c / (2 |^ m)) )
assume ( 0 <= c & m <= n ) ; :: thesis: d + (c / (2 |^ n)) <= d + (c / (2 |^ m))
then c / (2 |^ n) <= c / (2 |^ m) by Lm7;
hence d + (c / (2 |^ n)) <= d + (c / (2 |^ m)) by XREAL_1:8; :: thesis: verum

end;

Lm9: now

let m, n be Element of NAT ; :: thesis: for c, d being real number st 0 <= c & m <= n holds
d - (c / (2 |^ m)) <= d - (c / (2 |^ n))
let c, d be real number ; :: thesis: ( 0 <= c & m <= n implies d - (c / (2 |^ m)) <= d - (c / (2 |^ n)) )
assume ( 0 <= c & m <= n ) ; :: thesis: d - (c / (2 |^ m)) <= d - (c / (2 |^ n))
then c / (2 |^ n) <= c / (2 |^ m) by Lm7;
hence d - (c / (2 |^ m)) <= d - (c / (2 |^ n)) by XREAL_1:15; :: thesis: verum

end;

theorem Th57: :: JORDAN1A:57

for i, n, j, m being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (D,n)) & 1 <= j & j <= len (Gauge (D,m)) & ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) holds
((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1

proof end;

theorem :: JORDAN1A:58

for i, n, j, m being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (D,n)) & 1 <= j & j <= len (Gauge (D,m)) & ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) holds
((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2

proof end;

Lm10: now

let D be non empty Subset of (TOP-REAL 2); :: thesis: for n being Element of NAT
for e, w being real number holds w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))
let n be Element of NAT ; :: thesis: for e, w being real number holds w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))
let e, w be real number ; :: thesis: w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))
A1: 2 |^ n <> 0 by NEWTON:102;
thus w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = w + (((e - w) / (2 |^ n)) * (((2 |^ n) + 3) - 2)) by JORDAN8:def 1
.= (w + (((e - w) / (2 |^ n)) * (2 |^ n))) + ((e - w) / (2 |^ n))
.= (w + (e - w)) + ((e - w) / (2 |^ n)) by A1, XCMPLX_1:88
.= e + ((e - w) / (2 |^ n)) ; :: thesis: verum

end;

Lm11: now

let D be non empty Subset of (TOP-REAL 2); :: thesis: for n, i being Element of NAT st [i,1] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (i,1)) `2 = (S-bound D) - (((N-bound D) - (S-bound D)) / (2 |^ n))
let n, i be Element of NAT ; :: thesis: ( [i,1] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (i,1)) `2 = (S-bound D) - (((N-bound D) - (S-bound D)) / (2 |^ n)) )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
assume [i,1] in Indices (Gauge (D,n)) ; :: thesis: ((Gauge (D,n)) * (i,1)) `2 = (S-bound D) - (((N-bound D) - (S-bound D)) / (2 |^ n))
hence ((Gauge (D,n)) * (i,1)) `2 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * (i - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * (1 - 2)))]| `2 by JORDAN8:def 1
.= (S-bound D) - (((N-bound D) - (S-bound D)) / (2 |^ n)) by EUCLID:56 ;
:: thesis: verum

end;

Lm12: now

let D be non empty Subset of (TOP-REAL 2); :: thesis: for n, i being Element of NAT st [1,i] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (1,i)) `1 = (W-bound D) - (((E-bound D) - (W-bound D)) / (2 |^ n))
let n, i be Element of NAT ; :: thesis: ( [1,i] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (1,i)) `1 = (W-bound D) - (((E-bound D) - (W-bound D)) / (2 |^ n)) )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
assume [1,i] in Indices (Gauge (D,n)) ; :: thesis: ((Gauge (D,n)) * (1,i)) `1 = (W-bound D) - (((E-bound D) - (W-bound D)) / (2 |^ n))
hence ((Gauge (D,n)) * (1,i)) `1 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * (1 - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * (i - 2)))]| `1 by JORDAN8:def 1
.= (W-bound D) - (((E-bound D) - (W-bound D)) / (2 |^ n)) by EUCLID:56 ;
:: thesis: verum

end;

Lm13: now

let D be non empty Subset of (TOP-REAL 2); :: thesis: for n, i being Element of NAT st [i,(len (Gauge (D,n)))] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = (N-bound D) + (((N-bound D) - (S-bound D)) / (2 |^ n))
let n, i be Element of NAT ; :: thesis: ( [i,(len (Gauge (D,n)))] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = (N-bound D) + (((N-bound D) - (S-bound D)) / (2 |^ n)) )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
assume [i,(len (Gauge (D,n)))] in Indices (Gauge (D,n)) ; :: thesis: ((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = (N-bound D) + (((N-bound D) - (S-bound D)) / (2 |^ n))
hence ((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * (i - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)))]| `2 by JORDAN8:def 1
.= (S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) by EUCLID:56
.= (N-bound D) + (((N-bound D) - (S-bound D)) / (2 |^ n)) by Lm10 ;
:: thesis: verum

end;

Lm14: now

let D be non empty Subset of (TOP-REAL 2); :: thesis: for n, i being Element of NAT st [(len (Gauge (D,n))),i] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = (E-bound D) + (((E-bound D) - (W-bound D)) / (2 |^ n))
let n, i be Element of NAT ; :: thesis: ( [(len (Gauge (D,n))),i] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = (E-bound D) + (((E-bound D) - (W-bound D)) / (2 |^ n)) )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
assume [(len (Gauge (D,n))),i] in Indices (Gauge (D,n)) ; :: thesis: ((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = (E-bound D) + (((E-bound D) - (W-bound D)) / (2 |^ n))
hence ((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * ((len (Gauge (D,n))) - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * (i - 2)))]| `1 by JORDAN8:def 1
.= (W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) by EUCLID:56
.= (E-bound D) + (((E-bound D) - (W-bound D)) / (2 |^ n)) by Lm10 ;
:: thesis: verum

end;

theorem :: JORDAN1A:59

for i being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,1)) holds
((Gauge (C,1)) * ((Center (Gauge (C,1))),i)) `1 = ((W-bound C) + (E-bound C)) / 2

proof end;

theorem :: JORDAN1A:60

for i being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,1)) holds
((Gauge (C,1)) * (i,(Center (Gauge (C,1))))) `2 = ((S-bound C) + (N-bound C)) / 2

proof end;

theorem Th61: :: JORDAN1A:61

for i, n, j, m being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,n)) * (i,(len (Gauge (E,n))))) `2 <= ((Gauge (E,m)) * (j,(len (Gauge (E,m))))) `2

proof end;

theorem :: JORDAN1A:62

for i, n, j, m being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,n)) * ((len (Gauge (E,n))),i)) `1 <= ((Gauge (E,m)) * ((len (Gauge (E,m))),j)) `1

proof end;

theorem :: JORDAN1A:63

for i, n, j, m being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,m)) * (1,j)) `1 <= ((Gauge (E,n)) * (1,i)) `1

proof end;

theorem :: JORDAN1A:64

for i, n, j, m being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,m)) * (j,1)) `2 <= ((Gauge (E,n)) * (i,1)) `2

proof end;

theorem :: JORDAN1A:65

for m, n being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n)))))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))))

proof end;

theorem :: JORDAN1A:66

for m, n, j being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))

proof end;

theorem :: JORDAN1A:67

for m, n, j being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))))

proof end;

theorem Th68: :: JORDAN1A:68

for m, n, i, j being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st m <= n & 1 < i & i + 1 < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) holds
for i1, j1 being Element of NAT st ((2 |^ (n -' m)) * (i - 2)) + 2 <= i1 & i1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (j - 2)) + 2 <= j1 & j1 < ((2 |^ (n -' m)) * (j - 1)) + 2 holds
cell ((Gauge (E,n)),i1,j1) c= cell ((Gauge (E,m)),i,j)

proof end;

theorem :: JORDAN1A:69

for m, n, i, j being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st m <= n & 3 <= i & i < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) holds
for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
cell ((Gauge (E,n)),(i1 -' 1),j1) c= cell ((Gauge (E,m)),(i -' 1),j)

proof end;

theorem :: JORDAN1A:70

for i, n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) st i <= len (Gauge (C,n)) holds
cell ((Gauge (C,n)),i,0) c= UBD C

proof end;

theorem :: JORDAN1A:71

for i, n being Element of NAT
for E, C being compact non horizontal non vertical Subset of (TOP-REAL 2) st i <= len (Gauge (E,n)) holds
cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E

proof end;

begin

theorem Th72: :: JORDAN1A:72

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
north_halfline p meets L~ (Cage (C,n))

proof end;

theorem Th73: :: JORDAN1A:73

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
east_halfline p meets L~ (Cage (C,n))

proof end;

theorem Th74: :: JORDAN1A:74

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
south_halfline p meets L~ (Cage (C,n))

proof end;

theorem Th75: :: JORDAN1A:75

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
west_halfline p meets L~ (Cage (C,n))

proof end;

Lm15: for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )

proof end;

Lm16: for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )

proof end;

Lm17: for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )

proof end;

theorem Th76: :: JORDAN1A:76

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )

proof end;

theorem Th77: :: JORDAN1A:77

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )

proof end;

theorem Th78: :: JORDAN1A:78

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )

proof end;

theorem Th79: :: JORDAN1A:79

for k, n, t being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,(width (Gauge (C,n)))) holds
(Cage (C,n)) /. k in N-most (L~ (Cage (C,n)))

proof end;

theorem Th80: :: JORDAN1A:80

for k, n, t being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) holds
(Cage (C,n)) /. k in W-most (L~ (Cage (C,n)))

proof end;

theorem Th81: :: JORDAN1A:81

for k, n, t being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) holds
(Cage (C,n)) /. k in S-most (L~ (Cage (C,n)))

proof end;

theorem Th82: :: JORDAN1A:82

for k, n, t being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) holds
(Cage (C,n)) /. k in E-most (L~ (Cage (C,n)))

proof end;

theorem Th83: :: JORDAN1A:83

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds W-bound (L~ (Cage (C,n))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))

proof end;

theorem Th84: :: JORDAN1A:84

for n being 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))

proof end;

theorem Th85: :: JORDAN1A:85

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds E-bound (L~ (Cage (C,n))) = (E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))

proof end;

theorem :: JORDAN1A:86

for n, m being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-bound (L~ (Cage (C,n)))) + (S-bound (L~ (Cage (C,n)))) = (N-bound (L~ (Cage (C,m)))) + (S-bound (L~ (Cage (C,m))))

proof end;

theorem :: JORDAN1A:87

for n, m being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-bound (L~ (Cage (C,n)))) + (W-bound (L~ (Cage (C,n)))) = (E-bound (L~ (Cage (C,m)))) + (W-bound (L~ (Cage (C,m))))

proof end;

theorem :: JORDAN1A:88

for i, j being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st i < j holds
E-bound (L~ (Cage (C,j))) < E-bound (L~ (Cage (C,i)))

proof end;

theorem :: JORDAN1A:89

for i, j being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st i < j holds
W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,j)))

proof end;

theorem :: JORDAN1A:90

for i, j being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st i < j holds
S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,j)))

proof end;

theorem :: JORDAN1A:91

for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
N-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,(len (Gauge (C,n))))) `2

proof end;

theorem :: JORDAN1A:92

for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),i)) `1

proof end;

theorem :: JORDAN1A:93

for i, n being Element of NAT
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

proof end;

theorem :: JORDAN1A:94

for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
W-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (1,i)) `1

proof end;

theorem Th95: :: JORDAN1A:95

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (north_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 > x `2

proof end;

theorem Th96: :: JORDAN1A:96

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (east_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 > x `1

proof end;

theorem Th97: :: JORDAN1A:97

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 < x `2

proof end;

theorem Th98: :: JORDAN1A:98

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 < x `1

proof end;

theorem Th99: :: JORDAN1A:99

for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in N-most C & p in north_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is horizontal

proof end;

theorem Th100: :: JORDAN1A:100

for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in E-most C & p in east_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is vertical

proof end;

theorem Th101: :: JORDAN1A:101

for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in S-most C & p in south_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is horizontal

proof end;

theorem Th102: :: JORDAN1A:102

for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in W-most C & p in west_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is vertical

proof end;

theorem Th103: :: JORDAN1A:103

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in N-most C & p in (north_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = N-bound (L~ (Cage (C,n)))

proof end;

theorem Th104: :: JORDAN1A:104

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in E-most C & p in (east_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = E-bound (L~ (Cage (C,n)))

proof end;

theorem Th105: :: JORDAN1A:105

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in S-most C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = S-bound (L~ (Cage (C,n)))

proof end;

theorem Th106: :: JORDAN1A:106

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = W-bound (L~ (Cage (C,n)))

proof end;

theorem :: JORDAN1A:107

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in N-most C holds
ex p being Point of (TOP-REAL 2) st (north_halfline x) /\ (L~ (Cage (C,n))) = {p}

proof end;

theorem :: JORDAN1A:108

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in E-most C holds
ex p being Point of (TOP-REAL 2) st (east_halfline x) /\ (L~ (Cage (C,n))) = {p}

proof end;

theorem :: JORDAN1A:109

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in S-most C holds
ex p being Point of (TOP-REAL 2) st (south_halfline x) /\ (L~ (Cage (C,n))) = {p}

proof end;

theorem :: JORDAN1A:110

for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in W-most C holds
ex p being Point of (TOP-REAL 2) st (west_halfline x) /\ (L~ (Cage (C,n))) = {p}

proof end;