JORDAN: Jordan Curve Theorem

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 ;

proof end;

cluster LSeg ((NW-corner P),(SW-corner P)) -> vertical ;

proof end;

cluster LSeg ((NE-corner P),(SE-corner P)) -> vertical ;

proof end;

end;

registration

let P be Subset of (TOP-REAL 2);

cluster LSeg ((SE-corner P),(SW-corner P)) -> horizontal ;

coherence ;

cluster LSeg ((SW-corner P),(NW-corner P)) -> vertical ;

coherence ;

cluster LSeg ((SE-corner P),(NE-corner P)) -> vertical ;

coherence ;

end;

registration

cluster non empty compact vertical -> with_the_max_arc for Element of bool the carrier of (TOP-REAL 2);

proof end;

end;

theorem Th8: :: JORDAN:8

for r being Real
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
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 for Element of bool the carrier of (TOP-REAL n);

coherence ;

cluster non bounded -> non empty for Element of bool the carrier of (TOP-REAL n);

coherence ;

end;

registration

let n be non zero Nat;

cluster functional open closed non bounded convex for Element of bool the carrier of (TOP-REAL n);

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 2;

Lm6: now :: thesis:

let T be non empty TopSpace; :: thesis:
let a be Element of REAL ; :: thesis:
set c = the carrier of T;
set f = the carrier of T --> a;
thus the carrier of T --> a is continuous :: thesis:

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_1:8;
let Y be Subset of REAL; :: according to PSCOMP_1:def 3 :: thesis:
assume Y is closed ; :: thesis:

per cases ;

suppose a in Y ; :: thesis:

then A3: rng ( the carrier of T --> a) c= Y by A2, ZFMISC_1:31;
( the carrier of T --> a) " Y = ( the carrier of T --> a) " ((rng ( the carrier of T --> a)) /\ Y) by RELAT_1:133
.= ( the carrier of T --> a) " (rng ( the carrier of T --> a)) by A3, XBOOLE_1:28
.= [#] T by A1, RELAT_1:134 ;
hence ( the carrier of T --> a) " Y is closed ; :: thesis:

end;

suppose not a in Y ; :: thesis:

then A4: rng ( the carrier of T --> a) misses Y by A2, ZFMISC_1:50;
( the carrier of T --> a) " Y = ( the carrier of T --> a) " ((rng ( the carrier of T --> a)) /\ Y) by RELAT_1:133
.= ( the carrier of T --> a) " {} by A4
.= {} T ;
hence ( the carrier of T --> a) " Y is closed ; :: thesis:

end;

theorem Th16: :: JORDAN:16

for n being Nat
for r being positive Real
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
for p being Point of (TOP-REAL n) holds p is Point of (Tdisk (p,r))

proof end;

registration

let r be positive Real;
let n be non zero Element of NAT ;
let p, q be Point of (TOP-REAL n);

cluster (cl_Ball (p,r)) \ {q} -> non empty ;

proof end;

end;

theorem Th18: :: JORDAN:18

for r, s being Real
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
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
for n being Element of NAT
for x, y 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
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
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 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 holds Fr (Ball (x,r)) = Sphere (x,r)

proof end;

registration

let n be non zero Element of NAT ;

cluster bounded -> proper for Element of bool the carrier of (TOP-REAL n);

proof end;

end;

registration

let n be Element of NAT ;

cluster functional non empty closed bounded convex for Element of bool the carrier of (TOP-REAL n);

proof end;

cluster functional non empty open bounded convex for Element of bool the carrier of (TOP-REAL n);

proof end;

end;

registration

let n be Element of NAT ;
let A be bounded Subset of (TOP-REAL n);

cluster Cl A -> bounded ;

coherence by TOPREAL6:63;

end;

registration

let n be Element of NAT ;
let A be bounded Subset of (TOP-REAL n);

cluster Fr A -> bounded ;

coherence by TOPREAL6:89;

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 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 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 for SubSpace of T;

coherence ;

end;

registration

let p be Point of (TOP-REAL 2);
let r be Real;

cluster Tdisk (p,r) -> closed ;

proof end;

end;

registration

let p be Point of (TOP-REAL 2);
let r be Real;

cluster Tdisk (p,r) -> 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 pathwise_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 pathwise_connected TopSpace
for a, b being Point of T
for f being Path of a,b holds rng f = rng (- f) by Th31, BORSUK_2:def 3;

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 pathwise_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 pathwise_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 pathwise_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 pathwise_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 pathwise_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 pathwise_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 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 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 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;

cluster closed_inside_of_rectangle (a,b,c,d) -> closed ;

proof end;

end;

theorem Th48: :: JORDAN:48

for a, b, c, d being Real 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 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 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 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 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 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 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 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 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
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

proof end;

:: original: `2
redefine func x `2 -> Element of T;
coherence

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);

proof end;

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);

proof end;

proof end;

end;

:: deftheorem Def1 defines diffX2_1 JORDAN:def 1 :

:: deftheorem Def2 defines diffX2_2 JORDAN:def 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);

proof end;

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);

proof end;

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 ;

proof end;

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 ;

proof end;

proof end;

end;

:: deftheorem Def3 defines diffX1_X2_1 JORDAN:def 3 :

:: deftheorem Def4 defines diffX1_X2_2 JORDAN:def 4 :

:: deftheorem Def5 defines Proj2_1 JORDAN:def 5 :

:: deftheorem Def6 defines Proj2_2 JORDAN:def 6 :

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 ;

proof end;

cluster diffX2_2 o -> continuous ;

proof end;

end;

registration

cluster diffX1_X2_1 -> continuous ;

coherence by Th60, JORDAN5A:27;

cluster diffX1_X2_2 -> continuous ;

coherence by Th61, JORDAN5A:27;

cluster Proj2_1 -> continuous ;

coherence by Th62, JORDAN5A:27;

cluster Proj2_2 -> continuous ;

coherence by Th63, JORDAN5A:27;

end;

definition

let n be non zero Element of NAT ;
let o, p be Point of (TOP-REAL n);
let r be positive Real;
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) );

proof end;

proof end;

end;

:: deftheorem Def7 defines DiskProj JORDAN:def 7 :

theorem Th64: :: JORDAN:64

for o, p being Point of (TOP-REAL 2)
for r being positive Real 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 zero Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive Real 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 zero Element of NAT ;
let o, p be Point of (TOP-REAL n);
let r be positive Real;
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) );

proof end;

proof end;

end;

:: deftheorem Def8 defines RotateCircle JORDAN:def 8 :

theorem Th66: :: JORDAN:66

for o, p being Point of (TOP-REAL 2)
for r being positive Real st p in Ball (o,r) holds
RotateCircle (o,r,p) is continuous

proof end;

theorem Th67: :: JORDAN:67

for n being non zero Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive Real st p in Ball (o,r) holds
RotateCircle (o,r,p) is without_fixpoints

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 & V is a_component & 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 holds
((TOP-REAL 2) | (C `)) | U is pathwise_connected

proof end;

Lm12: for p, p1, p2 being Point of (TOP-REAL 2)
for A being Subset of (TOP-REAL 2)
for r being non negative Real 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 p, p1, p2 being Point of (TOP-REAL 2)
for C being Simple_closed_curve
for A, B, P being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C `))
for r being positive Real 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 & 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 B, P 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 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 & V is a_component & 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 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:52;

Lm17: |[1,0]| `1 = 1
by EUCLID:52;

Lm18: |[(- 1),0]| `2 = 0
by EUCLID:52;

Lm19: |[1,0]| `2 = 0
by EUCLID:52;

Lm20: |[0,3]| `1 = 0
by EUCLID:52;

Lm21: |[0,3]| `2 = 3
by EUCLID:52;

Lm22: |[0,(- 3)]| `1 = 0
by EUCLID:52;

Lm23: |[0,(- 3)]| `2 = - 3
by EUCLID:52;

Lm24: |[(- 1),3]| `1 = - 1
by EUCLID:52;

Lm25: |[(- 1),3]| `2 = 3
by EUCLID:52;

Lm26: |[(- 1),(- 3)]| `1 = - 1
by EUCLID:52;

Lm27: |[(- 1),(- 3)]| `2 = - 3
by EUCLID:52;

Lm28: |[1,3]| `1 = 1
by EUCLID:52;

Lm29: |[1,3]| `2 = 3
by EUCLID:52;

Lm30: |[1,(- 3)]| `1 = 1
by EUCLID:52;

Lm31: |[1,(- 3)]| `2 = - 3
by EUCLID:52;

Lm32: |[(- 1),(- 3)]| = |[(|[(- 1),(- 3)]| `1),(|[(- 1),(- 3)]| `2)]|
by EUCLID:53;

Lm33: |[(- 1),3]| = |[(|[(- 1),3]| `1),(|[(- 1),3]| `2)]|
by EUCLID:53;

Lm34: |[1,(- 3)]| = |[(|[1,(- 3)]| `1),(|[1,(- 3)]| `2)]|
by EUCLID:53;

Lm35: |[1,3]| = |[(|[1,3]| `1),(|[1,3]| `2)]|
by EUCLID:53;

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;

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;

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;

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;

Lm45: LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) is vertical
by Lm24, Lm26, SPPOL_1:16;

Lm46: LSeg (|[1,(- 3)]|,|[1,3]|) is vertical
by Lm28, Lm30, SPPOL_1:16;

Lm47: LSeg (|[(- 1),0]|,|[(- 1),3]|) is vertical
by Lm16, Lm24, SPPOL_1:16;

Lm48: LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) is vertical
by Lm16, Lm26, SPPOL_1:16;

Lm49: LSeg (|[1,0]|,|[1,3]|) is vertical
by Lm17, Lm28, SPPOL_1:16;

Lm50: LSeg (|[1,0]|,|[1,(- 3)]|) is vertical
by Lm17, Lm30, SPPOL_1:16;

Lm51: LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) is horizontal
by Lm23, Lm27, SPPOL_1:15;

Lm52: LSeg (|[1,(- 3)]|,|[0,(- 3)]|) is horizontal
by Lm23, Lm31, SPPOL_1:15;

Lm53: LSeg (|[(- 1),3]|,|[0,3]|) is horizontal
by Lm21, Lm25, SPPOL_1:15;

Lm54: LSeg (|[1,3]|,|[0,3]|) is horizontal
by Lm21, Lm29, SPPOL_1:15;

Lm55: LSeg (|[(- 1),3]|,|[1,3]|) is horizontal
by Lm25, Lm29, SPPOL_1:15;

Lm56: LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|) is horizontal
by Lm27, Lm31, SPPOL_1:15;

Lm57: LSeg (|[(- 1),0]|,|[(- 1),3]|) c= LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)
by Lm16, Lm18, Lm25, Lm26, Lm27, Lm45, Lm47, GOBOARD7:63;

Lm58: LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) c= LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)
by Lm18, Lm25, Lm26, Lm27, Lm45, Lm48, GOBOARD7:63;

Lm59: LSeg (|[1,0]|,|[1,3]|) c= LSeg (|[1,(- 3)]|,|[1,3]|)
by Lm17, Lm19, Lm29, Lm30, Lm31, Lm46, Lm49, GOBOARD7:63;

Lm60: LSeg (|[1,0]|,|[1,(- 3)]|) c= LSeg (|[1,(- 3)]|,|[1,3]|)
by Lm19, Lm29, Lm30, Lm31, Lm46, Lm50, GOBOARD7:63;

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_2:54;

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 2;

Lm66: dist (|[(- 1),0]|,|[1,0]|) = sqrt ((((|[(- 1),0]| `1) - (|[1,0]| `1)) ^2) + (((|[(- 1),0]| `2) - (|[1,0]| `2)) ^2)) by TOPREAL6:92
.= - (- 2) by Lm16, Lm17, Lm18, Lm19, SQUARE_1:23 ;

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:52;

|[0,3]| `1 <= |[1,3]| `1
by Lm28, EUCLID:52;

then LSeg (|[(- 1),3]|,|[0,3]|) c= LSeg (|[(- 1),3]|,|[1,3]|)
by Lm53, Lm55, Lm68, Lm69, GOBOARD7:64;

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

LSeg (|[1,3]|,|[0,3]|) c= LSeg (|[(- 1),3]|,|[1,3]|)
by Lm20, Lm21, Lm24, Lm25, Lm28, Lm54, Lm55, GOBOARD7:64;

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

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

Lm73: |[(- 1),(- 3)]| `1 <= |[0,(- 3)]| `1
by Lm26, EUCLID:52;

|[0,(- 3)]| `1 <= |[1,(- 3)]| `1
by Lm30, EUCLID:52;

then LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) c= LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|)
by Lm51, Lm56, Lm72, Lm73, GOBOARD7:64;

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

LSeg (|[1,(- 3)]|,|[0,(- 3)]|) c= LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|)
by Lm22, Lm23, Lm26, Lm27, Lm30, Lm52, Lm56, GOBOARD7:64;

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

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 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 A, P 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 ;