:: Gauges and Cages
:: by Artur Korni{\l}owicz, Robert Milewski, Adam Naumowicz and
::
:: Copyright (c) 2000-2017 Association of Mizar Users

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 -> Nat equals :: JORDAN1A:def 1
((len f) div 2) + 1;
coherence
((len f) div 2) + 1 is Nat
;
end;

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

theorem :: JORDAN1A:1
for f being FinSequence st len f is odd holds
len f = (2 * ()) - 1
proof end;

theorem :: JORDAN1A:2
for f being FinSequence st len f is even holds
len f = (2 * ()) - 2
proof end;

registration
cluster non empty being_simple_closed_curve compact non horizontal non vertical for Element of K16( the U1 of ());
existence
ex b1 being Subset of () 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 Th3: :: JORDAN1A:3
for D being non empty Subset of ()
for p being Point of () st p in N-most D holds
p `2 = N-bound D
proof end;

theorem Th4: :: JORDAN1A:4
for D being non empty Subset of ()
for p being Point of () st p in E-most D holds
p `1 = E-bound D
proof end;

theorem Th5: :: JORDAN1A:5
for D being non empty Subset of ()
for p being Point of () st p in S-most D holds
p `2 = S-bound D
proof end;

theorem Th6: :: JORDAN1A:6
for D being non empty Subset of ()
for p being Point of () st p in W-most D holds
p `1 = W-bound D
proof end;

theorem :: JORDAN1A:7
for D being Subset of () holds BDD D misses D
proof end;

theorem Th8: :: JORDAN1A:8
for p being Point of () holds p in Vertical_Line (p `1)
proof end;

theorem :: JORDAN1A:9
for r, s being Real holds |[r,s]| in Vertical_Line r
proof end;

theorem :: JORDAN1A:10
for s being Real
for A being Subset of () st A c= Vertical_Line s holds
A is vertical
proof end;

theorem :: JORDAN1A:11
for p, q being Point of ()
for r being Real st p `1 = q `1 & r in [.(),().] holds
|[(p `1),r]| in LSeg (p,q)
proof end;

theorem :: JORDAN1A:12
for p, q being Point of ()
for r being Real st p `2 = q `2 & r in [.(),().] holds
|[r,(p `2)]| in LSeg (p,q)
proof end;

theorem :: JORDAN1A:13
for p, q being Point of ()
for s being Real st p in Vertical_Line s & q in Vertical_Line s holds
LSeg (p,q) c= Vertical_Line s
proof end;

theorem :: JORDAN1A:14
for A, B being Subset of () st A meets B holds
proj2 .: A meets proj2 .: B
proof end;

theorem :: JORDAN1A:15
for s being Real
for A, B being Subset of () st A misses B & A c= Vertical_Line s & B c= Vertical_Line s holds
proj2 .: A misses proj2 .: B
proof end;

theorem :: JORDAN1A:16
for i, j being 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,())))
proof end;

theorem :: JORDAN1A:17
for i, j being 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 Th18: :: JORDAN1A:18
for i1, i2, j1, j2 being 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 Th19: :: JORDAN1A:19
for i1, i2, j1, j2 being 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 Th20: :: JORDAN1A:20
for t being 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,())) `2 >= N-bound (L~ f)
proof end;

theorem Th21: :: JORDAN1A:21
for t being 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 Th22: :: JORDAN1A:22
for t being 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 Th23: :: JORDAN1A:23
for t being 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 Th24: :: JORDAN1A:24
for i, j being Nat
for G being Go-board st i <= len G & j <= width G holds
not cell (G,i,j) is empty
proof end;

theorem Th25: :: JORDAN1A:25
for i, j being Nat
for G being Go-board st i <= len G & j <= width G holds
cell (G,i,j) is connected
proof end;

theorem Th26: :: JORDAN1A:26
for i being Nat
for G being Go-board st i <= len G holds
not cell (G,i,0) is bounded
proof end;

theorem Th27: :: JORDAN1A:27
for i being Nat
for G being Go-board st i <= len G holds
not cell (G,i,()) is bounded
proof end;

theorem :: JORDAN1A:28
for n being Nat
for D being non empty Subset of () holds width (Gauge (D,n)) = (2 |^ n) + 3
proof end;

theorem :: JORDAN1A:29
for i, j being Nat
for D being non empty Subset of () st i < j holds
len (Gauge (D,i)) < len (Gauge (D,j))
proof end;

theorem :: JORDAN1A:30
for i, j being Nat
for D being non empty Subset of () st i <= j holds
len (Gauge (D,i)) <= len (Gauge (D,j))
proof end;

theorem Th31: :: JORDAN1A:31
for i, m, n being Nat
for D being non empty Subset of () 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 Th32: :: JORDAN1A:32
for i, m, n being Nat
for D being non empty Subset of () 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 Th33: :: JORDAN1A:33
for i, j, m, n being Nat
for D being non empty Subset of () st m <= n & 1 < i & i < len (Gauge (D,m)) & 1 < j & j < width (Gauge (D,m)) holds
for i1, j1 being 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 Th34: :: JORDAN1A:34
for i, m, n being Nat
for D being non empty Subset of () 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 Th35: :: JORDAN1A:35
for i, m, n being Nat
for D being non empty Subset of () 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 :: thesis: for D being non empty Subset of ()
for n being Nat holds
( 1 <= Center (Gauge (D,n)) & Center (Gauge (D,n)) <= len (Gauge (D,n)) )
let D be non empty Subset of (); :: thesis: for n being Nat holds
( 1 <= Center (Gauge (D,n)) & Center (Gauge (D,n)) <= len (Gauge (D,n)) )

let n be 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_1:6;
hence 1 <= Center (Gauge (D,n)) ; :: thesis: Center (Gauge (D,n)) <= len (Gauge (D,n))
0 < len (Gauge (D,n)) by JORDAN8:10;
then (len (Gauge (D,n))) div 2 < len (Gauge (D,n)) by INT_1:56;
hence Center (Gauge (D,n)) <= len (Gauge (D,n)) by NAT_1:13; :: thesis: verum
end;

Lm4: now :: thesis: for D being non empty Subset of ()
for n, j being Nat st 1 <= j & j <= len (Gauge (D,n)) holds
[(Center (Gauge (D,n))),j] in Indices (Gauge (D,n))
let D be non empty Subset of (); :: thesis: for n, j being Nat st 1 <= j & j <= len (Gauge (D,n)) holds
[(Center (Gauge (D,n))),j] in Indices (Gauge (D,n))

let n, j be 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 ;
Center (Gauge (D,n)) <= len (Gauge (D,n)) by Lm3;
hence [(Center (Gauge (D,n))),j] in Indices (Gauge (D,n)) by ; :: thesis: verum
end;

Lm5: now :: thesis: for D being non empty Subset of ()
for n, j being Nat st 1 <= j & j <= len (Gauge (D,n)) holds
[j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n))
let D be non empty Subset of (); :: thesis: for n, j being Nat st 1 <= j & j <= len (Gauge (D,n)) holds
[j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n))

let n, j be 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 ;
Center (Gauge (D,n)) <= len (Gauge (D,n)) by Lm3;
hence [j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n)) by ; :: thesis: verum
end;

Lm6: for n being Nat
for D being non empty Subset of ()
for w being Real st n > 0 holds
(w / (2 |^ n)) * ((Center (Gauge (D,n))) - 2) = w / 2

proof end;

Lm7: now :: thesis: for m, n being Nat
for c, d being Real st 0 <= c & m <= n holds
c / (2 |^ n) <= c / (2 |^ m)
let m, n be Nat; :: thesis: for c, d being Real st 0 <= c & m <= n holds
c / (2 |^ n) <= c / (2 |^ m)

let c, d be Real; :: 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 ;
hence c / (2 |^ n) <= c / (2 |^ m) by ; :: thesis: verum
end;

Lm8: for m, n being Nat
for c, d being Real st 0 <= c & m <= n holds
d + (c / (2 |^ n)) <= d + (c / (2 |^ m))

by ;

Lm9: now :: thesis: for m, n being Nat
for c, d being Real st 0 <= c & m <= n holds
d - (c / (2 |^ m)) <= d - (c / (2 |^ n))
let m, n be Nat; :: thesis: for c, d being Real st 0 <= c & m <= n holds
d - (c / (2 |^ m)) <= d - (c / (2 |^ n))

let c, d be Real; :: 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_1:13; :: thesis: verum
end;

theorem Th36: :: JORDAN1A:36
for i, j, m, n being Nat
for D being non empty Subset of () 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:37
for i, j, m, n being Nat
for D being non empty Subset of () 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 :: thesis: for D being non empty Subset of ()
for n being Nat
for e, w being Real holds w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))
let D be non empty Subset of (); :: thesis: for n being Nat
for e, w being Real holds w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))

let n be Nat; :: thesis: for e, w being Real holds w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))
let e, w be Real; :: thesis: w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))
A1: 2 |^ n <> 0 by NEWTON:83;
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
.= e + ((e - w) / (2 |^ n)) ; :: thesis: verum
end;

Lm11: now :: thesis: for D being non empty Subset of ()
for n, i being Nat st [i,1] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (i,1)) `2 = () - ((() - ()) / (2 |^ n))
let D be non empty Subset of (); :: thesis: for n, i being Nat st [i,1] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (i,1)) `2 = () - ((() - ()) / (2 |^ n))

let n, i be Nat; :: thesis: ( [i,1] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (i,1)) `2 = () - ((() - ()) / (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 = () - ((() - ()) / (2 |^ n))
hence ((Gauge (D,n)) * (i,1)) `2 = |[(() + (((() - ()) / (2 |^ n)) * (i - 2))),(() + (((() - ()) / (2 |^ n)) * (1 - 2)))]| `2 by JORDAN8:def 1
.= () - ((() - ()) / (2 |^ n)) by EUCLID:52 ;
:: thesis: verum
end;

Lm12: now :: thesis: for D being non empty Subset of ()
for n, i being Nat st [1,i] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (1,i)) `1 = () - ((() - ()) / (2 |^ n))
let D be non empty Subset of (); :: thesis: for n, i being Nat st [1,i] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (1,i)) `1 = () - ((() - ()) / (2 |^ n))

let n, i be Nat; :: thesis: ( [1,i] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (1,i)) `1 = () - ((() - ()) / (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 = () - ((() - ()) / (2 |^ n))
hence ((Gauge (D,n)) * (1,i)) `1 = |[(() + (((() - ()) / (2 |^ n)) * (1 - 2))),(() + (((() - ()) / (2 |^ n)) * (i - 2)))]| `1 by JORDAN8:def 1
.= () - ((() - ()) / (2 |^ n)) by EUCLID:52 ;
:: thesis: verum
end;

Lm13: now :: thesis: for D being non empty Subset of ()
for n, i being Nat st [i,(len (Gauge (D,n)))] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = () + ((() - ()) / (2 |^ n))
let D be non empty Subset of (); :: thesis: for n, i being Nat st [i,(len (Gauge (D,n)))] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = () + ((() - ()) / (2 |^ n))

let n, i be Nat; :: thesis: ( [i,(len (Gauge (D,n)))] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = () + ((() - ()) / (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 = () + ((() - ()) / (2 |^ n))
hence ((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = |[(() + (((() - ()) / (2 |^ n)) * (i - 2))),(() + (((() - ()) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)))]| `2 by JORDAN8:def 1
.= () + (((() - ()) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) by EUCLID:52
.= () + ((() - ()) / (2 |^ n)) by Lm10 ;
:: thesis: verum
end;

Lm14: now :: thesis: for D being non empty Subset of ()
for n, i being Nat st [(len (Gauge (D,n))),i] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = () + ((() - ()) / (2 |^ n))
let D be non empty Subset of (); :: thesis: for n, i being Nat st [(len (Gauge (D,n))),i] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = () + ((() - ()) / (2 |^ n))

let n, i be Nat; :: thesis: ( [(len (Gauge (D,n))),i] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = () + ((() - ()) / (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 = () + ((() - ()) / (2 |^ n))
hence ((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = |[(() + (((() - ()) / (2 |^ n)) * ((len (Gauge (D,n))) - 2))),(() + (((() - ()) / (2 |^ n)) * (i - 2)))]| `1 by JORDAN8:def 1
.= () + (((() - ()) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) by EUCLID:52
.= () + ((() - ()) / (2 |^ n)) by Lm10 ;
:: thesis: verum
end;

theorem :: JORDAN1A:38
for i being Nat
for C being connected compact non horizontal non vertical Subset of () st 1 <= i & i <= len (Gauge (C,1)) holds
((Gauge (C,1)) * ((Center (Gauge (C,1))),i)) `1 = (() + ()) / 2
proof end;

theorem :: JORDAN1A:39
for i being Nat
for C being connected compact non horizontal non vertical Subset of () st 1 <= i & i <= len (Gauge (C,1)) holds
((Gauge (C,1)) * (i,(Center (Gauge (C,1))))) `2 = (() + ()) / 2
proof end;

theorem Th40: :: JORDAN1A:40
for i, j, m, n being Nat
for E being compact non horizontal non vertical Subset of () 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:41
for i, j, m, n being Nat
for E being compact non horizontal non vertical Subset of () 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:42
for i, j, m, n being Nat
for E being compact non horizontal non vertical Subset of () 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:43
for i, j, m, n being Nat
for E being compact non horizontal non vertical Subset of () 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:44
for m, n being Nat
for E being compact non horizontal non vertical Subset of () 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:45
for j, m, n being Nat
for E being compact non horizontal non vertical Subset of () 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:46
for j, m, n being Nat
for E being compact non horizontal non vertical Subset of () 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 Th47: :: JORDAN1A:47
for i, j, m, n being Nat
for E being compact non horizontal non vertical Subset of () st m <= n & 1 < i & i + 1 < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) holds
for i1, j1 being 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:48
for i, j, m, n being Nat
for E being compact non horizontal non vertical Subset of () st m <= n & 3 <= i & i < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) holds
for i1, j1 being 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:49
for i, n being Nat
for C being compact non horizontal non vertical Subset of () st i <= len (Gauge (C,n)) holds
cell ((Gauge (C,n)),i,0) c= UBD C
proof end;

theorem :: JORDAN1A:50
for i, n being Nat
for E, C being compact non horizontal non vertical Subset of () st i <= len (Gauge (E,n)) holds
cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E
proof end;

theorem Th51: :: JORDAN1A:51
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p being Point of () st p in C holds
north_halfline p meets L~ (Cage (C,n))
proof end;

theorem Th52: :: JORDAN1A:52
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p being Point of () st p in C holds
east_halfline p meets L~ (Cage (C,n))
proof end;

theorem Th53: :: JORDAN1A:53
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p being Point of () st p in C holds
south_halfline p meets L~ (Cage (C,n))
proof end;

theorem Th54: :: JORDAN1A:54
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p being Point of () st p in C holds
west_halfline p meets L~ (Cage (C,n))
proof end;

Lm15: for n being Nat
for C being connected compact non horizontal non vertical Subset of () ex k, t being 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 Nat
for C being connected compact non horizontal non vertical Subset of () ex k, t being 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 Nat
for C being connected compact non horizontal non vertical Subset of () ex k, t being 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 Th55: :: JORDAN1A:55
for n being Nat
for C being connected compact non horizontal non vertical Subset of () ex k, t being 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 Th56: :: JORDAN1A:56
for n being Nat
for C being connected compact non horizontal non vertical Subset of () ex k, t being 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 Th57: :: JORDAN1A:57
for n being Nat
for C being connected compact non horizontal non vertical Subset of () ex k, t being 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 Th58: :: JORDAN1A:58
for k, n, t being Nat
for C being connected compact non horizontal non vertical Subset of () 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 Th59: :: JORDAN1A:59
for k, n, t being Nat
for C being connected compact non horizontal non vertical Subset of () 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 Th60: :: JORDAN1A:60
for k, n, t being Nat
for C being connected compact non horizontal non vertical Subset of () 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 Th61: :: JORDAN1A:61
for k, n, t being Nat
for C being connected compact non horizontal non vertical Subset of () 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 Th62: :: JORDAN1A:62
for n being Nat
for C being connected compact non horizontal non vertical Subset of () holds W-bound (L~ (Cage (C,n))) = () - ((() - ()) / (2 |^ n))
proof end;

theorem Th63: :: JORDAN1A:63
for n being Nat
for C being connected compact non horizontal non vertical Subset of () holds S-bound (L~ (Cage (C,n))) = () - ((() - ()) / (2 |^ n))
proof end;

theorem Th64: :: JORDAN1A:64
for n being Nat
for C being connected compact non horizontal non vertical Subset of () holds E-bound (L~ (Cage (C,n))) = () + ((() - ()) / (2 |^ n))
proof end;

theorem :: JORDAN1A:65
for m, n being Nat
for C being connected compact non horizontal non vertical Subset of () 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:66
for m, n being Nat
for C being connected compact non horizontal non vertical Subset of () 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:67
for i, j being Nat
for C being connected compact non horizontal non vertical Subset of () st i < j holds
E-bound (L~ (Cage (C,j))) < E-bound (L~ (Cage (C,i)))
proof end;

theorem :: JORDAN1A:68
for i, j being Nat
for C being connected compact non horizontal non vertical Subset of () st i < j holds
W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,j)))
proof end;

theorem :: JORDAN1A:69
for i, j being Nat
for C being connected compact non horizontal non vertical Subset of () st i < j holds
S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,j)))
proof end;

theorem :: JORDAN1A:70
for i, n being Nat
for C being connected compact non horizontal non vertical Subset of () 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:71
for i, n being Nat
for C being connected compact non horizontal non vertical Subset of () 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:72
for i, n being Nat
for C being connected compact non horizontal non vertical Subset of () 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:73
for i, n being Nat
for C being connected compact non horizontal non vertical Subset of () st 1 <= i & i <= len (Gauge (C,n)) holds
W-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (1,i)) `1
proof end;

theorem Th74: :: JORDAN1A:74
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () st x in C & p in () /\ (L~ (Cage (C,n))) holds
p `2 > x `2
proof end;

theorem Th75: :: JORDAN1A:75
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () st x in C & p in () /\ (L~ (Cage (C,n))) holds
p `1 > x `1
proof end;

theorem Th76: :: JORDAN1A:76
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () st x in C & p in () /\ (L~ (Cage (C,n))) holds
p `2 < x `2
proof end;

theorem Th77: :: JORDAN1A:77
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () st x in C & p in () /\ (L~ (Cage (C,n))) holds
p `1 < x `1
proof end;

theorem Th78: :: JORDAN1A:78
for i, n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () 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 Th79: :: JORDAN1A:79
for i, n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () 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 Th80: :: JORDAN1A:80
for i, n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () 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 Th81: :: JORDAN1A:81
for i, n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () 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 Th82: :: JORDAN1A:82
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () st x in N-most C & p in () /\ (L~ (Cage (C,n))) holds
p `2 = N-bound (L~ (Cage (C,n)))
proof end;

theorem Th83: :: JORDAN1A:83
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () st x in E-most C & p in () /\ (L~ (Cage (C,n))) holds
p `1 = E-bound (L~ (Cage (C,n)))
proof end;

theorem Th84: :: JORDAN1A:84
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () st x in S-most C & p in () /\ (L~ (Cage (C,n))) holds
p `2 = S-bound (L~ (Cage (C,n)))
proof end;

theorem Th85: :: JORDAN1A:85
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for p, x being Point of () st x in W-most C & p in () /\ (L~ (Cage (C,n))) holds
p `1 = W-bound (L~ (Cage (C,n)))
proof end;

theorem :: JORDAN1A:86
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for x being Point of () st x in N-most C holds
ex p being Point of () st () /\ (L~ (Cage (C,n))) = {p}
proof end;

theorem :: JORDAN1A:87
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for x being Point of () st x in E-most C holds
ex p being Point of () st () /\ (L~ (Cage (C,n))) = {p}
proof end;

theorem :: JORDAN1A:88
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for x being Point of () st x in S-most C holds
ex p being Point of () st () /\ (L~ (Cage (C,n))) = {p}
proof end;

theorem :: JORDAN1A:89
for n being Nat
for C being connected compact non horizontal non vertical Subset of ()
for x being Point of () st x in W-most C holds
ex p being Point of () st () /\ (L~ (Cage (C,n))) = {p}
proof end;