:: Jordan's Curve Theorem
:: by Artur Korni{\l}owicz
::
:: Received September 15, 2005
:: Copyright (c) 2005 Association of Mizar Users



set T2 = TOP-REAL 2;

Lm1: for A, B, C, Z being set st A c= Z & B c= Z & C c= Z holds
(A \/ B) \/ C c= Z
proof end;

Lm2: for A, B, C, D, Z being set st A c= Z & B c= Z & C c= Z & D c= Z holds
((A \/ B) \/ C) \/ D c= Z
proof end;

Lm3: for A, B, C, D, Z being set st A misses Z & B misses Z & C misses Z & D misses Z holds
((A \/ B) \/ C) \/ D misses Z
proof end;

registration
let M be Reflexive symmetric triangle MetrStruct ;
let x, y be Point of M;
cluster dist x,y -> non negative ;
coherence
not dist x,y is negative by METRIC_1:5;
end;

registration
let n be Element of NAT ;
let x, y be Point of (TOP-REAL n);
cluster dist x,y -> non negative ;
coherence
not dist x,y is negative
proof end;
end;

theorem Th1: :: JORDAN:1
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n) st p1 <> p2 holds
(1 / 2) * (p1 + p2) <> p1
proof end;

theorem Th2: :: JORDAN:2
for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
p1 `2 < ((1 / 2) * (p1 + p2)) `2
proof end;

theorem Th3: :: JORDAN:3
for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
((1 / 2) * (p1 + p2)) `2 < p2 `2
proof end;

theorem Th4: :: JORDAN:4
for B being Subset of (TOP-REAL 2)
for A being vertical Subset of (TOP-REAL 2) holds A /\ B is vertical
proof end;

theorem :: JORDAN:5
for B being Subset of (TOP-REAL 2)
for A being horizontal Subset of (TOP-REAL 2) holds A /\ B is horizontal
proof end;

theorem :: JORDAN:6
for p, p1, p2 being Point of (TOP-REAL 2) st p in LSeg p1,p2 & LSeg p1,p2 is vertical holds
LSeg p,p2 is vertical
proof end;

theorem :: JORDAN:7
for p, p1, p2 being Point of (TOP-REAL 2) st p in LSeg p1,p2 & LSeg p1,p2 is horizontal holds
LSeg p,p2 is horizontal
proof end;

registration
let P be Subset of (TOP-REAL 2);
cluster LSeg (SW-corner P),(SE-corner P) -> horizontal ;
coherence
LSeg (SW-corner P),(SE-corner P) is horizontal
proof end;
cluster LSeg (NW-corner P),(SW-corner P) -> vertical ;
coherence
LSeg (NW-corner P),(SW-corner P) is vertical
proof end;
cluster LSeg (NE-corner P),(SE-corner P) -> vertical ;
coherence
LSeg (NE-corner P),(SE-corner P) is vertical
proof end;
end;

registration
let P be Subset of (TOP-REAL 2);
cluster LSeg (SE-corner P),(SW-corner P) -> horizontal ;
coherence
LSeg (SE-corner P),(SW-corner P) is horizontal ;
cluster LSeg (SW-corner P),(NW-corner P) -> vertical ;
coherence
LSeg (SW-corner P),(NW-corner P) is vertical ;
cluster LSeg (SE-corner P),(NE-corner P) -> vertical ;
coherence
LSeg (SE-corner P),(NE-corner P) is vertical ;
end;

registration
cluster non empty compact vertical -> with_the_max_arc Element of bool the carrier of (TOP-REAL 2);
coherence
for b1 being Subset of (TOP-REAL 2) st b1 is vertical & not b1 is empty & b1 is compact holds
b1 is with_the_max_arc
proof end;
end;

theorem Th8: :: JORDAN:8
for r being real number
for p1, p2 being Point of (TOP-REAL 2) st p1 `1 <= r & r <= p2 `1 holds
LSeg p1,p2 meets Vertical_Line r
proof end;

theorem :: JORDAN:9
for r being real number
for p1, p2 being Point of (TOP-REAL 2) st p1 `2 <= r & r <= p2 `2 holds
LSeg p1,p2 meets Horizontal_Line r
proof end;

registration
let n be Element of NAT ;
cluster empty -> Bounded Element of bool the carrier of (TOP-REAL n);
coherence
for b1 being Subset of (TOP-REAL n) st b1 is empty holds
b1 is Bounded ;
cluster non Bounded -> non empty Element of bool the carrier of (TOP-REAL n);
coherence
for b1 being Subset of (TOP-REAL n) st not b1 is Bounded holds
not b1 is empty ;
end;

registration
let n be non empty Element of NAT ;
cluster open closed non Bounded convex Element of bool the carrier of (TOP-REAL n);
existence
ex b1 being Subset of (TOP-REAL n) st
( b1 is open & b1 is closed & not b1 is Bounded & b1 is convex )
proof end;
end;

theorem Th10: :: JORDAN:10
for C being compact Subset of (TOP-REAL 2) holds (north_halfline (UMP C)) \ {(UMP C)} misses C
proof end;

theorem Th11: :: JORDAN:11
for C being compact Subset of (TOP-REAL 2) holds (south_halfline (LMP C)) \ {(LMP C)} misses C
proof end;

theorem Th12: :: JORDAN:12
for C being compact Subset of (TOP-REAL 2) holds (north_halfline (UMP C)) \ {(UMP C)} c= UBD C
proof end;

theorem Th13: :: JORDAN:13
for C being compact Subset of (TOP-REAL 2) holds (south_halfline (LMP C)) \ {(LMP C)} c= UBD C
proof end;

theorem Th14: :: JORDAN:14
for A, B being Subset of (TOP-REAL 2) st A is_inside_component_of B holds
UBD B misses A
proof end;

theorem :: JORDAN:15
for A, B being Subset of (TOP-REAL 2) st A is_outside_component_of B holds
BDD B misses A
proof end;

Lm4: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve
for U being Subset of ((TOP-REAL 2) | (C ` )) st p in C holds
{p} misses U
proof end;

set C0 = Closed-Interval-TSpace 0 ,1;

set C1 = Closed-Interval-TSpace (- 1),1;

set l0 = (#) (- 1),1;

set l1 = (- 1),1 (#) ;

set h1 = L[01] ((#) (- 1),1),((- 1),1 (#) );

Lm5: the carrier of [:(TOP-REAL 2),(TOP-REAL 2):] = [:the carrier of (TOP-REAL 2),the carrier of (TOP-REAL 2):]
by BORSUK_1:def 5;

Lm6: now
let T be non empty TopSpace; :: thesis: for a being Real holds the carrier of T --> a is continuous
let a be Real; :: thesis: the carrier of T --> a is continuous
set c = the carrier of T;
set f = the carrier of T --> a;
thus the carrier of T --> a is continuous :: thesis: verum
proof
A1: dom (the carrier of T --> a) = the carrier of T by FUNCT_2:def 1;
A2: rng (the carrier of T --> a) = {a} by FUNCOP_1:14;
let Y be Subset of REAL ; :: according to PSCOMP_1:def 25 :: thesis: ( not Y is closed or (the carrier of T --> a) " Y is closed )
assume Y is closed ; :: thesis: (the carrier of T --> a) " Y is closed
per cases ( a in Y or not a in Y ) ;
suppose a in Y ; :: thesis: (the carrier of T --> a) " Y is closed
then A3: rng (the carrier of T --> a) c= Y by A2, ZFMISC_1:37;
(the carrier of T --> a) " Y = (the carrier of T --> a) " ((rng (the carrier of T --> a)) /\ Y) by RELAT_1:168
.= (the carrier of T --> a) " (rng (the carrier of T --> a)) by A3, XBOOLE_1:28
.= [#] T by A1, RELAT_1:169 ;
hence (the carrier of T --> a) " Y is closed ; :: thesis: verum
end;
suppose not a in Y ; :: thesis: (the carrier of T --> a) " Y is closed
then A4: rng (the carrier of T --> a) misses Y by A2, ZFMISC_1:56;
(the carrier of T --> a) " Y = (the carrier of T --> a) " ((rng (the carrier of T --> a)) /\ Y) by RELAT_1:168
.= (the carrier of T --> a) " {} by A4, XBOOLE_0:def 7
.= {} T by RELAT_1:171 ;
hence (the carrier of T --> a) " Y is closed ; :: thesis: verum
end;
end;
end;
end;

theorem Th16: :: JORDAN:16
for n being Element of NAT
for r being positive real number
for a being Point of (TOP-REAL n) holds a in Ball a,r
proof end;

theorem Th17: :: JORDAN:17
for n being Element of NAT
for r being non negative real number
for p being Point of (TOP-REAL n) holds p is Point of (Tdisk p,r)
proof end;

registration
let r be positive real number ;
let n be non empty Element of NAT ;
let p, q be Point of (TOP-REAL n);
cluster (cl_Ball p,r) \ {q} -> non empty ;
coherence
not (cl_Ball p,r) \ {q} is empty
proof end;
end;

theorem Th18: :: JORDAN:18
for r, s being real number
for n being Element of NAT
for x being Point of (TOP-REAL n) st r <= s holds
Ball x,r c= Ball x,s
proof end;

theorem Th19: :: JORDAN:19
for r being real number
for n being Element of NAT
for x being Point of (TOP-REAL n) holds (cl_Ball x,r) \ (Ball x,r) = Sphere x,r
proof end;

theorem Th20: :: JORDAN:20
for r being real number
for n being Element of NAT
for y, x being Point of (TOP-REAL n) st y in Sphere x,r holds
(LSeg x,y) \ {x,y} c= Ball x,r
proof end;

theorem Th21: :: JORDAN:21
for r, s being real number
for n being Element of NAT
for x being Point of (TOP-REAL n) st r < s holds
cl_Ball x,r c= Ball x,s
proof end;

theorem Th22: :: JORDAN:22
for r, s being real number
for n being Element of NAT
for x being Point of (TOP-REAL n) st r < s holds
Sphere x,r c= Ball x,s
proof end;

theorem Th23: :: JORDAN:23
for n being Element of NAT
for x being Point of (TOP-REAL n)
for r being non zero real number holds Cl (Ball x,r) = cl_Ball x,r
proof end;

theorem Th24: :: JORDAN:24
for n being Element of NAT
for x being Point of (TOP-REAL n)
for r being non zero real number holds Fr (Ball x,r) = Sphere x,r
proof end;

registration
let n be non empty Element of NAT ;
cluster Bounded -> proper Element of bool the carrier of (TOP-REAL n);
coherence
for b1 being Subset of (TOP-REAL n) st b1 is Bounded holds
b1 is proper
proof end;
end;

registration
let n be Element of NAT ;
cluster non empty closed Bounded convex Element of bool the carrier of (TOP-REAL n);
existence
ex b1 being Subset of (TOP-REAL n) st
( not b1 is empty & b1 is closed & b1 is convex & b1 is Bounded )
proof end;
cluster non empty open Bounded convex Element of bool the carrier of (TOP-REAL n);
existence
ex b1 being Subset of (TOP-REAL n) st
( not b1 is empty & b1 is open & b1 is convex & b1 is Bounded )
proof end;
end;

registration
let n be Element of NAT ;
let A be Bounded Subset of (TOP-REAL n);
cluster Cl A -> Bounded ;
coherence
Cl A is Bounded by TOPREAL6:71;
end;

registration
let n be Element of NAT ;
let A be Bounded Subset of (TOP-REAL n);
cluster Fr A -> Bounded ;
coherence
Fr A is Bounded by TOPREAL6:98;
end;

theorem Th25: :: JORDAN:25
for n being Element of NAT
for A being closed Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n) st not p in A holds
ex r being positive real number st Ball p,r misses A
proof end;

theorem Th26: :: JORDAN:26
for n being Element of NAT
for A being Bounded Subset of (TOP-REAL n)
for a being Point of (TOP-REAL n) ex r being positive real number st A c= Ball a,r
proof end;

theorem :: JORDAN:27
for S, T being TopStruct
for f being Function of S,T st f is being_homeomorphism holds
f is onto ;

registration
let T be non empty T_2 TopSpace;
cluster non empty -> non empty T_2 SubSpace of T;
coherence
for b1 being non empty SubSpace of T holds b1 is T_2 ;
end;

registration
let p be Point of (TOP-REAL 2);
let r be real number ;
cluster Tdisk p,r -> closed ;
coherence
Tdisk p,r is closed
proof end;
end;

registration
let p be Point of (TOP-REAL 2);
let r be real number ;
cluster Tdisk p,r -> compact ;
coherence
Tdisk p,r is compact
proof end;
end;


theorem :: JORDAN:28
for T being non empty TopSpace
for a, b being Point of T
for f being Path of a,b st a,b are_connected holds
rng f is connected
proof end;

theorem Th29: :: JORDAN:29
for X being non empty TopSpace
for Y being non empty SubSpace of X
for x1, x2 being Point of X
for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & x1,x2 are_connected & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
proof end;

theorem Th30: :: JORDAN:30
for X being non empty arcwise_connected TopSpace
for Y being non empty SubSpace of X
for x1, x2 being Point of X
for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
proof end;

Lm7: for T being non empty TopSpace
for a, b being Point of T
for f being Path of a,b st a,b are_connected holds
rng f c= rng (- f)
proof end;

theorem Th31: :: JORDAN:31
for T being non empty TopSpace
for a, b being Point of T
for f being Path of a,b st a,b are_connected holds
rng f = rng (- f)
proof end;

theorem Th32: :: JORDAN:32
for T being non empty arcwise_connected TopSpace
for a, b being Point of T
for f being Path of a,b holds rng f = rng (- f)
proof end;

theorem Th33: :: JORDAN:33
for T being non empty TopSpace
for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng f c= rng (f + g)
proof end;

theorem :: JORDAN:34
for T being non empty arcwise_connected TopSpace
for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c holds rng f c= rng (f + g)
proof end;

theorem Th35: :: JORDAN:35
for T being non empty TopSpace
for a, b, c being Point of T
for f being Path of b,c
for g being Path of a,b st a,b are_connected & b,c are_connected holds
rng f c= rng (g + f)
proof end;

theorem :: JORDAN:36
for T being non empty arcwise_connected TopSpace
for a, b, c being Point of T
for f being Path of b,c
for g being Path of a,b holds rng f c= rng (g + f)
proof end;

theorem Th37: :: JORDAN:37
for T being non empty TopSpace
for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng (f + g) = (rng f) \/ (rng g)
proof end;

theorem :: JORDAN:38
for T being non empty arcwise_connected TopSpace
for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c holds rng (f + g) = (rng f) \/ (rng g)
proof end;

theorem Th39: :: JORDAN:39
for T being non empty TopSpace
for a, b, c, d being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d st a,b are_connected & b,c are_connected & c,d are_connected holds
rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
proof end;

theorem Th40: :: JORDAN:40
for T being non empty arcwise_connected TopSpace
for a, b, c, d being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
proof end;

Lm8: for T being non empty TopSpace
for a, b, c, d, e being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected holds
rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
proof end;

Lm9: for T being non empty arcwise_connected TopSpace
for a, b, c, d, e being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e holds rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
proof end;

Lm10: for T being non empty TopSpace
for a, b, c, d, e, z being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected & e,z are_connected holds
rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
proof end;

Lm11: for T being non empty arcwise_connected TopSpace
for a, b, c, d, e, z being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z holds rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
proof end;

theorem Th41: :: JORDAN:41
for T being non empty TopSpace
for a being Point of T holds I[01] --> a is Path of a,a
proof end;

theorem Th42: :: JORDAN:42
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
ex F being Path of p1,p2 ex f being Function of I[01] ,((TOP-REAL n) | P) st
( rng f = P & F = f )
proof end;

theorem Th43: :: JORDAN:43
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n) ex F being Path of p1,p2 ex f being Function of I[01] ,((TOP-REAL n) | (LSeg p1,p2)) st
( rng f = LSeg p1,p2 & F = f )
proof end;

theorem Th44: :: JORDAN:44
for P being Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & q2 in P & q1 <> p1 & q1 <> p2 & q2 <> p1 & q2 <> p2 holds
ex f being Path of q1,q2 st
( rng f c= P & rng f misses {p1,p2} )
proof end;


theorem Th45: :: JORDAN:45
for a, b, c, d being real number st a <= b & c <= d holds
rectangle a,b,c,d c= closed_inside_of_rectangle a,b,c,d
proof end;

theorem Th46: :: JORDAN:46
for a, b, c, d being real number holds inside_of_rectangle a,b,c,d c= closed_inside_of_rectangle a,b,c,d
proof end;

theorem Th47: :: JORDAN:47
for a, b, c, d being real number holds closed_inside_of_rectangle a,b,c,d = (outside_of_rectangle a,b,c,d) `
proof end;

registration
let a, b, c, d be real number ;
cluster closed_inside_of_rectangle a,b,c,d -> closed ;
coherence
closed_inside_of_rectangle a,b,c,d is closed
proof end;
end;

theorem Th48: :: JORDAN:48
for a, b, c, d being real number holds closed_inside_of_rectangle a,b,c,d misses outside_of_rectangle a,b,c,d
proof end;

theorem Th49: :: JORDAN:49
for a, b, c, d being real number holds (closed_inside_of_rectangle a,b,c,d) /\ (inside_of_rectangle a,b,c,d) = inside_of_rectangle a,b,c,d
proof end;

theorem Th50: :: JORDAN:50
for a, b, c, d being real number st a < b & c < d holds
Int (closed_inside_of_rectangle a,b,c,d) = inside_of_rectangle a,b,c,d
proof end;

theorem Th51: :: JORDAN:51
for a, b, c, d being real number st a <= b & c <= d holds
(closed_inside_of_rectangle a,b,c,d) \ (inside_of_rectangle a,b,c,d) = rectangle a,b,c,d
proof end;

theorem Th52: :: JORDAN:52
for a, b, c, d being real number st a < b & c < d holds
Fr (closed_inside_of_rectangle a,b,c,d) = rectangle a,b,c,d
proof end;

theorem :: JORDAN:53
for a, b, c, d being real number st a <= b & c <= d holds
W-bound (closed_inside_of_rectangle a,b,c,d) = a
proof end;

theorem :: JORDAN:54
for a, b, c, d being real number st a <= b & c <= d holds
S-bound (closed_inside_of_rectangle a,b,c,d) = c
proof end;

theorem :: JORDAN:55
for a, b, c, d being real number st a <= b & c <= d holds
E-bound (closed_inside_of_rectangle a,b,c,d) = b
proof end;

theorem :: JORDAN:56
for a, b, c, d being real number st a <= b & c <= d holds
N-bound (closed_inside_of_rectangle a,b,c,d) = d
proof end;

theorem Th57: :: JORDAN:57
for a, b, c, d being real number
for p1, p2 being Point of (TOP-REAL 2)
for P being Subset of (TOP-REAL 2) st a < b & c < d & p1 in closed_inside_of_rectangle a,b,c,d & not p2 in closed_inside_of_rectangle a,b,c,d & P is_an_arc_of p1,p2 holds
Segment P,p1,p2,p1,(First_Point P,p1,p2,(rectangle a,b,c,d)) c= closed_inside_of_rectangle a,b,c,d
proof end;


definition
let S, T be non empty TopSpace;
let x be Point of [:S,T:];
:: original: `1
redefine func x `1 -> Element of S;
coherence
x `1 is Element of S
proof end;
:: original: `2
redefine func x `2 -> Element of T;
coherence
x `2 is Element of T
proof end;
end;

definition
let o be Point of (TOP-REAL 2);
func diffX2_1 o -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def1: :: JORDAN:def 1
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = ((x `2 ) `1 ) - (o `1 );
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `2 ) `1 ) - (o `1 )
proof end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `2 ) `1 ) - (o `1 ) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `2 ) `1 ) - (o `1 ) ) holds
b1 = b2
proof end;
func diffX2_2 o -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def2: :: JORDAN:def 2
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = ((x `2 ) `2 ) - (o `2 );
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `2 ) `2 ) - (o `2 )
proof end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `2 ) `2 ) - (o `2 ) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `2 ) `2 ) - (o `2 ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines diffX2_1 JORDAN:def 1 :
for o being Point of (TOP-REAL 2)
for b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b2 = diffX2_1 o iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `2 ) `1 ) - (o `1 ) );

:: deftheorem Def2 defines diffX2_2 JORDAN:def 2 :
for o being Point of (TOP-REAL 2)
for b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b2 = diffX2_2 o iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `2 ) `2 ) - (o `2 ) );

definition
func diffX1_X2_1 -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def3: :: JORDAN:def 3
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = ((x `1 ) `1 ) - ((x `2 ) `1 );
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1 ) `1 ) - ((x `2 ) `1 )
proof end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1 ) `1 ) - ((x `2 ) `1 ) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `1 ) `1 ) - ((x `2 ) `1 ) ) holds
b1 = b2
proof end;
func diffX1_X2_2 -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def4: :: JORDAN:def 4
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = ((x `1 ) `2 ) - ((x `2 ) `2 );
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1 ) `2 ) - ((x `2 ) `2 )
proof end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1 ) `2 ) - ((x `2 ) `2 ) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `1 ) `2 ) - ((x `2 ) `2 ) ) holds
b1 = b2
proof end;
func Proj2_1 -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def5: :: JORDAN:def 5
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = (x `2 ) `1 ;
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2 ) `1
proof end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2 ) `1 ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = (x `2 ) `1 ) holds
b1 = b2
proof end;
func Proj2_2 -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def6: :: JORDAN:def 6
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = (x `2 ) `2 ;
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2 ) `2
proof end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2 ) `2 ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = (x `2 ) `2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines diffX1_X2_1 JORDAN:def 3 :
for b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b1 = diffX1_X2_1 iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1 ) `1 ) - ((x `2 ) `1 ) );

:: deftheorem Def4 defines diffX1_X2_2 JORDAN:def 4 :
for b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b1 = diffX1_X2_2 iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1 ) `2 ) - ((x `2 ) `2 ) );

:: deftheorem Def5 defines Proj2_1 JORDAN:def 5 :
for b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b1 = Proj2_1 iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2 ) `1 );

:: deftheorem Def6 defines Proj2_2 JORDAN:def 6 :
for b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b1 = Proj2_2 iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2 ) `2 );

theorem Th58: :: JORDAN:58
for o being Point of (TOP-REAL 2) holds diffX2_1 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof end;

theorem Th59: :: JORDAN:59
for o being Point of (TOP-REAL 2) holds diffX2_2 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof end;

theorem Th60: :: JORDAN:60
diffX1_X2_1 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof end;

theorem Th61: :: JORDAN:61
diffX1_X2_2 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof end;

theorem Th62: :: JORDAN:62
Proj2_1 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof end;

theorem Th63: :: JORDAN:63
Proj2_2 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof end;

registration
let o be Point of (TOP-REAL 2);
cluster diffX2_1 o -> continuous ;
coherence
diffX2_1 o is continuous
proof end;
cluster diffX2_2 o -> continuous ;
coherence
diffX2_2 o is continuous
proof end;
end;

registration
cluster diffX1_X2_1 -> continuous ;
coherence
diffX1_X2_1 is continuous by Th60, TOPREAL6:83;
cluster diffX1_X2_2 -> continuous ;
coherence
diffX1_X2_2 is continuous by Th61, TOPREAL6:83;
cluster Proj2_1 -> continuous ;
coherence
Proj2_1 is continuous by Th62, TOPREAL6:83;
cluster Proj2_2 -> continuous ;
coherence
Proj2_2 is continuous by Th63, TOPREAL6:83;
end;

definition
let n be non empty Element of NAT ;
let o, p be Point of (TOP-REAL n);
let r be positive real number ;
assume A1: p is Point of (Tdisk o,r) ;
set X = (TOP-REAL n) | ((cl_Ball o,r) \ {p});
func DiskProj o,r,p -> Function of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})),(Tcircle o,r) means :Def7: :: JORDAN:def 7
for x being Point of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & it . x = HC p,y,o,r );
existence
ex b1 being Function of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})),(Tcircle o,r) st
for x being Point of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & b1 . x = HC p,y,o,r )
proof end;
uniqueness
for b1, b2 being Function of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})),(Tcircle o,r) st ( for x being Point of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & b1 . x = HC p,y,o,r ) ) & ( for x being Point of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & b2 . x = HC p,y,o,r ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines DiskProj JORDAN:def 7 :
for n being non empty Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive real number st p is Point of (Tdisk o,r) holds
for b5 being Function of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})),(Tcircle o,r) holds
( b5 = DiskProj o,r,p iff for x being Point of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & b5 . x = HC p,y,o,r ) );

theorem Th64: :: JORDAN:64
for o, p being Point of (TOP-REAL 2)
for r being positive real number st p is Point of (Tdisk o,r) holds
DiskProj o,r,p is continuous
proof end;

theorem Th65: :: JORDAN:65
for n being non empty Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive real number st p in Ball o,r holds
(DiskProj o,r,p) | (Sphere o,r) = id (Sphere o,r)
proof end;

definition
let n be non empty Element of NAT ;
let o, p be Point of (TOP-REAL n);
let r be positive real number ;
assume A1: p in Ball o,r ;
set X = Tcircle o,r;
func RotateCircle o,r,p -> Function of (Tcircle o,r),(Tcircle o,r) means :Def8: :: JORDAN:def 8
for x being Point of (Tcircle o,r) ex y being Point of (TOP-REAL n) st
( x = y & it . x = HC y,p,o,r );
existence
ex b1 being Function of (Tcircle o,r),(Tcircle o,r) st
for x being Point of (Tcircle o,r) ex y being Point of (TOP-REAL n) st
( x = y & b1 . x = HC y,p,o,r )
proof end;
uniqueness
for b1, b2 being Function of (Tcircle o,r),(Tcircle o,r) st ( for x being Point of (Tcircle o,r) ex y being Point of (TOP-REAL n) st
( x = y & b1 . x = HC y,p,o,r ) ) & ( for x being Point of (Tcircle o,r) ex y being Point of (TOP-REAL n) st
( x = y & b2 . x = HC y,p,o,r ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def8 defines RotateCircle JORDAN:def 8 :
for n being non empty Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive real number st p in Ball o,r holds
for b5 being Function of (Tcircle o,r),(Tcircle o,r) holds
( b5 = RotateCircle o,r,p iff for x being Point of (Tcircle o,r) ex y being Point of (TOP-REAL n) st
( x = y & b5 . x = HC y,p,o,r ) );

theorem Th66: :: JORDAN:66
for o, p being Point of (TOP-REAL 2)
for r being positive real number st p in Ball o,r holds
RotateCircle o,r,p is continuous
proof end;

theorem Th67: :: JORDAN:67
for n being non empty Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive real number st p in Ball o,r holds
RotateCircle o,r,p has_no_fixpoint
proof end;


theorem Th68: :: JORDAN:68
for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2)
for U, V being Subset of ((TOP-REAL 2) | (C ` )) st U = P & U is_a_component_of (TOP-REAL 2) | (C ` ) & V is_a_component_of (TOP-REAL 2) | (C ` ) & U <> V holds
Cl P misses V
proof end;

theorem Th69: :: JORDAN:69
for C being Simple_closed_curve
for U being Subset of ((TOP-REAL 2) | (C ` )) st U is_a_component_of (TOP-REAL 2) | (C ` ) holds
((TOP-REAL 2) | (C ` )) | U is arcwise_connected
proof end;

Lm12: for p1, p2, p being Point of (TOP-REAL 2)
for A being Subset of (TOP-REAL 2)
for r being non negative real number st A is_an_arc_of p1,p2 & A is Subset of (Tdisk p,r) holds
ex f being Function of (Tdisk p,r),((TOP-REAL 2) | A) st
( f is continuous & f | A = id A )
proof end;

Lm13: for p1, p2, p being Point of (TOP-REAL 2)
for C being Simple_closed_curve
for A, P, B being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C ` ))
for r being positive real number st A is_an_arc_of p1,p2 & A c= C & C c= Ball p,r & p in U & (Cl P) /\ (P ` ) c= A & P c= Ball p,r holds
for f being Function of (Tdisk p,r),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is_a_component_of (TOP-REAL 2) | (C ` ) & B = (cl_Ball p,r) \ {p} holds
ex g being Function of (Tdisk p,r),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk p,r) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) )
proof end;

Lm14: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve
for P, B being Subset of (TOP-REAL 2)
for U, V being Subset of ((TOP-REAL 2) | (C ` ))
for A being non empty Subset of (TOP-REAL 2) st U <> V holds
for r being positive real number st A c= C & C c= Ball p,r & p in V & (Cl P) /\ (P ` ) c= A & Ball p,r meets P holds
for f being Function of (Tdisk p,r),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is_a_component_of (TOP-REAL 2) | (C ` ) & V is_a_component_of (TOP-REAL 2) | (C ` ) & B = (cl_Ball p,r) \ {p} holds
ex g being Function of (Tdisk p,r),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk p,r) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
proof end;

Lm15: for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C ` )) st not BDD C is empty & U = P & U is_a_component_of (TOP-REAL 2) | (C ` ) holds
C = Fr P
proof end;

set rp = 1;

set rl = - 1;

set rg = 3;

set rd = - 3;

set a = |[(- 1),0 ]|;

set b = |[1,0 ]|;

set c = |[0 ,3]|;

set d = |[0 ,(- 3)]|;

set lg = |[(- 1),3]|;

set pg = |[1,3]|;

set ld = |[(- 1),(- 3)]|;

set pd = |[1,(- 3)]|;

set R = closed_inside_of_rectangle (- 1),1,(- 3),3;

set dR = rectangle (- 1),1,(- 3),3;

set TR = Trectangle (- 1),1,(- 3),3;

Lm16: |[(- 1),0 ]| `1 = - 1
by EUCLID:56;

Lm17: |[1,0 ]| `1 = 1
by EUCLID:56;

Lm18: |[(- 1),0 ]| `2 = 0
by EUCLID:56;

Lm19: |[1,0 ]| `2 = 0
by EUCLID:56;

Lm20: |[0 ,3]| `1 = 0
by EUCLID:56;

Lm21: |[0 ,3]| `2 = 3
by EUCLID:56;

Lm22: |[0 ,(- 3)]| `1 = 0
by EUCLID:56;

Lm23: |[0 ,(- 3)]| `2 = - 3
by EUCLID:56;

Lm24: |[(- 1),3]| `1 = - 1
by EUCLID:56;

Lm25: |[(- 1),3]| `2 = 3
by EUCLID:56;

Lm26: |[(- 1),(- 3)]| `1 = - 1
by EUCLID:56;

Lm27: |[(- 1),(- 3)]| `2 = - 3
by EUCLID:56;

Lm28: |[1,3]| `1 = 1
by EUCLID:56;

Lm29: |[1,3]| `2 = 3
by EUCLID:56;

Lm30: |[1,(- 3)]| `1 = 1
by EUCLID:56;

Lm31: |[1,(- 3)]| `2 = - 3
by EUCLID:56;

Lm32: |[(- 1),(- 3)]| = |[(|[(- 1),(- 3)]| `1 ),(|[(- 1),(- 3)]| `2 )]|
by EUCLID:57;

Lm33: |[(- 1),3]| = |[(|[(- 1),3]| `1 ),(|[(- 1),3]| `2 )]|
by EUCLID:57;

Lm34: |[1,(- 3)]| = |[(|[1,(- 3)]| `1 ),(|[1,(- 3)]| `2 )]|
by EUCLID:57;

Lm35: |[1,3]| = |[(|[1,3]| `1 ),(|[1,3]| `2 )]|
by EUCLID:57;

Lm36: rectangle (- 1),1,(- 3),3 = ((LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|)) \/ ((LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|))
by SPPOL_2:def 3;

Lm37: LSeg |[(- 1),(- 3)]|,|[(- 1),3]| c= (LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|)
by XBOOLE_1:7;

(LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|) c= rectangle (- 1),1,(- 3),3
by Lm36, XBOOLE_1:7;

then Lm38: LSeg |[(- 1),(- 3)]|,|[(- 1),3]| c= rectangle (- 1),1,(- 3),3
by Lm37, XBOOLE_1:1;

Lm39: LSeg |[(- 1),3]|,|[1,3]| c= (LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|)
by XBOOLE_1:7;

(LSeg |[(- 1),(- 3)]|,|[(- 1),3]|) \/ (LSeg |[(- 1),3]|,|[1,3]|) c= rectangle (- 1),1,(- 3),3
by Lm36, XBOOLE_1:7;

then Lm40: LSeg |[(- 1),3]|,|[1,3]| c= rectangle (- 1),1,(- 3),3
by Lm39, XBOOLE_1:1;

Lm41: LSeg |[1,3]|,|[1,(- 3)]| c= (LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|)
by XBOOLE_1:7;

(LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|) c= rectangle (- 1),1,(- 3),3
by Lm36, XBOOLE_1:7;

then Lm42: LSeg |[1,3]|,|[1,(- 3)]| c= rectangle (- 1),1,(- 3),3
by Lm41, XBOOLE_1:1;

Lm43: LSeg |[1,(- 3)]|,|[(- 1),(- 3)]| c= (LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|)
by XBOOLE_1:7;

(LSeg |[1,3]|,|[1,(- 3)]|) \/ (LSeg |[1,(- 3)]|,|[(- 1),(- 3)]|) c= rectangle (- 1),1,(- 3),3
by Lm36, XBOOLE_1:7;

then Lm44: LSeg |[1,(- 3)]|,|[(- 1),(- 3)]| c= rectangle (- 1),1,(- 3),3
by Lm43, XBOOLE_1:1;

Lm45: LSeg |[(- 1),(- 3)]|,|[(- 1),3]| is vertical
by Lm24, Lm26, SPPOL_1:37;

Lm46: LSeg |[1,(- 3)]|,|[1,3]| is vertical
by Lm28, Lm30, SPPOL_1:37;

Lm47: LSeg |[(- 1),0 ]|,|[(- 1),3]| is vertical
by Lm16, Lm24, SPPOL_1:37;

Lm48: LSeg |[(- 1),0 ]|,|[(- 1),(- 3)]| is vertical
by Lm16, Lm26, SPPOL_1:37;

Lm49: LSeg |[1,0 ]|,|[1,3]| is vertical
by Lm17, Lm28, SPPOL_1:37;

Lm50: LSeg |[1,0 ]|,|[1,(- 3)]| is vertical
by Lm17, Lm30, SPPOL_1:37;

Lm51: LSeg |[(- 1),(- 3)]|,|[0 ,(- 3)]| is horizontal
by Lm23, Lm27, SPPOL_1:36;

Lm52: LSeg |[1,(- 3)]|,|[0 ,(- 3)]| is horizontal
by Lm23, Lm31, SPPOL_1:36;

Lm53: LSeg |[(- 1),3]|,|[0 ,3]| is horizontal
by Lm21, Lm25, SPPOL_1:36;

Lm54: LSeg |[1,3]|,|[0 ,3]| is horizontal
by Lm21, Lm29, SPPOL_1:36;

Lm55: LSeg |[(- 1),3]|,|[1,3]| is horizontal
by Lm25, Lm29, SPPOL_1:36;

Lm56: LSeg |[(- 1),(- 3)]|,|[1,(- 3)]| is horizontal
by Lm27, Lm31, SPPOL_1:36;

Lm57: LSeg |[(- 1),0 ]|,|[(- 1),3]| c= LSeg |[(- 1),(- 3)]|,|[(- 1),3]|
by Lm16, Lm18, Lm25, Lm26, Lm27, Lm45, Lm47, GOBOARD7:65;

Lm58: LSeg |[(- 1),0 ]|,|[(- 1),(- 3)]| c= LSeg |[(- 1),(- 3)]|,|[(- 1),3]|
by Lm18, Lm25, Lm26, Lm27, Lm45, Lm48, GOBOARD7:65;

Lm59: LSeg |[1,0 ]|,|[1,3]| c= LSeg |[1,(- 3)]|,|[1,3]|
by Lm17, Lm19, Lm29, Lm30, Lm31, Lm46, Lm49, GOBOARD7:65;

Lm60: LSeg |[1,0 ]|,|[1,(- 3)]| c= LSeg |[1,(- 3)]|,|[1,3]|
by Lm19, Lm29, Lm30, Lm31, Lm46, Lm50, GOBOARD7:65;

Lm61: rectangle (- 1),1,(- 3),3 = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & p `2 <= 3 & p `2 >= - 3 ) or ( p `1 <= 1 & p `1 >= - 1 & p `2 = 3 ) or ( p `1 <= 1 & p `1 >= - 1 & p `2 = - 3 ) or ( p `1 = 1 & p `2 <= 3 & p `2 >= - 3 ) ) }
by SPPOL_2:58;

then Lm62: |[0 ,3]| in rectangle (- 1),1,(- 3),3
by Lm20, Lm21;

Lm63: |[0 ,(- 3)]| in rectangle (- 1),1,(- 3),3
by Lm22, Lm23, Lm61;

Lm64: (2 + 1) ^2 = (4 + 4) + 1
;

then Lm65: sqrt 9 = 3
by SQUARE_1:def 4;

Lm66: dist |[(- 1),0 ]|,|[1,0 ]| = sqrt ((((|[(- 1),0 ]| `1 ) - (|[1,0 ]| `1 )) ^2 ) + (((|[(- 1),0 ]| `2 ) - (|[1,0 ]| `2 )) ^2 )) by TOPREAL6:101
.= - (- 2) by Lm16, Lm17, Lm18, Lm19, SQUARE_1:90 ;

theorem Th70: :: JORDAN:70
for C being Simple_closed_curve
for h being Homeomorphism of TOP-REAL 2 holds h .: C is being_simple_closed_curve
proof end;

theorem Th71: :: JORDAN:71
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
P c= closed_inside_of_rectangle (- 1),1,(- 3),3
proof end;

Lm67: rectangle (- 1),1,(- 3),3 c= closed_inside_of_rectangle (- 1),1,(- 3),3
by Th45;

Lm68: |[(- 1),3]| `2 = |[(- 1),3]| `2
;

Lm69: |[(- 1),3]| `1 <= |[0 ,3]| `1
by Lm24, EUCLID:56;

|[0 ,3]| `1 <= |[1,3]| `1
by Lm28, EUCLID:56;

then LSeg |[(- 1),3]|,|[0 ,3]| c= LSeg |[(- 1),3]|,|[1,3]|
by Lm53, Lm55, Lm68, Lm69, GOBOARD7:66;

then Lm70: LSeg |[(- 1),3]|,|[0 ,3]| c= rectangle (- 1),1,(- 3),3
by Lm40, XBOOLE_1:1;

LSeg |[1,3]|,|[0 ,3]| c= LSeg |[(- 1),3]|,|[1,3]|
by Lm20, Lm21, Lm24, Lm25, Lm28, Lm54, Lm55, GOBOARD7:66;

then Lm71: LSeg |[1,3]|,|[0 ,3]| c= rectangle (- 1),1,(- 3),3
by Lm40, XBOOLE_1:1;

Lm72: |[(- 1),(- 3)]| `2 = |[(- 1),(- 3)]| `2
;

Lm73: |[(- 1),(- 3)]| `1 <= |[0 ,(- 3)]| `1
by Lm26, EUCLID:56;

|[0 ,(- 3)]| `1 <= |[1,(- 3)]| `1
by Lm30, EUCLID:56;

then LSeg |[(- 1),(- 3)]|,|[0 ,(- 3)]| c= LSeg |[(- 1),(- 3)]|,|[1,(- 3)]|
by Lm51, Lm56, Lm72, Lm73, GOBOARD7:66;

then Lm74: LSeg |[(- 1),(- 3)]|,|[0 ,(- 3)]| c= rectangle (- 1),1,(- 3),3
by Lm44, XBOOLE_1:1;

LSeg |[1,(- 3)]|,|[0 ,(- 3)]| c= LSeg |[(- 1),(- 3)]|,|[1,(- 3)]|
by Lm22, Lm23, Lm26, Lm27, Lm30, Lm52, Lm56, GOBOARD7:66;

then Lm75: LSeg |[1,(- 3)]|,|[0 ,(- 3)]| c= rectangle (- 1),1,(- 3),3
by Lm44, XBOOLE_1:1;

Lm76: for p being Point of (TOP-REAL 2) st 0 <= p `2 & p in rectangle (- 1),1,(- 3),3 & not p in LSeg |[(- 1),0 ]|,|[(- 1),3]| & not p in LSeg |[(- 1),3]|,|[0 ,3]| & not p in LSeg |[0 ,3]|,|[1,3]| holds
p in LSeg |[1,3]|,|[1,0 ]|
proof end;

Lm77: for p being Point of (TOP-REAL 2) st p `2 <= 0 & p in rectangle (- 1),1,(- 3),3 & not p in LSeg |[(- 1),0 ]|,|[(- 1),(- 3)]| & not p in LSeg |[(- 1),(- 3)]|,|[0 ,(- 3)]| & not p in LSeg |[0 ,(- 3)]|,|[1,(- 3)]| holds
p in LSeg |[1,(- 3)]|,|[1,0 ]|
proof end;

theorem Th72: :: JORDAN:72
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
P misses LSeg |[(- 1),3]|,|[1,3]|
proof end;

theorem Th73: :: JORDAN:73
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
P misses LSeg |[(- 1),(- 3)]|,|[1,(- 3)]|
proof end;

theorem Th74: :: JORDAN:74
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
P /\ (rectangle (- 1),1,(- 3),3) = {|[(- 1),0 ]|,|[1,0 ]|}
proof end;

Lm78: for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C holds
LSeg |[(- 1),3]|,|[0 ,3]| misses C
proof end;

Lm79: for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C holds
LSeg |[1,3]|,|[0 ,3]| misses C
proof end;

Lm80: for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C holds
LSeg |[(- 1),(- 3)]|,|[0 ,(- 3)]| misses C
proof end;

Lm81: for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C holds
LSeg |[1,(- 3)]|,|[0 ,(- 3)]| misses C
proof end;

Lm82: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C & p in C ` & p in LSeg |[(- 1),0 ]|,|[(- 1),3]| holds
LSeg p,|[(- 1),3]| misses C
proof end;

Lm83: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C & p in C ` & p in LSeg |[1,0 ]|,|[1,3]| holds
LSeg p,|[1,3]| misses C
proof end;

Lm84: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C & p in C ` & p in LSeg |[(- 1),0 ]|,|[(- 1),(- 3)]| holds
LSeg p,|[(- 1),(- 3)]| misses C
proof end;

Lm85: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C & p in C ` & p in LSeg |[1,0 ]|,|[1,(- 3)]| holds
LSeg p,|[1,(- 3)]| misses C
proof end;

Lm86: for r being real number holds
( not |[0 ,r]| in rectangle (- 1),1,(- 3),3 or r = - 3 or r = 3 )
proof end;

theorem Th75: :: JORDAN:75
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
W-bound P = - 1
proof end;

theorem Th76: :: JORDAN:76
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
E-bound P = 1
proof end;

theorem Th77: :: JORDAN:77
for P being compact Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
W-most P = {|[(- 1),0 ]|}
proof end;

theorem Th78: :: JORDAN:78
for P being compact Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
E-most P = {|[1,0 ]|}
proof end;

theorem Th79: :: JORDAN:79
for P being compact Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
( W-min P = |[(- 1),0 ]| & W-max P = |[(- 1),0 ]| )
proof end;

theorem Th80: :: JORDAN:80
for P being compact Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
( E-min P = |[1,0 ]| & E-max P = |[1,0 ]| )
proof end;

Lm87: for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
|[0 ,3]| `1 = ((W-bound P) + (E-bound P)) / 2
proof end;

Lm88: for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
|[0 ,(- 3)]| `1 = ((W-bound P) + (E-bound P)) / 2
proof end;

theorem Th81: :: JORDAN:81
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
LSeg |[0 ,3]|,(UMP P) is vertical
proof end;

theorem Th82: :: JORDAN:82
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P holds
LSeg (LMP P),|[0 ,(- 3)]| is vertical
proof end;

theorem Th83: :: JORDAN:83
for p being Point of (TOP-REAL 2)
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P & p in P holds
p `2 < 3
proof end;

theorem Th84: :: JORDAN:84
for p being Point of (TOP-REAL 2)
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P & p in P holds
- 3 < p `2
proof end;

theorem Th85: :: JORDAN:85
for p being Point of (TOP-REAL 2)
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in D & p in LSeg |[0 ,3]|,(UMP D) holds
(UMP D) `2 <= p `2
proof end;

theorem Th86: :: JORDAN:86
for p being Point of (TOP-REAL 2)
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in D & p in LSeg (LMP D),|[0 ,(- 3)]| holds
p `2 <= (LMP D) `2
proof end;

theorem Th87: :: JORDAN:87
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in D holds
LSeg |[0 ,3]|,(UMP D) c= north_halfline (UMP D)
proof end;

theorem Th88: :: JORDAN:88
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in D holds
LSeg (LMP D),|[0 ,(- 3)]| c= south_halfline (LMP D)
proof end;

theorem Th89: :: JORDAN:89
for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C & P is_inside_component_of C holds
LSeg |[0 ,3]|,(UMP C) misses P
proof end;

theorem Th90: :: JORDAN:90
for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C & P is_inside_component_of C holds
LSeg (LMP C),|[0 ,(- 3)]| misses P
proof end;

theorem Th91: :: JORDAN:91
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in D holds
(LSeg |[0 ,3]|,(UMP D)) /\ D = {(UMP D)}
proof end;

theorem :: JORDAN:92
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in D holds
(LSeg |[0 ,(- 3)]|,(LMP D)) /\ D = {(LMP D)}
proof end;

theorem Th93: :: JORDAN:93
for P, A being Subset of (TOP-REAL 2) st P is compact & |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in P & A is_inside_component_of P holds
A c= closed_inside_of_rectangle (- 1),1,(- 3),3
proof end;

Lm89: for p being Point of (TOP-REAL 2) st p in closed_inside_of_rectangle (- 1),1,(- 3),3 holds
closed_inside_of_rectangle (- 1),1,(- 3),3 c= Ball p,10
proof end;

theorem :: JORDAN:94
for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C holds
LSeg |[0 ,3]|,|[0 ,(- 3)]| meets C
proof end;

Lm90: for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C holds
ex Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st
( Jc is_an_arc_of |[(- 1),0 ]|,|[1,0 ]| & Jd is_an_arc_of |[(- 1),0 ]|,|[1,0 ]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0 ]|,|[1,0 ]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc )
proof end;

theorem Th95: :: JORDAN:95
for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C holds
for Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st Jc is_an_arc_of |[(- 1),0 ]|,|[1,0 ]| & Jd is_an_arc_of |[(- 1),0 ]|,|[1,0 ]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0 ]|,|[1,0 ]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
for Ux being Subset of (TOP-REAL 2) st Ux = Component_of (Down ((1 / 2) * ((UMP ((LSeg (LMP Jc),|[0 ,(- 3)]|) /\ Jd)) + (LMP Jc))),(C ` )) holds
( Ux is_inside_component_of C & ( for V being Subset of (TOP-REAL 2) st V is_inside_component_of C holds
V = Ux ) )
proof end;

theorem Th96: :: JORDAN:96
for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C holds
for Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st Jc is_an_arc_of |[(- 1),0 ]|,|[1,0 ]| & Jd is_an_arc_of |[(- 1),0 ]|,|[1,0 ]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0 ]|,|[1,0 ]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
BDD C = Component_of (Down ((1 / 2) * ((UMP ((LSeg (LMP Jc),|[0 ,(- 3)]|) /\ Jd)) + (LMP Jc))),(C ` ))
proof end;

Lm91: for C being Simple_closed_curve st |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in C holds
C is Jordan
proof end;

Lm92: for C being Simple_closed_curve holds C is Jordan
proof end;

registration
let C be Simple_closed_curve;
cluster BDD C -> non empty ;
coherence
not BDD C is empty
proof end;
end;

theorem :: JORDAN:97
for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C ` )) st U = P & U is_a_component_of (TOP-REAL 2) | (C ` ) holds
C = Fr P
proof end;

theorem :: JORDAN:98
for C being Simple_closed_curve ex A1, A2 being Subset of (TOP-REAL 2) st
( C ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (C ` )) st C1 = A1 & C2 = A2 holds
( C1 is_a_component_of (TOP-REAL 2) | (C ` ) & C2 is_a_component_of (TOP-REAL 2) | (C ` ) ) ) )
proof end;

theorem :: JORDAN:99
for C being Simple_closed_curve holds C is Jordan by Lm92;