:: Some Properties of Circles on the Plane
:: by Artur Korni{\l}owicz and Yasunari Shidama
::
:: Received October 18, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

set P2 = 2 * PI;

set o = |[0,0]|;

set R = the carrier of R^1;

Lm1: 0 in INT
by INT_1:def 1;

reconsider p0 = - 1 as negative Real ;

reconsider p1 = 1 as positive Real ;

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

set cCIT = the carrier of (Closed-Interval-TSpace ((- 1),1));

Lm2: the carrier of (Closed-Interval-TSpace ((- 1),1)) = [.(- 1),1.]
by TOPMETR:18;

Lm3: 1 - 0 <= 1
;

Lm4: (3 / 2) - (1 / 2) <= 1
;

registration

cluster K100(0,1) -> non empty ;

coherence
not ].0,1.[ is empty ;

cluster K97((- 1),1) -> non empty ;

coherence
not [.(- 1),1.] is empty ;

cluster K100((1 / 2),(3 / 2)) -> non empty ;

coherence
not ].(1 / 2),(3 / 2).[ is empty

proof end;

end;

Lm5: PI / 2 < PI / 1
by XREAL_1:76;

Lm6: 1 * PI < (3 / 2) * PI
by XREAL_1:68;

Lm7: (3 / 2) * PI < 2 * PI
by XREAL_1:68;

Lm8: for X being non empty TopSpace
for Y being non empty SubSpace of X
for Z being non empty SubSpace of Y
for p being Point of Z holds p is Point of X

proof end;

Lm9: for X being TopSpace
for Y being SubSpace of X
for Z being SubSpace of Y
for A being Subset of Z holds A is Subset of X

proof end;

registration

cluster sin -> continuous ;

coherence
sin is continuous ;

cluster cos -> continuous ;

coherence
cos is continuous ;

cluster arcsin -> continuous ;

coherence
arcsin is continuous by RELAT_1:69, SIN_COS6:63, SIN_COS6:84;

cluster arccos -> continuous ;

coherence
arccos is continuous by RELAT_1:69, SIN_COS6:86, SIN_COS6:107;

end;

theorem Th1: :: TOPREALB:1

for a, b, r being Real holds sin ((a * r) + b) = (sin * (AffineMap (a,b))) . r

proof end;

theorem Th2: :: TOPREALB:2

for a, b, r being Real holds cos ((a * r) + b) = (cos * (AffineMap (a,b))) . r

proof end;

registration

let a be non zero Real;
let b be Real;

cluster AffineMap (a,b) -> one-to-one onto ;

coherence
( AffineMap (a,b) is onto & AffineMap (a,b) is one-to-one ) by FCONT_1:55, FCONT_1:50;

end;

definition

let a, b be Real;

func IntIntervals (a,b) -> set equals :: TOPREALB:def 1
{ ].(a + n),(b + n).[ where n is Element of INT : verum } ;

coherence
{ ].(a + n),(b + n).[ where n is Element of INT : verum } is set ;

end;

:: deftheorem defines IntIntervals TOPREALB:def 1 :
for a, b being Real holds IntIntervals (a,b) = { ].(a + n),(b + n).[ where n is Element of INT : verum } ;

theorem :: TOPREALB:3

for a, b being Real
for x being set holds
( x in IntIntervals (a,b) iff ex n being Element of INT st x = ].(a + n),(b + n).[ ) ;

registration

let a, b be Real;

cluster IntIntervals (a,b) -> non empty ;

coherence
not IntIntervals (a,b) is empty

proof end;

end;

theorem :: TOPREALB:4

for a, b being Real st b - a <= 1 holds
IntIntervals (a,b) is mutually-disjoint

proof end;

definition

let a, b be Real;
:: original: IntIntervals
redefine func IntIntervals (a,b) -> Subset-Family of R^1;
coherence
IntIntervals (a,b) is Subset-Family of R^1

proof end;

end;

definition

let a, b be Real;
:: original: IntIntervals
redefine func IntIntervals (a,b) -> open Subset-Family of R^1;
coherence
IntIntervals (a,b) is open Subset-Family of R^1

proof end;

end;

definition

let r be Real;

func R^1 r -> Point of R^1 equals :: TOPREALB:def 2
r;

coherence
r is Point of R^1 by TOPMETR:17, XREAL_0:def 1;

end;

:: deftheorem defines R^1 TOPREALB:def 2 :
for r being Real holds R^1 r = r;

definition

let A be Subset of REAL;

func R^1 A -> Subset of R^1 equals :: TOPREALB:def 3
A;

coherence
A is Subset of R^1 by TOPMETR:17;

end;

:: deftheorem defines R^1 TOPREALB:def 3 :
for A being Subset of REAL holds R^1 A = A;

registration

let A be non empty Subset of REAL;

cluster R^1 A -> non empty ;

coherence
not R^1 A is empty ;

end;

registration

let A be open Subset of REAL;

cluster R^1 A -> open ;

coherence
R^1 A is open by BORSUK_5:39;

end;

registration

let A be closed Subset of REAL;

cluster R^1 A -> closed ;

coherence
R^1 A is closed by JORDAN5A:23;

end;

registration

let A be open Subset of REAL;

cluster R^1 | (R^1 A) -> open ;

coherence
R^1 | (R^1 A) is open by PRE_TOPC:8;

end;

registration

let A be closed Subset of REAL;

cluster R^1 | (R^1 A) -> closed ;

coherence
R^1 | (R^1 A) is closed by PRE_TOPC:8;

end;

definition

let f be PartFunc of REAL,REAL;

func R^1 f -> Function of (R^1 | (R^1 (dom f))),(R^1 | (R^1 (rng f))) equals :: TOPREALB:def 4
f;

coherence
f is Function of (R^1 | (R^1 (dom f))),(R^1 | (R^1 (rng f)))

proof end;

end;

:: deftheorem defines R^1 TOPREALB:def 4 :
for f being PartFunc of REAL,REAL holds R^1 f = f;

registration

let f be PartFunc of REAL,REAL;

cluster R^1 f -> onto ;

coherence
R^1 f is onto by PRE_TOPC:8;

end;

registration

let f be one-to-one PartFunc of REAL,REAL;

cluster R^1 f -> one-to-one ;

coherence
R^1 f is one-to-one ;

end;

theorem Th5: :: TOPREALB:5

R^1 | (R^1 ([#] REAL)) = R^1

proof end;

theorem Th6: :: TOPREALB:6

for f being PartFunc of REAL,REAL st dom f = REAL holds
R^1 | (R^1 (dom f)) = R^1

proof end;

theorem Th7: :: TOPREALB:7

for f being Function of REAL,REAL holds f is Function of R^1,(R^1 | (R^1 (rng f)))

proof end;

theorem Th8: :: TOPREALB:8

for f being Function of REAL,REAL holds f is Function of R^1,R^1

proof end;

Lm10: sin is Function of R^1,R^1

proof end;

Lm11: cos is Function of R^1,R^1

proof end;

registration

let f be continuous PartFunc of REAL,REAL;

cluster R^1 f -> continuous ;

coherence
R^1 f is continuous

proof end;

end;

set A = R^1 ].0,1.[;

Lm12: now :: thesis: for a being non zero Real
for b being Real holds
( R^1 = R^1 | (R^1 (dom (AffineMap (a,b)))) & R^1 = R^1 | (R^1 (rng (AffineMap (a,b)))) )

let a be non zero Real; :: thesis: for b being Real holds
( R^1 = R^1 | (R^1 (dom (AffineMap (a,b)))) & R^1 = R^1 | (R^1 (rng (AffineMap (a,b)))) )
let b be Real; :: thesis: ( R^1 = R^1 | (R^1 (dom (AffineMap (a,b)))) & R^1 = R^1 | (R^1 (rng (AffineMap (a,b)))) )
A1: rng (AffineMap (a,b)) = REAL by FCONT_1:55;
A2: [#] R^1 = REAL by TOPMETR:17;
dom (AffineMap (a,b)) = REAL by FUNCT_2:def 1;
hence ( R^1 = R^1 | (R^1 (dom (AffineMap (a,b)))) & R^1 = R^1 | (R^1 (rng (AffineMap (a,b)))) ) by A1, A2, TSEP_1:3; :: thesis: verum

end;

registration

let a be non zero Real;
let b be Real;

cluster R^1 (AffineMap (a,b)) -> open ;

coherence
R^1 (AffineMap (a,b)) is open

proof end;

end;

definition

let S be SubSpace of TOP-REAL 2;

attr S is being_simple_closed_curve means :Def5: :: TOPREALB:def 5
the carrier of S is Simple_closed_curve;

end;

:: deftheorem Def5 defines being_simple_closed_curve TOPREALB:def 5 :
for S being SubSpace of TOP-REAL 2 holds
( S is being_simple_closed_curve iff the carrier of S is Simple_closed_curve );

registration

cluster being_simple_closed_curve -> non empty compact pathwise_connected for SubSpace of TOP-REAL 2;

coherence
for b₁ being SubSpace of TOP-REAL 2 st b₁ is being_simple_closed_curve holds
( not b₁ is empty & b₁ is pathwise_connected & b₁ is compact )

proof end;

end;

registration

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

cluster Sphere (x,r) -> being_simple_closed_curve ;

coherence
Sphere (x,r) is being_simple_closed_curve

proof end;

end;

definition

let n be Nat;
let x be Point of (TOP-REAL n);
let r be Real;

func Tcircle (x,r) -> SubSpace of TOP-REAL n equals :: TOPREALB:def 6
(TOP-REAL n) | (Sphere (x,r));

coherence
(TOP-REAL n) | (Sphere (x,r)) is SubSpace of TOP-REAL n ;

end;

:: deftheorem defines Tcircle TOPREALB:def 6 :
for n being Nat
for x being Point of (TOP-REAL n)
for r being Real holds Tcircle (x,r) = (TOP-REAL n) | (Sphere (x,r));

registration

let n be non zero Nat;
let x be Point of (TOP-REAL n);
let r be non negative Real;

cluster Tcircle (x,r) -> non empty strict ;

coherence
( Tcircle (x,r) is strict & not Tcircle (x,r) is empty ) ;

end;

theorem Th9: :: TOPREALB:9

for n being Element of NAT
for r being Real
for x being Point of (TOP-REAL n) holds the carrier of (Tcircle (x,r)) = Sphere (x,r)

proof end;

registration

let n be Nat;
let x be Point of (TOP-REAL n);
let r be zero Real;

cluster Tcircle (x,r) -> trivial ;

coherence
Tcircle (x,r) is trivial

proof end;

end;

theorem Th10: :: TOPREALB:10

for r being Real holds Tcircle ((0. (TOP-REAL 2)),r) is SubSpace of Trectangle ((- r),r,(- r),r)

proof end;

registration

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

cluster Tcircle (x,r) -> being_simple_closed_curve ;

coherence
Tcircle (x,r) is being_simple_closed_curve by Th9;

end;

registration

cluster strict TopSpace-like V225() V226() being_simple_closed_curve for SubSpace of TOP-REAL 2;

existence
ex b₁ being SubSpace of TOP-REAL 2 st
( b₁ is being_simple_closed_curve & b₁ is strict )

proof end;

end;

theorem :: TOPREALB:11

for S, T being being_simple_closed_curve SubSpace of TOP-REAL 2 holds S,T are_homeomorphic

proof end;

:: Unit circle

definition

let n be Nat;

func Tunit_circle n -> SubSpace of TOP-REAL n equals :: TOPREALB:def 7
Tcircle ((0. (TOP-REAL n)),1);

coherence
Tcircle ((0. (TOP-REAL n)),1) is SubSpace of TOP-REAL n ;

end;

:: deftheorem defines Tunit_circle TOPREALB:def 7 :
for n being Nat holds Tunit_circle n = Tcircle ((0. (TOP-REAL n)),1);

set TUC = Tunit_circle 2;

set cS1 = the carrier of (Tunit_circle 2);

Lm13: the carrier of (Tunit_circle 2) = Sphere (|[0,0]|,1)
by Th9, EUCLID:54;

registration

let n be non zero Nat;

cluster Tunit_circle n -> non empty ;

coherence
not Tunit_circle n is empty ;

end;

theorem Th12: :: TOPREALB:12

for n being non zero Element of NAT
for x being Point of (TOP-REAL n) st x is Point of (Tunit_circle n) holds
|.x.| = 1

proof end;

theorem Th13: :: TOPREALB:13

for x being Point of (TOP-REAL 2) st x is Point of (Tunit_circle 2) holds
( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 )

proof end;

theorem Th14: :: TOPREALB:14

for x being Point of (TOP-REAL 2) st x is Point of (Tunit_circle 2) & x `1 = 1 holds
x `2 = 0

proof end;

theorem Th15: :: TOPREALB:15

for x being Point of (TOP-REAL 2) st x is Point of (Tunit_circle 2) & x `1 = - 1 holds
x `2 = 0

proof end;

theorem :: TOPREALB:16

for x being Point of (TOP-REAL 2) st x is Point of (Tunit_circle 2) & x `2 = 1 holds
x `1 = 0

proof end;

theorem :: TOPREALB:17

for x being Point of (TOP-REAL 2) st x is Point of (Tunit_circle 2) & x `2 = - 1 holds
x `1 = 0

proof end;

set TREC = Trectangle (p0,p1,p0,p1);

theorem :: TOPREALB:18

Tunit_circle 2 is SubSpace of Trectangle ((- 1),1,(- 1),1) by Th10;

theorem Th19: :: TOPREALB:19

for n being non zero Element of NAT
for r being positive Real
for x being Point of (TOP-REAL n)
for f being Function of (Tunit_circle n),(Tcircle (x,r)) st ( for a being Point of (Tunit_circle n)
for b being Point of (TOP-REAL n) st a = b holds
f . a = (r * b) + x ) holds
f is being_homeomorphism

proof end;

registration

cluster Tunit_circle 2 -> being_simple_closed_curve ;

coherence
Tunit_circle 2 is being_simple_closed_curve ;

end;

Lm14: for n being non zero Element of NAT
for r being positive Real
for x being Point of (TOP-REAL n) holds Tunit_circle n, Tcircle (x,r) are_homeomorphic

proof end;

theorem :: TOPREALB:20

for n being non zero Element of NAT
for r, s being positive Real
for x, y being Point of (TOP-REAL n) holds Tcircle (x,r), Tcircle (y,s) are_homeomorphic

proof end;

registration

let x be Point of (TOP-REAL 2);
let r be non negative Real;

cluster Tcircle (x,r) -> pathwise_connected ;

coherence
Tcircle (x,r) is pathwise_connected

proof end;

end;

:: Open unit circle

definition

func c[10] -> Point of (Tunit_circle 2) equals :: TOPREALB:def 8
|[1,0]|;

coherence
|[1,0]| is Point of (Tunit_circle 2)

proof end;

func c[-10] -> Point of (Tunit_circle 2) equals :: TOPREALB:def 9
|[(- 1),0]|;

coherence
|[(- 1),0]| is Point of (Tunit_circle 2)

proof end;

end;

:: deftheorem defines c[10] TOPREALB:def 8 :
c[10] = |[1,0]|;

:: deftheorem defines c[-10] TOPREALB:def 9 :
c[-10] = |[(- 1),0]|;

Lm15: c[10] <> c[-10]
by SPPOL_2:1;

definition

let p be Point of (Tunit_circle 2);

func Topen_unit_circle p -> strict SubSpace of Tunit_circle 2 means :Def10: :: TOPREALB:def 10
the carrier of it = the carrier of (Tunit_circle 2) \ {p};

existence
ex b₁ being strict SubSpace of Tunit_circle 2 st the carrier of b₁ = the carrier of (Tunit_circle 2) \ {p}

proof end;

uniqueness
for b₁, b₂ being strict SubSpace of Tunit_circle 2 st the carrier of b₁ = the carrier of (Tunit_circle 2) \ {p} & the carrier of b₂ = the carrier of (Tunit_circle 2) \ {p} holds
b₁ = b₂ by TSEP_1:5;

end;

:: deftheorem Def10 defines Topen_unit_circle TOPREALB:def 10 :
for p being Point of (Tunit_circle 2)
for b₂ being strict SubSpace of Tunit_circle 2 holds
( b₂ = Topen_unit_circle p iff the carrier of b₂ = the carrier of (Tunit_circle 2) \ {p} );

registration

let p be Point of (Tunit_circle 2);

cluster Topen_unit_circle p -> non empty strict ;

coherence
not Topen_unit_circle p is empty

proof end;

end;

theorem Th21: :: TOPREALB:21

for p being Point of (Tunit_circle 2) holds p is not Point of (Topen_unit_circle p)

proof end;

theorem Th22: :: TOPREALB:22

for p being Point of (Tunit_circle 2) holds Topen_unit_circle p = (Tunit_circle 2) | (([#] (Tunit_circle 2)) \ {p})

proof end;

theorem Th23: :: TOPREALB:23

for p, q being Point of (Tunit_circle 2) st p <> q holds
q is Point of (Topen_unit_circle p)

proof end;

theorem Th24: :: TOPREALB:24

for p being Point of (TOP-REAL 2) st p is Point of (Topen_unit_circle c[10]) & p `2 = 0 holds
p = c[-10]

proof end;

theorem Th25: :: TOPREALB:25

for p being Point of (TOP-REAL 2) st p is Point of (Topen_unit_circle c[-10]) & p `2 = 0 holds
p = c[10]

proof end;

set TOUC = Topen_unit_circle c[10];

set TOUCm = Topen_unit_circle c[-10];

set X = the carrier of (Topen_unit_circle c[10]);

set Xm = the carrier of (Topen_unit_circle c[-10]);

set Y = the carrier of (R^1 | (R^1 ].0,(0 + p1).[));

set Ym = the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[));

Lm16: the carrier of (Topen_unit_circle c[10]) = [#] (Topen_unit_circle c[10])
;

Lm17: the carrier of (Topen_unit_circle c[-10]) = [#] (Topen_unit_circle c[-10])
;

theorem Th26: :: TOPREALB:26

for p being Point of (Tunit_circle 2)
for x being Point of (TOP-REAL 2) st x is Point of (Topen_unit_circle p) holds
( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 )

proof end;

theorem Th27: :: TOPREALB:27

for x being Point of (TOP-REAL 2) st x is Point of (Topen_unit_circle c[10]) holds
( - 1 <= x `1 & x `1 < 1 )

proof end;

theorem Th28: :: TOPREALB:28

for x being Point of (TOP-REAL 2) st x is Point of (Topen_unit_circle c[-10]) holds
( - 1 < x `1 & x `1 <= 1 )

proof end;

registration

let p be Point of (Tunit_circle 2);

cluster Topen_unit_circle p -> strict open ;

coherence
Topen_unit_circle p is open

proof end;

end;

theorem :: TOPREALB:29

for p being Point of (Tunit_circle 2) holds Topen_unit_circle p, I(01) are_homeomorphic

proof end;

theorem :: TOPREALB:30

for p, q being Point of (Tunit_circle 2) holds Topen_unit_circle p, Topen_unit_circle q are_homeomorphic

proof end;

definition

func CircleMap -> Function of R^1,(Tunit_circle 2) means :Def11: :: TOPREALB:def 11
for x being Real holds it . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]|;

existence
ex b₁ being Function of R^1,(Tunit_circle 2) st
for x being Real holds b₁ . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]|

proof end;

uniqueness
for b₁, b₂ being Function of R^1,(Tunit_circle 2) st ( for x being Real holds b₁ . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ) & ( for x being Real holds b₂ . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def11 defines CircleMap TOPREALB:def 11 :
for b₁ being Function of R^1,(Tunit_circle 2) holds
( b₁ = CircleMap iff for x being Real holds b₁ . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| );

Lm18: dom CircleMap = REAL
by FUNCT_2:def 1, TOPMETR:17;

theorem Th31: :: TOPREALB:31

for i being Integer
for r being Real holds CircleMap . r = CircleMap . (r + i)

proof end;

theorem Th32: :: TOPREALB:32

for i being Integer holds CircleMap . i = c[10]

proof end;

theorem Th33: :: TOPREALB:33

CircleMap " {c[10]} = INT

proof end;

Lm19: CircleMap . (1 / 2) = |[(- 1),0]|

proof end;

theorem Th34: :: TOPREALB:34

for r being Real st frac r = 1 / 2 holds
CircleMap . r = |[(- 1),0]|

proof end;

theorem :: TOPREALB:35

for r being Real st frac r = 1 / 4 holds
CircleMap . r = |[0,1]|

proof end;

theorem :: TOPREALB:36

for r being Real st frac r = 3 / 4 holds
CircleMap . r = |[0,(- 1)]|

proof end;

Lm20: for r being Real holds CircleMap . r = |[((cos * (AffineMap ((2 * PI),0))) . r),((sin * (AffineMap ((2 * PI),0))) . r)]|

proof end;

theorem :: TOPREALB:37

for r being Real
for i, j being Integer holds CircleMap . r = |[((cos * (AffineMap ((2 * PI),((2 * PI) * i)))) . r),((sin * (AffineMap ((2 * PI),((2 * PI) * j)))) . r)]|

proof end;

registration

cluster CircleMap -> continuous ;

coherence
CircleMap is continuous

proof end;

end;

Lm21: for A being Subset of R^1 holds CircleMap | A is Function of (R^1 | A),(Tunit_circle 2)

proof end;

Lm22: for r being Real st - 1 <= r & r <= 1 holds
( 0 <= (arccos r) / (2 * PI) & (arccos r) / (2 * PI) <= 1 / 2 )

proof end;

theorem Th38: :: TOPREALB:38

for A being Subset of R^1
for f being Function of (R^1 | A),(Tunit_circle 2) st [.0,1.[ c= A & f = CircleMap | A holds
f is onto

proof end;

registration

cluster CircleMap -> onto ;

coherence
CircleMap is onto

proof end;

end;

Lm23: CircleMap | [.0,1.[ is one-to-one

proof end;

registration

let r be Real;

cluster K5(CircleMap,[.r,(r + 1).[) -> one-to-one ;

coherence
CircleMap | [.r,(r + 1).[ is one-to-one

proof end;

end;

registration

let r be Real;

cluster K5(CircleMap,].r,(r + 1).[) -> one-to-one ;

coherence
CircleMap | ].r,(r + 1).[ is one-to-one

proof end;

end;

theorem Th39: :: TOPREALB:39

for a, b being Real st b - a <= 1 holds
for d being set st d in IntIntervals (a,b) holds
CircleMap | d is one-to-one

proof end;

theorem Th40: :: TOPREALB:40

for a, b being Real
for d being set st d in IntIntervals (a,b) holds
CircleMap .: d = CircleMap .: (union (IntIntervals (a,b)))

proof end;

definition

let r be Point of R^1;

func CircleMap r -> Function of (R^1 | (R^1 ].r,(r + 1).[)),(Topen_unit_circle (CircleMap . r)) equals :: TOPREALB:def 12
CircleMap | ].r,(r + 1).[;

coherence
CircleMap | ].r,(r + 1).[ is Function of (R^1 | (R^1 ].r,(r + 1).[)),(Topen_unit_circle (CircleMap . r))

proof end;

end;

:: deftheorem defines CircleMap TOPREALB:def 12 :
for r being Point of R^1 holds CircleMap r = CircleMap | ].r,(r + 1).[;

Lm24: for a, r being Real holds rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) = ].r,(r + 1).[

proof end;

theorem Th41: :: TOPREALB:41

for i being Integer
for a being Real holds CircleMap (R^1 (a + i)) = (CircleMap (R^1 a)) * ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[)

proof end;

registration

let r be Point of R^1;

cluster CircleMap r -> one-to-one onto continuous ;

coherence
( CircleMap r is one-to-one & CircleMap r is onto & CircleMap r is continuous )

proof end;

end;

definition

func Circle2IntervalR -> Function of (Topen_unit_circle c[10]),(R^1 | (R^1 ].0,1.[)) means :Def13: :: TOPREALB:def 13
for p being Point of (Topen_unit_circle c[10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies it . p = (arccos x) / (2 * PI) ) & ( y <= 0 implies it . p = 1 - ((arccos x) / (2 * PI)) ) );

existence
ex b₁ being Function of (Topen_unit_circle c[10]),(R^1 | (R^1 ].0,1.[)) st
for p being Point of (Topen_unit_circle c[10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b₁ . p = (arccos x) / (2 * PI) ) & ( y <= 0 implies b₁ . p = 1 - ((arccos x) / (2 * PI)) ) )

proof end;

uniqueness
for b₁, b₂ being Function of (Topen_unit_circle c[10]),(R^1 | (R^1 ].0,1.[)) st ( for p being Point of (Topen_unit_circle c[10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b₁ . p = (arccos x) / (2 * PI) ) & ( y <= 0 implies b₁ . p = 1 - ((arccos x) / (2 * PI)) ) ) ) & ( for p being Point of (Topen_unit_circle c[10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b₂ . p = (arccos x) / (2 * PI) ) & ( y <= 0 implies b₂ . p = 1 - ((arccos x) / (2 * PI)) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def13 defines Circle2IntervalR TOPREALB:def 13 :
for b₁ being Function of (Topen_unit_circle c[10]),(R^1 | (R^1 ].0,1.[)) holds
( b₁ = Circle2IntervalR iff for p being Point of (Topen_unit_circle c[10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b₁ . p = (arccos x) / (2 * PI) ) & ( y <= 0 implies b₁ . p = 1 - ((arccos x) / (2 * PI)) ) ) );

set A1 = R^1 ].(1 / 2),((1 / 2) + p1).[;

definition

func Circle2IntervalL -> Function of (Topen_unit_circle c[-10]),(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) means :Def14: :: TOPREALB:def 14
for p being Point of (Topen_unit_circle c[-10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies it . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies it . p = 1 - ((arccos x) / (2 * PI)) ) );

existence
ex b₁ being Function of (Topen_unit_circle c[-10]),(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) st
for p being Point of (Topen_unit_circle c[-10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b₁ . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies b₁ . p = 1 - ((arccos x) / (2 * PI)) ) )

proof end;

uniqueness
for b₁, b₂ being Function of (Topen_unit_circle c[-10]),(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) st ( for p being Point of (Topen_unit_circle c[-10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b₁ . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies b₁ . p = 1 - ((arccos x) / (2 * PI)) ) ) ) & ( for p being Point of (Topen_unit_circle c[-10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b₂ . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies b₂ . p = 1 - ((arccos x) / (2 * PI)) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def14 defines Circle2IntervalL TOPREALB:def 14 :
for b₁ being Function of (Topen_unit_circle c[-10]),(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) holds
( b₁ = Circle2IntervalL iff for p being Point of (Topen_unit_circle c[-10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b₁ . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies b₁ . p = 1 - ((arccos x) / (2 * PI)) ) ) );

set C = Circle2IntervalR ;

set Cm = Circle2IntervalL ;

theorem Th42: :: TOPREALB:42

(CircleMap (R^1 0)) " = Circle2IntervalR

proof end;

theorem Th43: :: TOPREALB:43

(CircleMap (R^1 (1 / 2))) " = Circle2IntervalL

proof end;

set A = ].0,1.[;

set Q = [.(- 1),1.[;

set E = ].0,PI.];

set j = 1 / (2 * PI);

reconsider Q = [.(- 1),1.[, E = ].0,PI.] as non empty Subset of REAL ;

Lm25: the carrier of (R^1 | (R^1 Q)) = R^1 Q
by PRE_TOPC:8;

Lm26: the carrier of (R^1 | (R^1 E)) = R^1 E
by PRE_TOPC:8;

Lm27: the carrier of (R^1 | (R^1 ].0,1.[)) = R^1 ].0,1.[
by PRE_TOPC:8;

set Af = (AffineMap ((1 / (2 * PI)),0)) | (R^1 E);

dom (AffineMap ((1 / (2 * PI)),0)) = the carrier of R^1
by FUNCT_2:def 1, TOPMETR:17;

then Lm28: dom ((AffineMap ((1 / (2 * PI)),0)) | (R^1 E)) = R^1 E
by RELAT_1:62;

rng ((AffineMap ((1 / (2 * PI)),0)) | (R^1 E)) c= ].0,1.[

proof end;

then reconsider Af = (AffineMap ((1 / (2 * PI)),0)) | (R^1 E) as Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].0,1.[)) by Lm26, Lm27, Lm28, FUNCT_2:2;

Lm29: R^1 (AffineMap ((1 / (2 * PI)),0)) = AffineMap ((1 / (2 * PI)),0)
;

Lm30: dom (AffineMap ((1 / (2 * PI)),0)) = REAL
by FUNCT_2:def 1;

Lm31: rng (AffineMap ((1 / (2 * PI)),0)) = REAL
by FCONT_1:55;

R^1 | ([#] R^1) = R^1
by TSEP_1:3;

then reconsider Af = Af as continuous Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].0,1.[)) by Lm29, Lm30, Lm31, TOPMETR:17, TOPREALA:8;

set L = (R^1 (AffineMap ((- 1),1))) | (R^1 ].0,1.[);

Lm32: dom (AffineMap ((- 1),1)) = REAL
by FUNCT_2:def 1;

then Lm33: dom ((R^1 (AffineMap ((- 1),1))) | (R^1 ].0,1.[)) = ].0,1.[
by RELAT_1:62;

rng ((R^1 (AffineMap ((- 1),1))) | (R^1 ].0,1.[)) c= ].0,1.[

proof end;

then reconsider L = (R^1 (AffineMap ((- 1),1))) | (R^1 ].0,1.[) as Function of (R^1 | (R^1 ].0,1.[)),(R^1 | (R^1 ].0,1.[)) by Lm27, Lm33, FUNCT_2:2;

Lm34: rng (AffineMap ((- 1),1)) = REAL
by FCONT_1:55;

Lm35: R^1 | ([#] R^1) = R^1
by TSEP_1:3;

then reconsider L = L as continuous Function of (R^1 | (R^1 ].0,1.[)),(R^1 | (R^1 ].0,1.[)) by Lm32, Lm34, TOPMETR:17, TOPREALA:8;

reconsider ac = R^1 arccos as continuous Function of (R^1 | (R^1 [.(- 1),1.])),(R^1 | (R^1 [.0,PI.])) by SIN_COS6:85, SIN_COS6:86;

set c = ac | (R^1 Q);

Lm36: dom (ac | (R^1 Q)) = Q
by RELAT_1:62, SIN_COS6:86, XXREAL_1:35;

Lm37: rng (ac | (R^1 Q)) c= E

proof end;

then reconsider c = ac | (R^1 Q) as Function of (R^1 | (R^1 Q)),(R^1 | (R^1 E)) by Lm25, Lm26, Lm36, FUNCT_2:2;

the carrier of (R^1 | (R^1 [.(- 1),1.])) = [.(- 1),1.]
by PRE_TOPC:8;

then reconsider QQ = R^1 Q as Subset of (R^1 | (R^1 [.(- 1),1.])) by XXREAL_1:35;

the carrier of (R^1 | (R^1 [.0,PI.])) = [.0,PI.]
by PRE_TOPC:8;

then reconsider EE = R^1 E as Subset of (R^1 | (R^1 [.0,PI.])) by XXREAL_1:36;

Lm38: (R^1 | (R^1 [.(- 1),1.])) | QQ = R^1 | (R^1 Q)
by GOBOARD9:2;

(R^1 | (R^1 [.0,PI.])) | EE = R^1 | (R^1 E)
by GOBOARD9:2;

then Lm39: c is continuous
by Lm38, TOPREALA:8;

reconsider p = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;

Lm40: dom p = the carrier of (TOP-REAL 2)
by FUNCT_2:def 1;

Lm41: p is continuous
by JORDAN5A:27;

Lm42: for aX1 being Subset of (Topen_unit_circle c[10]) st aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) } holds
Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is continuous

proof end;

Lm43: for aX1 being Subset of (Topen_unit_circle c[10]) st aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) } holds
Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is continuous

proof end;

Lm44: for p being Point of (Topen_unit_circle c[10]) st p = c[-10] holds
Circle2IntervalR is_continuous_at p

proof end;

reconsider jj = 1 as Element of REAL by XREAL_0:def 1;

set h1 = REAL --> jj;

reconsider h1 = REAL --> jj as PartFunc of REAL,REAL ;

Lm45: Circle2IntervalR is continuous

proof end;

set A = ].(1 / 2),((1 / 2) + p1).[;

set Q = ].(- 1),1.];

set E = [.0,PI.[;

reconsider Q = ].(- 1),1.], E = [.0,PI.[ as non empty Subset of REAL ;

Lm46: the carrier of (R^1 | (R^1 Q)) = R^1 Q
by PRE_TOPC:8;

Lm47: the carrier of (R^1 | (R^1 E)) = R^1 E
by PRE_TOPC:8;

Lm48: the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) = R^1 ].(1 / 2),((1 / 2) + p1).[
by PRE_TOPC:8;

set Af = (AffineMap ((- (1 / (2 * PI))),1)) | (R^1 E);

dom (AffineMap ((- (1 / (2 * PI))),1)) = the carrier of R^1
by FUNCT_2:def 1, TOPMETR:17;

then Lm49: dom ((AffineMap ((- (1 / (2 * PI))),1)) | (R^1 E)) = R^1 E
by RELAT_1:62;

rng ((AffineMap ((- (1 / (2 * PI))),1)) | (R^1 E)) c= ].(1 / 2),((1 / 2) + p1).[

proof end;

then reconsider Af = (AffineMap ((- (1 / (2 * PI))),1)) | (R^1 E) as Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by Lm47, Lm48, Lm49, FUNCT_2:2;

Lm50: R^1 (AffineMap ((- (1 / (2 * PI))),1)) = AffineMap ((- (1 / (2 * PI))),1)
;

Lm51: dom (AffineMap ((- (1 / (2 * PI))),1)) = REAL
by FUNCT_2:def 1;

rng (AffineMap ((- (1 / (2 * PI))),1)) = REAL
by FCONT_1:55;

then reconsider Af = Af as continuous Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by Lm35, Lm50, Lm51, TOPMETR:17, TOPREALA:8;

set Af1 = (AffineMap ((1 / (2 * PI)),1)) | (R^1 E);

dom (AffineMap ((1 / (2 * PI)),1)) = the carrier of R^1
by FUNCT_2:def 1, TOPMETR:17;

then Lm52: dom ((AffineMap ((1 / (2 * PI)),1)) | (R^1 E)) = R^1 E
by RELAT_1:62;

rng ((AffineMap ((1 / (2 * PI)),1)) | (R^1 E)) c= ].(1 / 2),((1 / 2) + p1).[

proof end;

then reconsider Af1 = (AffineMap ((1 / (2 * PI)),1)) | (R^1 E) as Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by Lm47, Lm48, Lm52, FUNCT_2:2;

Lm53: R^1 (AffineMap ((1 / (2 * PI)),1)) = AffineMap ((1 / (2 * PI)),1)
;

Lm54: dom (AffineMap ((1 / (2 * PI)),1)) = REAL
by FUNCT_2:def 1;

rng (AffineMap ((1 / (2 * PI)),1)) = REAL
by FCONT_1:55;

then reconsider Af1 = Af1 as continuous Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by Lm35, Lm53, Lm54, TOPMETR:17, TOPREALA:8;

set c = ac | (R^1 Q);

Lm55: dom (ac | (R^1 Q)) = Q
by RELAT_1:62, SIN_COS6:86, XXREAL_1:36;

Lm56: rng (ac | (R^1 Q)) c= E

proof end;

then reconsider c = ac | (R^1 Q) as Function of (R^1 | (R^1 Q)),(R^1 | (R^1 E)) by Lm46, Lm47, Lm55, FUNCT_2:2;

the carrier of (R^1 | (R^1 [.(- 1),1.])) = [.(- 1),1.]
by PRE_TOPC:8;

then reconsider QQ = R^1 Q as Subset of (R^1 | (R^1 [.(- 1),1.])) by XXREAL_1:36;

the carrier of (R^1 | (R^1 [.0,PI.])) = [.0,PI.]
by PRE_TOPC:8;

then reconsider EE = R^1 E as Subset of (R^1 | (R^1 [.0,PI.])) by XXREAL_1:35;

Lm57: (R^1 | (R^1 [.(- 1),1.])) | QQ = R^1 | (R^1 Q)
by GOBOARD9:2;

(R^1 | (R^1 [.0,PI.])) | EE = R^1 | (R^1 E)
by GOBOARD9:2;

then Lm58: c is continuous
by Lm57, TOPREALA:8;

Lm59: for aX1 being Subset of (Topen_unit_circle c[-10]) st aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) } holds
Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is continuous

proof end;

Lm60: for aX1 being Subset of (Topen_unit_circle c[-10]) st aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) } holds
Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is continuous

proof end;

Lm61: for p being Point of (Topen_unit_circle c[-10]) st p = c[10] holds
Circle2IntervalL is_continuous_at p

proof end;

Lm62: Circle2IntervalL is continuous

proof end;

registration

cluster Circle2IntervalR -> one-to-one onto continuous ;

coherence
( Circle2IntervalR is one-to-one & Circle2IntervalR is onto & Circle2IntervalR is continuous ) by Lm45, Th42, GRCAT_1:41;

cluster Circle2IntervalL -> one-to-one onto continuous ;

coherence
( Circle2IntervalL is one-to-one & Circle2IntervalL is onto & Circle2IntervalL is continuous ) by Lm62, Th43, GRCAT_1:41;

end;

Lm63: CircleMap (R^1 0) is open

proof end;

Lm64: CircleMap (R^1 (1 / 2)) is open
by Lm19, Th43, TOPREALA:14;

registration

let i be Integer;

cluster CircleMap (R^1 i) -> open ;

coherence
CircleMap (R^1 i) is open

proof end;

cluster CircleMap (R^1 ((1 / 2) + i)) -> open ;

coherence
CircleMap (R^1 ((1 / 2) + i)) is open

proof end;

end;

registration

cluster Circle2IntervalR -> open ;

coherence
Circle2IntervalR is open

proof end;

cluster Circle2IntervalL -> open ;

coherence
Circle2IntervalL is open by Lm19, Th43, TOPREALA:13;

end;

theorem :: TOPREALB:44

CircleMap (R^1 (1 / 2)) is being_homeomorphism

proof end;

theorem :: TOPREALB:45

ex F being Subset-Family of (Tunit_circle 2) st
( F = {(CircleMap .: ].0,1.[),(CircleMap .: ].(1 / 2),(3 / 2).[)} & F is Cover of (Tunit_circle 2) & F is open & ( for U being Subset of (Tunit_circle 2) holds
( ( U = CircleMap .: ].0,1.[ implies ( union (IntIntervals (0,1)) = CircleMap " U & ( for d being Subset of R^1 st d in IntIntervals (0,1) holds
for f being Function of (R^1 | d),((Tunit_circle 2) | U) st f = CircleMap | d holds
f is being_homeomorphism ) ) ) & ( U = CircleMap .: ].(1 / 2),(3 / 2).[ implies ( union (IntIntervals ((1 / 2),(3 / 2))) = CircleMap " U & ( for d being Subset of R^1 st d in IntIntervals ((1 / 2),(3 / 2)) holds
for f being Function of (R^1 | d),((Tunit_circle 2) | U) st f = CircleMap | d holds
f is being_homeomorphism ) ) ) ) ) )

proof end;