:: Bounded Domains and Unbounded Domains
:: by Yatsuka Nakamura , Andrzej Trybulec and Czeslaw Bylinski
::
:: Received January 7, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users


registration
let n be Nat;
cluster TOP-REAL n -> add-continuous Mult-continuous ;
coherence
( TOP-REAL n is add-continuous & TOP-REAL n is Mult-continuous )
proof end;
end;

theorem :: JORDAN2C:1
canceled;

theorem :: JORDAN2C:2
canceled;

theorem :: JORDAN2C:3
canceled;

theorem :: JORDAN2C:4
canceled;

theorem :: JORDAN2C:5
canceled;

::$CT 5
theorem Th1: :: JORDAN2C:6
for r, s being Real
for f being increasing FinSequence of REAL st rng f = {r,s} & len f = 2 & r <= s holds
( f . 1 = r & f . 2 = s )
proof end;

theorem :: JORDAN2C:7
canceled;

::$CT
theorem :: JORDAN2C:8
for n being Nat
for q being Point of (TOP-REAL n) holds |.|.q.|.| = |.q.| by ABSVALUE:def 1;

theorem Th3: :: JORDAN2C:9
for n being Nat
for q1, q2 being Point of (TOP-REAL n) holds |.(|.q1.| - |.q2.|).| <= |.(q1 - q2).|
proof end;

theorem Th4: :: JORDAN2C:10
for r being Real holds |.|[r]|.| = |.r.|
proof end;

Lm1: for n being Nat
for r being Real st r > 0 holds
for x, y, z being Element of (Euclid n) st x = 0* n holds
for p being Element of (TOP-REAL n) st p = y & r * p = z holds
r * (dist (x,y)) = dist (x,z)
proof end;

Lm2: for n being Nat
for r, s being Real st r > 0 holds
for x being Element of (Euclid n) st x = 0* n holds
for A being Subset of (TOP-REAL n) st A = Ball (x,s) holds
r * A = Ball (x,(r * s))
proof end;

Lm3: for n being Nat
for r, s, t being Real st 0 < s & s <= t holds
for x being Element of (Euclid n) st x = 0* n holds
for BA being Subset of (TOP-REAL n) st BA = Ball (x,r) holds
s * BA c= t * BA
proof end;

theorem Th5: :: JORDAN2C:11
for n being Nat
for A being Subset of (TOP-REAL n) holds
( A is bounded iff A is bounded Subset of (Euclid n) )
proof end;

theorem :: JORDAN2C:12
for n being Nat
for A, B being Subset of (TOP-REAL n) st B is bounded & A c= B holds
A is bounded by RLTOPSP1:42;

definition
let n be Nat;
let A, B be Subset of (TOP-REAL n);
pred B is_inside_component_of A means :: JORDAN2C:def 2
( B is_a_component_of A ` & B is bounded );
end;

:: deftheorem JORDAN2C:def 1 :
canceled;

:: deftheorem defines is_inside_component_of JORDAN2C:def 2 :
for n being Nat
for A, B being Subset of (TOP-REAL n) holds
( B is_inside_component_of A iff ( B is_a_component_of A ` & B is bounded ) );

registration
let M be non empty MetrStruct ;
cluster bounded for Element of bool the carrier of M;
existence
ex b1 being Subset of M st b1 is bounded
proof end;
end;

theorem Th7: :: JORDAN2C:13
for n being Nat
for A, B being Subset of (TOP-REAL n) holds
( B is_inside_component_of A iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) )
proof end;

definition
let n be Nat;
let A, B be Subset of (TOP-REAL n);
pred B is_outside_component_of A means :: JORDAN2C:def 3
( B is_a_component_of A ` & not B is bounded );
end;

:: deftheorem defines is_outside_component_of JORDAN2C:def 3 :
for n being Nat
for A, B being Subset of (TOP-REAL n) holds
( B is_outside_component_of A iff ( B is_a_component_of A ` & not B is bounded ) );

theorem Th8: :: JORDAN2C:14
for n being Nat
for A, B being Subset of (TOP-REAL n) holds
( B is_outside_component_of A iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) )
proof end;

theorem :: JORDAN2C:15
for n being Nat
for A, B being Subset of (TOP-REAL n) st B is_inside_component_of A holds
B c= A ` by SPRECT_1:5;

theorem :: JORDAN2C:16
for n being Nat
for A, B being Subset of (TOP-REAL n) st B is_outside_component_of A holds
B c= A ` by SPRECT_1:5;

definition
let n be Nat;
let A be Subset of (TOP-REAL n);
func BDD A -> Subset of (TOP-REAL n) equals :: JORDAN2C:def 4
 union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } ;
correctness
coherence
union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } is Subset of (TOP-REAL n);
proof end;
end;

:: deftheorem defines BDD JORDAN2C:def 4 :
for n being Nat
for A being Subset of (TOP-REAL n) holds BDD A = union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } ;

definition
let n be Nat;
let A be Subset of (TOP-REAL n);
func UBD A -> Subset of (TOP-REAL n) equals :: JORDAN2C:def 5
 union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } ;
correctness
coherence
union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } is Subset of (TOP-REAL n);
proof end;
end;

:: deftheorem defines UBD JORDAN2C:def 5 :
for n being Nat
for A being Subset of (TOP-REAL n) holds UBD A = union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } ;

registration
let n be Nat;
cluster [#] (TOP-REAL n) -> convex ;
coherence
[#] (TOP-REAL n) is convex ;
end;

registration
let n be Nat;
cluster [#] (TOP-REAL n) -> a_component ;
coherence
[#] (TOP-REAL n) is a_component
proof end;
end;

theorem :: JORDAN2C:17
canceled;

theorem :: JORDAN2C:18
canceled;

theorem :: JORDAN2C:19
canceled;

::$CT 3
theorem Th11: :: JORDAN2C:20
for n being Nat
for A being Subset of (TOP-REAL n) holds BDD A is a_union_of_components of (TOP-REAL n) | (A `)
proof end;

theorem Th12: :: JORDAN2C:21
for n being Nat
for A being Subset of (TOP-REAL n) holds UBD A is a_union_of_components of (TOP-REAL n) | (A `)
proof end;

theorem Th13: :: JORDAN2C:22
for n being Nat
for A, B being Subset of (TOP-REAL n) st B is_inside_component_of A holds
B c= BDD A
proof end;

theorem Th14: :: JORDAN2C:23
for n being Nat
for A, B being Subset of (TOP-REAL n) st B is_outside_component_of A holds
B c= UBD A
proof end;

theorem Th15: :: JORDAN2C:24
for n being Nat
for A being Subset of (TOP-REAL n) holds BDD A misses UBD A
proof end;

theorem Th16: :: JORDAN2C:25
for n being Nat
for A being Subset of (TOP-REAL n) holds BDD A c= A `
proof end;

theorem Th17: :: JORDAN2C:26
for n being Nat
for A being Subset of (TOP-REAL n) holds UBD A c= A `
proof end;

theorem Th18: :: JORDAN2C:27
for n being Nat
for A being Subset of (TOP-REAL n) holds (BDD A) \/ (UBD A) = A `
proof end;

theorem Th19: :: JORDAN2C:28
for n being Nat
for P being Subset of (TOP-REAL n) st P = REAL n holds
P is connected
proof end;

theorem :: JORDAN2C:29
canceled;

theorem :: JORDAN2C:30
canceled;

theorem :: JORDAN2C:31
canceled;

theorem :: JORDAN2C:32
canceled;

::$CT 4
theorem Th20: :: JORDAN2C:33
for n being Nat
for W being Subset of (Euclid n) st n >= 1 & W = REAL n holds
not W is bounded
proof end;

theorem Th21: :: JORDAN2C:34
for n being Nat
for A being Subset of (TOP-REAL n) holds
( A is bounded iff ex r being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r )
proof end;

theorem Th22: :: JORDAN2C:35
for n being Nat st n >= 1 holds
not [#] (TOP-REAL n) is bounded
proof end;

theorem Th23: :: JORDAN2C:36
for n being Nat st n >= 1 holds
UBD ({} (TOP-REAL n)) = REAL n
proof end;

theorem Th24: :: JORDAN2C:37
for n being Nat
for w1, w2, w3 being Point of (TOP-REAL n)
for P being non empty Subset of (TOP-REAL n)
for h1, h2 being Function of I[01],((TOP-REAL n) | P) st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01],((TOP-REAL n) | P) st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 )
proof end;

theorem Th25: :: JORDAN2C:38
for n being Nat
for P being Subset of (TOP-REAL n)
for w1, w2, w3 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
proof end;

theorem Th26: :: JORDAN2C:39
for n being Nat
for P being Subset of (TOP-REAL n)
for w1, w2, w3, w4 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 )
proof end;

theorem Th27: :: JORDAN2C:40
for n being Nat
for P being Subset of (TOP-REAL n)
for w1, w2, w3, w4, w5, w6, w7 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & w5 in P & w6 in P & w7 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P & LSeg (w4,w5) c= P & LSeg (w5,w6) c= P & LSeg (w6,w7) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w7 = h . 1 )
proof end;

theorem Th28: :: JORDAN2C:41
for n being Nat
for w1, w2 being Point of (TOP-REAL n)
for P being Subset of (TopSpaceMetr (Euclid n)) st P = LSeg (w1,w2) & not 0. (TOP-REAL n) in LSeg (w1,w2) holds
ex w0 being Point of (TOP-REAL n) st
( w0 in LSeg (w1,w2) & |.w0.| > 0 & |.w0.| = (dist_min P) . (0. (TOP-REAL n)) )
proof end;

theorem Th29: :: JORDAN2C:42
for n being Nat
for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w4 being Point of (TOP-REAL n) st Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
proof end;

theorem Th30: :: JORDAN2C:43
for n being Nat
for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w4 being Point of (TOP-REAL n) st Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
proof end;

theorem :: JORDAN2C:44
for f being FinSequence of REAL holds
( f is Element of REAL (len f) & f is Point of (TOP-REAL (len f)) ) by EUCLID:76;

theorem Th32: :: JORDAN2C:45
for n being Nat
for x being Element of REAL n
for f, g being FinSequence of REAL
for r being Real st f = x & g = r * x holds
( len f = len g & ( for i being Element of NAT st 1 <= i & i <= len f holds
g /. i = r * (f /. i) ) )
proof end;

theorem Th33: :: JORDAN2C:46
for n being Nat
for x being Element of REAL n
for f being FinSequence st x <> 0* n & x = f holds
ex i being Element of NAT st
( 1 <= i & i <= n & f . i <> 0 )
proof end;

theorem Th34: :: JORDAN2C:47
for n being Nat
for x being Element of REAL n st n >= 2 & x <> 0* n holds
ex y being Element of REAL n st
for r being Real holds
( not y = r * x & not x = r * y )
proof end;

theorem Th35: :: JORDAN2C:48
for n being Nat
for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
proof end;

theorem Th36: :: JORDAN2C:49
for n being Nat
for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
proof end;

theorem Th37: :: JORDAN2C:50
for n being Nat
for a being Real st n >= 1 holds
{ q where q is Point of (TOP-REAL n) : |.q.| > a } <> {}
proof end;

theorem Th38: :: JORDAN2C:51
for n being Nat
for a being Real
for P being Subset of (TOP-REAL n) st n >= 2 & P = { q where q is Point of (TOP-REAL n) : |.q.| > a } holds
P is connected
proof end;

theorem Th39: :: JORDAN2C:52
for n being Nat
for a being Real st n >= 1 holds
(REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } <> {}
proof end;

theorem Th40: :: JORDAN2C:53
for n being Nat
for a being Real
for P being Subset of (TOP-REAL n) st n >= 2 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
P is connected
proof end;

theorem Th41: :: JORDAN2C:54
for a being Real
for n being Nat
for P being Subset of (TOP-REAL n) st n >= 1 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
not P is bounded
proof end;

theorem Th42: :: JORDAN2C:55
for a being Real
for P being Subset of (TOP-REAL 1) st P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } holds
P is convex
proof end;

theorem Th43: :: JORDAN2C:56
for a being Real
for P being Subset of (TOP-REAL 1) st P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } holds
P is convex
proof end;

theorem :: JORDAN2C:57
canceled;

theorem :: JORDAN2C:58
canceled;

::$CT 2
theorem Th44: :: JORDAN2C:59
for W being Subset of (Euclid 1)
for a being Real st W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } holds
not W is bounded
proof end;

theorem Th45: :: JORDAN2C:60
for W being Subset of (Euclid 1)
for a being Real st W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } holds
not W is bounded
proof end;

theorem Th46: :: JORDAN2C:61
for n being Nat
for W being Subset of (Euclid n)
for a being Real st n >= 2 & W = { q where q is Point of (TOP-REAL n) : |.q.| > a } holds
not W is bounded
proof end;

theorem Th47: :: JORDAN2C:62
for n being Nat
for W being Subset of (Euclid n)
for a being Real st n >= 2 & W = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
not W is bounded
proof end;

theorem Th48: :: JORDAN2C:63
for n being Nat
for P, P1, Q being Subset of (TOP-REAL n)
for W being Subset of (Euclid n) st P = W & P is connected & not W is bounded & P1 = Component_of (Down (P,(Q `))) & W misses Q holds
P1 is_outside_component_of Q
proof end;

theorem Th49: :: JORDAN2C:64
for n being Nat
for A being Subset of (Euclid n)
for B being non empty Subset of (Euclid n)
for C being Subset of ((Euclid n) | B) st A = C & C is bounded holds
A is bounded
proof end;

theorem Th50: :: JORDAN2C:65
for n being Nat
for A being Subset of (TOP-REAL n) st A is compact holds
A is bounded
proof end;

registration
let n be Element of NAT ;
cluster compact -> bounded for Element of bool the carrier of (TOP-REAL n);
coherence
for b1 being Subset of (TOP-REAL n) st b1 is compact holds
b1 is bounded by Th50;
end;

theorem Th51: :: JORDAN2C:66
for n being Nat
for A being Subset of (TOP-REAL n) st 1 <= n & A is bounded holds
A ` <> {}
proof end;

theorem Th52: :: JORDAN2C:67
for n being Nat
for r being Real holds
( ex B being Subset of (Euclid n) st B = { q where q is Point of (TOP-REAL n) : |.q.| < r } & ( for A being Subset of (Euclid n) st A = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < r } holds
A is bounded ) )
proof end;

theorem Th53: :: JORDAN2C:68
for n being Nat
for A being Subset of (TOP-REAL n) st n >= 2 & A is bounded holds
UBD A is_outside_component_of A
proof end;

theorem Th54: :: JORDAN2C:69
for n being Nat
for a being Real
for P being Subset of (TOP-REAL n) st P = { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
P is convex
proof end;

theorem Th55: :: JORDAN2C:70
for n being Nat
for u being Point of (Euclid n)
for a being Real
for P being Subset of (TOP-REAL n) st P = Ball (u,a) holds
P is convex
proof end;

theorem :: JORDAN2C:71
canceled;

::$CT
theorem Th56: :: JORDAN2C:72
for n being Nat
for r being Real
for p, q being Point of (TOP-REAL n)
for u being Point of (Euclid n) st p <> q & p in Ball (u,r) & q in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p & h . 1 = q & rng h c= Ball (u,r) )
proof end;

theorem Th57: :: JORDAN2C:73
for n being Nat
for r being Real
for p, p1, p2 being Point of (TOP-REAL n)
for u being Point of (Euclid n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) )
proof end;

theorem Th58: :: JORDAN2C:74
for n being Nat
for r being Real
for p, p1, p2 being Point of (TOP-REAL n)
for u being Point of (Euclid n)
for P being Subset of (TOP-REAL n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & rng f c= P & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) & Ball (u,r) c= P holds
ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p )
proof end;

theorem Th59: :: JORDAN2C:75
for n being Nat
for R being Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st R is connected & R is open & P = { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } holds
P is open
proof end;

theorem Th60: :: JORDAN2C:76
for n being Nat
for p being Point of (TOP-REAL n)
for R, P being Subset of (TOP-REAL n) st R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
P is open
proof end;

theorem Th61: :: JORDAN2C:77
for n being Nat
for p being Point of (TOP-REAL n)
for P, R being Subset of (TOP-REAL n) st p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
P c= R
proof end;

theorem Th62: :: JORDAN2C:78
for n being Nat
for P, R being Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n) st R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
R c= P
proof end;

theorem Th63: :: JORDAN2C:79
for n being Nat
for R being Subset of (TOP-REAL n)
for p, q being Point of (TOP-REAL n) st R is connected & R is open & p in R & q in R & p <> q holds
ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q )
proof end;

theorem Th64: :: JORDAN2C:80
for n being Nat
for A being Subset of (TOP-REAL n)
for a being Real st A = { q where q is Point of (TOP-REAL n) : |.q.| = a } holds
( A ` is open & A is closed )
proof end;

theorem Th65: :: JORDAN2C:81
for n being Nat
for B being non empty Subset of (TOP-REAL n) st B is open holds
(TOP-REAL n) | B is locally_connected
proof end;

theorem Th66: :: JORDAN2C:82
for n being Nat
for B being non empty Subset of (TOP-REAL n)
for A being Subset of (TOP-REAL n)
for a being Real st A = { q where q is Point of (TOP-REAL n) : |.q.| = a } & A ` = B holds
(TOP-REAL n) | B is locally_connected
proof end;

theorem Th67: :: JORDAN2C:83
for n being Nat
for f being Function of (TOP-REAL n),R^1 st ( for q being Point of (TOP-REAL n) holds f . q = |.q.| ) holds
f is continuous
proof end;

theorem Th68: :: JORDAN2C:84
for n being Nat ex f being Function of (TOP-REAL n),R^1 st
( ( for q being Point of (TOP-REAL n) holds f . q = |.q.| ) & f is continuous )
proof end;

theorem Th69: :: JORDAN2C:85
for n being Nat
for g being Function of I[01],(TOP-REAL n) st g is continuous holds
ex f being Function of I[01],R^1 st
( ( for t being Point of I[01] holds f . t = |.(g . t).| ) & f is continuous )
proof end;

theorem Th70: :: JORDAN2C:86
for n being Nat
for g being Function of I[01],(TOP-REAL n)
for a being Real st g is continuous & |.(g /. 0).| <= a & a <= |.(g /. 1).| holds
ex s being Point of I[01] st |.(g /. s).| = a
proof end;

theorem Th71: :: JORDAN2C:87
for n being Nat
for r being Real
for q being Point of (TOP-REAL n) st q = <*r*> holds
|.q.| = |.r.|
proof end;

theorem :: JORDAN2C:88
for n being Nat
for A being Subset of (TOP-REAL n)
for a being Real st n >= 1 & a > 0 & A = { q where q is Point of (TOP-REAL n) : |.q.| = a } holds
BDD A is_inside_component_of A
proof end;

theorem Th73: :: JORDAN2C:89
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( len (GoB (SpStSeq D)) = 2 & width (GoB (SpStSeq D)) = 2 & (SpStSeq D) /. 1 = (GoB (SpStSeq D)) * (1,2) & (SpStSeq D) /. 2 = (GoB (SpStSeq D)) * (2,2) & (SpStSeq D) /. 3 = (GoB (SpStSeq D)) * (2,1) & (SpStSeq D) /. 4 = (GoB (SpStSeq D)) * (1,1) & (SpStSeq D) /. 5 = (GoB (SpStSeq D)) * (1,2) )
proof end;

theorem Th74: :: JORDAN2C:90
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds not LeftComp (SpStSeq D) is bounded
proof end;

theorem Th75: :: JORDAN2C:91
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds LeftComp (SpStSeq D) c= UBD (L~ (SpStSeq D))
proof end;

theorem Th76: :: JORDAN2C:92
for G being TopSpace
for A, B, C being Subset of G st A is a_component & B is a_component & C is connected & A meets C & B meets C holds
A = B
proof end;

theorem Th77: :: JORDAN2C:93
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for B being Subset of (TOP-REAL 2) st B is_a_component_of (L~ (SpStSeq D)) ` & not B is bounded holds
B = LeftComp (SpStSeq D)
proof end;

theorem Th78: :: JORDAN2C:94
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( RightComp (SpStSeq D) c= BDD (L~ (SpStSeq D)) & RightComp (SpStSeq D) is bounded )
proof end;

theorem Th79: :: JORDAN2C:95
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( LeftComp (SpStSeq D) = UBD (L~ (SpStSeq D)) & RightComp (SpStSeq D) = BDD (L~ (SpStSeq D)) )
proof end;

theorem Th80: :: JORDAN2C:96
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( UBD (L~ (SpStSeq D)) <> {} & UBD (L~ (SpStSeq D)) is_outside_component_of L~ (SpStSeq D) & BDD (L~ (SpStSeq D)) <> {} & BDD (L~ (SpStSeq D)) is_inside_component_of L~ (SpStSeq D) )
proof end;

theorem Th81: :: JORDAN2C:97
for G being non empty TopSpace
for A being Subset of G st A ` <> {} holds
( A is boundary iff for x being set
for V being Subset of G st x in A & x in V & V is open holds
ex B being Subset of G st
( B is_a_component_of A ` & V meets B ) )
proof end;

theorem Th82: :: JORDAN2C:98
for A being Subset of (TOP-REAL 2) st A ` <> {} holds
( ( A is boundary & A is Jordan ) iff ex A1, A2 being Subset of (TOP-REAL 2) st
( A ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A = (Cl A1) \ A1 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) ) )
proof end;

theorem Th83: :: JORDAN2C:99
for n being Nat
for p being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st n >= 1 & P = {p} holds
P is boundary
proof end;

theorem Th84: :: JORDAN2C:100
for p, q being Point of (TOP-REAL 2)
for r being Real st p `1 = q `2 & - (p `2) = q `1 & p = r * q holds
( p `1 = 0 & p `2 = 0 & p = 0. (TOP-REAL 2) )
proof end;

theorem Th85: :: JORDAN2C:101
for q1, q2 being Point of (TOP-REAL 2) holds LSeg (q1,q2) is boundary
proof end;

registration
let q1, q2 be Point of (TOP-REAL 2);
cluster LSeg (q1,q2) -> boundary ;
coherence
LSeg (q1,q2) is boundary by Th85;
end;

theorem Th86: :: JORDAN2C:102
for f being FinSequence of (TOP-REAL 2) holds L~ f is boundary
proof end;

registration
let f be FinSequence of (TOP-REAL 2);
cluster L~ f -> boundary ;
coherence
L~ f is boundary by Th86;
end;

theorem Th87: :: JORDAN2C:103
for n being Nat
for r being Real
for ep being Point of (Euclid n)
for p, q being Point of (TOP-REAL n) st p = ep & q in Ball (ep,r) holds
( |.(p - q).| < r & |.(q - p).| < r )
proof end;

theorem :: JORDAN2C:104
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for a being Real
for p being Point of (TOP-REAL 2) st a > 0 & p in L~ (SpStSeq D) holds
ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a )
proof end;

theorem :: JORDAN2C:105
REAL 0 = {(0. (TOP-REAL 0))} by EUCLID:77;

theorem Th90: :: JORDAN2C:106
for n being Nat
for A being Subset of (TOP-REAL n) st A is bounded holds
BDD A is bounded
proof end;

theorem Th91: :: JORDAN2C:107
for G being non empty TopSpace
for A, B, C, D being Subset of G st B is a_component & C is a_component & A \/ B = the carrier of G & C misses A holds
C = B
proof end;

theorem Th92: :: JORDAN2C:108
for A being Subset of (TOP-REAL 2) st A is bounded & A is Jordan holds
BDD A is_inside_component_of A
proof end;

theorem :: JORDAN2C:109
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for a being Real
for p being Point of (TOP-REAL 2) st a > 0 & p in L~ (SpStSeq D) holds
ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a )
proof end;

theorem :: JORDAN2C:110
for f being V23() standard clockwise_oriented special_circular_sequence
for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `1 < W-bound (L~ f) holds
p in LeftComp f
proof end;

theorem :: JORDAN2C:111
for f being V23() standard clockwise_oriented special_circular_sequence
for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `1 > E-bound (L~ f) holds
p in LeftComp f
proof end;

theorem :: JORDAN2C:112
for f being V23() standard clockwise_oriented special_circular_sequence
for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `2 < S-bound (L~ f) holds
p in LeftComp f
proof end;

theorem :: JORDAN2C:113
for f being V23() standard clockwise_oriented special_circular_sequence
for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `2 > N-bound (L~ f) holds
p in LeftComp f
proof end;

:: Moved from GOBRD14, AK, 22.02.2006
theorem :: JORDAN2C:114
for T being TopSpace
for A, B being Subset of T st B is_a_component_of A holds
B is connected
proof end;

theorem :: JORDAN2C:115
for n being Nat
for A, B being Subset of (TOP-REAL n) st B is_inside_component_of A holds
B is connected
proof end;

theorem Th100: :: JORDAN2C:116
for n being Nat
for A, B being Subset of (TOP-REAL n) st B is_outside_component_of A holds
B is connected
proof end;

theorem :: JORDAN2C:117
for n being Nat
for A, B being Subset of (TOP-REAL n) st B is_a_component_of A ` holds
A misses B by SPRECT_1:5, SUBSET_1:23;

theorem :: JORDAN2C:118
for n being Nat
for R, P, Q being Subset of (TOP-REAL n) st P is_outside_component_of Q & R is_inside_component_of Q holds
P misses R
proof end;

theorem :: JORDAN2C:119
for n being Nat st 2 <= n holds
for A, B, P being Subset of (TOP-REAL n) st P is bounded & A is_outside_component_of P & B is_outside_component_of P holds
A = B
proof end;

registration
let C be closed Subset of (TOP-REAL 2);
cluster BDD C -> open ;
coherence
BDD C is open
proof end;
cluster UBD C -> open ;
coherence
UBD C is open
proof end;
end;

registration
let C be compact Subset of (TOP-REAL 2);
cluster UBD C -> connected ;
coherence
UBD C is connected by Th53, Th100;
end;

theorem Th104: :: JORDAN2C:120
for p being Point of (TOP-REAL 2) holds not west_halfline p is bounded
proof end;

theorem Th105: :: JORDAN2C:121
for p being Point of (TOP-REAL 2) holds not east_halfline p is bounded
proof end;

theorem Th106: :: JORDAN2C:122
for p being Point of (TOP-REAL 2) holds not north_halfline p is bounded
proof end;

theorem Th107: :: JORDAN2C:123
for p being Point of (TOP-REAL 2) holds not south_halfline p is bounded
proof end;

registration
let C be compact Subset of (TOP-REAL 2);
cluster UBD C -> non empty ;
coherence
not UBD C is empty
proof end;
end;

theorem Th108: :: JORDAN2C:124
for C being compact Subset of (TOP-REAL 2) holds UBD C is_a_component_of C `
proof end;

theorem Th109: :: JORDAN2C:125
for C being compact Subset of (TOP-REAL 2)
for WH being connected Subset of (TOP-REAL 2) st not WH is bounded & WH misses C holds
WH c= UBD C
proof end;

theorem :: JORDAN2C:126
for C being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st west_halfline p misses C holds
west_halfline p c= UBD C
proof end;

theorem :: JORDAN2C:127
for C being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st east_halfline p misses C holds
east_halfline p c= UBD C
proof end;

theorem :: JORDAN2C:128
for C being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st south_halfline p misses C holds
south_halfline p c= UBD C
proof end;

theorem :: JORDAN2C:129
for C being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st north_halfline p misses C holds
north_halfline p c= UBD C
proof end;

theorem :: JORDAN2C:130
for n being Nat
for r being Real st r > 0 holds
for x, y, z being Element of (Euclid n) st x = 0* n holds
for p being Element of (TOP-REAL n) st p = y & r * p = z holds
r * (dist (x,y)) = dist (x,z) by Lm1;

theorem :: JORDAN2C:131
for n being Nat
for r, s being Real st r > 0 holds
for x being Element of (Euclid n) st x = 0* n holds
for A being Subset of (TOP-REAL n) st A = Ball (x,s) holds
r * A = Ball (x,(r * s)) by Lm2;

theorem :: JORDAN2C:132
for n being Nat
for r, s, t being Real st 0 < s & s <= t holds
for x being Element of (Euclid n) st x = 0* n holds
for BA being Subset of (TOP-REAL n) st BA = Ball (x,r) holds
s * BA c= t * BA by Lm3;