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

begin

set P2 = 2 * PI;

set o = ;

set R = the carrier of R^1;

Lm1: 0 in INT
by INT_1:def 1;

reconsider p0 = - 1 as real negative number ;

reconsider p1 = 1 as real positive number ;

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:25;

Lm3: 1 - 0 <= 1
;

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

registration
cluster K107(0,1) -> non empty ;
coherence
not ].0,1.[ is empty
;
cluster K104((- 1),1) -> non empty ;
coherence
not [.(- 1),1.] is empty
;
cluster K107((1 / 2),(3 / 2)) -> non empty ;
coherence
not ].(1 / 2),(3 / 2).[ is empty
proof end;
end;

Lm5: PI / 2 < PI / 1
by XREAL_1:78;

Lm6: 1 * PI < (3 / 2) * PI
by XREAL_1:70;

Lm7: (3 / 2) * PI < 2 * PI
by XREAL_1:70;

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 end;

theorem Th1: :: TOPREALB:1
for a, r, b being real number holds sin ((a * r) + b) = (sin * (AffineMap (a,b))) . r
proof end;

theorem Th2: :: TOPREALB:2
for a, r, b being real number holds cos ((a * r) + b) = (cos * (AffineMap (a,b))) . r
proof end;

registration
let a be non zero real number ;
let b be real number ;
cluster AffineMap (a,b) -> one-to-one onto ;
coherence
( AffineMap (a,b) is onto & AffineMap (a,b) is one-to-one )
proof end;
end;

definition
let a, b be real number ;
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 number holds IntIntervals (a,b) = { ].(a + n),(b + n).[ where n is Element of INT : verum } ;

theorem :: TOPREALB:3
for a, b being real number
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 number ;
cluster IntIntervals (a,b) -> non empty ;
coherence
not IntIntervals (a,b) is empty
proof end;
end;

theorem :: TOPREALB:4
for b, a being real number st b - a <= 1 holds
IntIntervals (a,b) is mutually-disjoint
proof end;

definition
let a, b be real number ;
:: 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 number ;
:: 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;

begin

definition
let r be real number ;
func R^1 r -> Point of R^1 equals :: TOPREALB:def 2
r;
coherence
r is Point of R^1
by ;
end;

:: deftheorem defines R^1 TOPREALB:def 2 :
for r being real number 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:24;
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_5:62;
end;

registration
let A be closed Subset of REAL;
cluster R^1 A -> closed ;
coherence
R^1 A is closed
by TOPREAL6:79;
end;

registration
let A be open Subset of REAL;
cluster R^1 | (R^1 A) -> open ;
coherence
R^1 | (R^1 A) is open
proof end;
end;

registration
let A be closed Subset of REAL;
cluster R^1 | (R^1 A) -> closed ;
coherence
R^1 | (R^1 A) is closed
proof end;
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
proof end;
end;

registration
let f be one-to-one PartFunc of REAL,REAL;
cluster R^1 f -> one-to-one ;
coherence ;
end;

theorem Th5: :: TOPREALB:5
R^1 | (R^1 ()) = 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
proof end;
end;

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

Lm12: now
let a be non zero real number ; :: thesis: for b being real number 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 number ; :: 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 JORDAN16:32;
A2: [#] R^1 = REAL by TOPMETR:24;
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 ; :: thesis: verum
end;

registration
let a be non zero real number ;
let b be real number ;
cluster R^1 (AffineMap (a,b)) -> open ;
coherence
R^1 (AffineMap (a,b)) is open
proof end;
end;

begin

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 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 real positive number ;
let x be Point of ();
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 ();
let r be real number ;
func Tcircle (x,r) -> SubSpace of TOP-REAL n equals :: TOPREALB:def 6
() | (Sphere (x,r));
coherence
() | (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 ()
for r being real number holds Tcircle (x,r) = () | (Sphere (x,r));

registration
let n be non empty Nat;
let x be Point of ();
let r be real non negative number ;
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 number
for x being Point of () holds the carrier of (Tcircle (x,r)) = Sphere (x,r)
proof end;

registration
let n be Nat;
let x be Point of ();
let r be empty real number ;
cluster Tcircle (x,r) -> trivial ;
coherence
Tcircle (x,r) is trivial
proof end;
end;

theorem Th10: :: TOPREALB:10
for r being real number holds Tcircle ((0. ()),r) is SubSpace of Trectangle ((- r),r,(- r),r)
proof end;

registration
let x be Point of ();
let r be real positive number ;
cluster Tcircle (x,r) -> being_simple_closed_curve ;
coherence
Tcircle (x,r) is being_simple_closed_curve
proof end;
end;

registration
cluster strict TopSpace-like V210() V211() being_simple_closed_curve 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;

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

set TUC = Tunit_circle 2;

set cS1 = the carrier of ();

Lm13: the carrier of () = Sphere (,1)
by ;

registration
let n be non empty Nat;
cluster Tunit_circle n -> non empty ;
coherence
not Tunit_circle n is empty
;
end;

theorem Th12: :: TOPREALB:12
for n being non empty Element of NAT
for x being Point of () st x is Point of () holds
|.x.| = 1
proof end;

theorem Th13: :: TOPREALB:13
for x being Point of () st x is Point of () holds
( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 )
proof end;

theorem Th14: :: TOPREALB:14
for x being Point of () st x is Point of () & x `1 = 1 holds
x `2 = 0
proof end;

theorem Th15: :: TOPREALB:15
for x being Point of () st x is Point of () & x `1 = - 1 holds
x `2 = 0
proof end;

theorem :: TOPREALB:16
for x being Point of () st x is Point of () & x `2 = 1 holds
x `1 = 0
proof end;

theorem :: TOPREALB:17
for x being Point of () st x is Point of () & 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 empty Element of NAT
for r being real positive number
for x being Point of ()
for f being Function of (),(Tcircle (x,r)) st ( for a being Point of ()
for b being Point of () st a = b holds
f . a = (r * b) + x ) holds
f is being_homeomorphism
proof end;

registration end;

Lm14: for n being non empty Element of NAT
for r being real positive number
for x being Point of () holds Tunit_circle n, Tcircle (x,r) are_homeomorphic
proof end;

theorem :: TOPREALB:20
for n being non empty Element of NAT
for r, s being real positive number
for x, y being Point of () holds Tcircle (x,r), Tcircle (y,s) are_homeomorphic
proof end;

registration
let x be Point of ();
let r be real non negative number ;
cluster Tcircle (x,r) -> pathwise_connected ;
coherence
Tcircle (x,r) is pathwise_connected
proof end;
end;

definition
func c[10] -> Point of () equals :: TOPREALB:def 8
|[1,0]|;
coherence
|[1,0]| is Point of ()
proof end;
func c[-10] -> Point of () equals :: TOPREALB:def 9
|[(- 1),0]|;
coherence
|[(- 1),0]| is Point of ()
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 ();
func Topen_unit_circle p -> strict SubSpace of Tunit_circle 2 means :Def10: :: TOPREALB:def 10
the carrier of it = the carrier of () \ {p};
existence
ex b1 being strict SubSpace of Tunit_circle 2 st the carrier of b1 = the carrier of () \ {p}
proof end;
uniqueness
for b1, b2 being strict SubSpace of Tunit_circle 2 st the carrier of b1 = the carrier of () \ {p} & the carrier of b2 = the carrier of () \ {p} holds
b1 = b2
by TSEP_1:5;
end;

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

registration
let p be Point of ();
cluster Topen_unit_circle p -> non empty strict ;
coherence
not Topen_unit_circle p is empty
proof end;
end;

theorem :: TOPREALB:21
canceled;

theorem Th22: :: TOPREALB:22
for p being Point of () holds p is not Point of
proof end;

theorem Th23: :: TOPREALB:23
for p being Point of () holds Topen_unit_circle p = () | (([#] ()) \ {p})
proof end;

theorem Th24: :: TOPREALB:24
for p, q being Point of () st p <> q holds
q is Point of
proof end;

theorem Th25: :: TOPREALB:25
for p being Point of () st p is Point of & p `2 = 0 holds
p = c[-10]
proof end;

theorem Th26: :: TOPREALB:26
for p being Point of () st p is Point of & 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 ;

set Xm = the carrier of ;

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 = [#]
;

Lm17: the carrier of = [#]
;

theorem Th27: :: TOPREALB:27
for p being Point of ()
for x being Point of () st x is Point of holds
( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 )
proof end;

theorem Th28: :: TOPREALB:28
for x being Point of () st x is Point of holds
( - 1 <= x `1 & x `1 < 1 )
proof end;

theorem Th29: :: TOPREALB:29
for x being Point of () st x is Point of holds
( - 1 < x `1 & x `1 <= 1 )
proof end;

registration
let p be Point of ();
cluster Topen_unit_circle p -> strict open ;
coherence
proof end;
end;

theorem :: TOPREALB:30
for p being Point of () holds Topen_unit_circle p, I(01) are_homeomorphic
proof end;

theorem :: TOPREALB:31
for p, q being Point of () holds Topen_unit_circle p, Topen_unit_circle q are_homeomorphic
proof end;

begin

definition
func CircleMap -> Function of R^1,() means :Def11: :: TOPREALB:def 11
for x being real number holds it . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]|;
existence
ex b1 being Function of R^1,() st
for x being real number holds b1 . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]|
proof end;
uniqueness
for b1, b2 being Function of R^1,() st ( for x being real number holds b1 . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ) & ( for x being real number 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,() holds
( b1 = CircleMap iff for x being real number holds b1 . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| );

Lm18: dom CircleMap = REAL
by ;

theorem Th32: :: TOPREALB:32
for i being Integer
for r being real number holds CircleMap . r = CircleMap . (r + i)
proof end;

theorem Th33: :: TOPREALB:33
for i being Integer holds CircleMap . i = c[10]
proof end;

theorem Th34: :: TOPREALB:34
proof end;

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

theorem Th35: :: TOPREALB:35
for r being real number st frac r = 1 / 2 holds
CircleMap . r = |[(- 1),0]|
proof end;

theorem :: TOPREALB:36
for r being real number st frac r = 1 / 4 holds
CircleMap . r = |[0,1]|
proof end;

theorem :: TOPREALB:37
for r being real number st frac r = 3 / 4 holds
CircleMap . r = |[0,(- 1)]|
proof end;

Lm20: for r being real number holds CircleMap . r = |[((cos * (AffineMap ((2 * PI),0))) . r),((sin * (AffineMap ((2 * PI),0))) . r)]|
proof end;

theorem :: TOPREALB:38
for r being real number
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 end;

Lm21: for A being Subset of R^1 holds CircleMap | A is Function of (R^1 | A),()
proof end;

Lm22: for r being real number st - 1 <= r & r <= 1 holds
( 0 <= () / (2 * PI) & () / (2 * PI) <= 1 / 2 )
proof end;

theorem Th39: :: TOPREALB:39
for A being Subset of R^1
for f being Function of (R^1 | A),() st [.0,1.[ c= A & f = CircleMap | A holds
f is onto
proof end;

registration
cluster CircleMap -> onto ;
coherence
proof end;
end;

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

registration
let r be real number ;
cluster K76(CircleMap,[.r,(r + 1).[) -> one-to-one ;
coherence
CircleMap | [.r,(r + 1).[ is one-to-one
proof end;
end;

registration
let r be real number ;
cluster K76(CircleMap,].r,(r + 1).[) -> one-to-one ;
coherence
CircleMap | ].r,(r + 1).[ is one-to-one
proof end;
end;

theorem Th40: :: TOPREALB:40
for b, a being real number 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 Th41: :: TOPREALB:41
for a, b being real number
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).[)),() equals :: TOPREALB:def 12
CircleMap | ].r,(r + 1).[;
coherence
CircleMap | ].r,(r + 1).[ is Function of (R^1 | (R^1 ].r,(r + 1).[)),()
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 number holds rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) = ].r,(r + 1).[
proof end;

theorem Th42: :: TOPREALB:42
for i being Integer
for a being real number 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 ,(R^1 | (R^1 ].0,1.[)) means :Def13: :: TOPREALB:def 13
for p being Point of ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies it . p = () / (2 * PI) ) & ( y <= 0 implies it . p = 1 - (() / (2 * PI)) ) );
existence
ex b1 being Function of ,(R^1 | (R^1 ].0,1.[)) st
for p being Point of ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = () / (2 * PI) ) & ( y <= 0 implies b1 . p = 1 - (() / (2 * PI)) ) )
proof end;
uniqueness
for b1, b2 being Function of ,(R^1 | (R^1 ].0,1.[)) st ( for p being Point of ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = () / (2 * PI) ) & ( y <= 0 implies b1 . p = 1 - (() / (2 * PI)) ) ) ) & ( for p being Point of ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b2 . p = () / (2 * PI) ) & ( y <= 0 implies b2 . p = 1 - (() / (2 * PI)) ) ) ) holds
b1 = b2
proof end;
end;

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

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

definition
func Circle2IntervalL -> Function of ,(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) means :Def14: :: TOPREALB:def 14
for p being Point of ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies it . p = 1 + (() / (2 * PI)) ) & ( y <= 0 implies it . p = 1 - (() / (2 * PI)) ) );
existence
ex b1 being Function of ,(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) st
for p being Point of ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = 1 + (() / (2 * PI)) ) & ( y <= 0 implies b1 . p = 1 - (() / (2 * PI)) ) )
proof end;
uniqueness
for b1, b2 being Function of ,(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) st ( for p being Point of ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = 1 + (() / (2 * PI)) ) & ( y <= 0 implies b1 . p = 1 - (() / (2 * PI)) ) ) ) & ( for p being Point of ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies b2 . p = 1 + (() / (2 * PI)) ) & ( y <= 0 implies b2 . p = 1 - (() / (2 * PI)) ) ) ) holds
b1 = b2
proof end;
end;

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

set C = Circle2IntervalR ;

set Cm = Circle2IntervalL ;

theorem Th43: :: TOPREALB:43
proof end;

theorem Th44: :: TOPREALB:44
(CircleMap (R^1 (1 / 2))) " = Circle2IntervalL
proof end;

set A = ].0,1.[;

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

set E = ;

set j = 1 / (2 * PI);

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

Lm25: the carrier of (R^1 | (R^1 Q)) = R^1 Q
by PRE_TOPC:29;

Lm26: the carrier of (R^1 | (R^1 E)) = R^1 E
by PRE_TOPC:29;

Lm27: the carrier of (R^1 | (R^1 ].0,1.[)) = R^1 ].0,1.[
by PRE_TOPC:29;

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

dom (AffineMap ((1 / (2 * PI)),0)) = the carrier of R^1
by ;

then Lm28: dom ((AffineMap ((1 / (2 * PI)),0)) | (R^1 E)) = R^1 E
by RELAT_1:91;

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 ;

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 JORDAN16:32;

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 ;

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_1:91;

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 ;

Lm34: rng (AffineMap ((- 1),1)) = REAL
by JORDAN16:32;

Lm35: 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 ;

reconsider ac = R^1 arccos as continuous Function of (R^1 | (R^1 [.(- 1),1.])),(R^1 | ()) by ;

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

Lm36: dom (ac | (R^1 Q)) = Q
by ;

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 ;

the carrier of (R^1 | (R^1 [.(- 1),1.])) = [.(- 1),1.]
by PRE_TOPC:29;

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

the carrier of (R^1 | ()) =
by PRE_TOPC:29;

then reconsider EE = R^1 E as Subset of (R^1 | ()) by XXREAL_1:36;

Lm38: (R^1 | (R^1 [.(- 1),1.])) | QQ = R^1 | (R^1 Q)
by GOBOARD9:4;

(R^1 | ()) | EE = R^1 | (R^1 E)
by GOBOARD9:4;

then Lm39: c is continuous
by ;

reconsider p = proj1 as Function of (),R^1 by TOPMETR:24;

Lm40: dom p = the carrier of ()
by FUNCT_2:def 1;

Lm41: p is continuous
by TOPREAL6:83;

Lm42: for aX1 being Subset of st aX1 = { q where q is Point of () : ( q in the carrier of & 0 <= q `2 ) } holds
Circle2IntervalR | ( | aX1) is continuous
proof end;

Lm43: for aX1 being Subset of st aX1 = { q where q is Point of () : ( q in the carrier of & 0 >= q `2 ) } holds
Circle2IntervalR | ( | aX1) is continuous
proof end;

Lm44: for p being Point of st p = c[-10] holds
Circle2IntervalR is_continuous_at p
proof end;

set h1 = REAL --> 1;

reconsider h1 = REAL --> 1 as PartFunc of REAL,REAL ;

Lm45: Circle2IntervalR is continuous
proof end;

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

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

set E = ;

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

Lm46: the carrier of (R^1 | (R^1 Q)) = R^1 Q
by PRE_TOPC:29;

Lm47: the carrier of (R^1 | (R^1 E)) = R^1 E
by PRE_TOPC:29;

Lm48: the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) = R^1 ].(1 / 2),((1 / 2) + p1).[
by PRE_TOPC:29;

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

dom (AffineMap ((- (1 / (2 * PI))),1)) = the carrier of R^1
by ;

then Lm49: dom ((AffineMap ((- (1 / (2 * PI))),1)) | (R^1 E)) = R^1 E
by RELAT_1:91;

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 ;

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 JORDAN16:32;

then reconsider Af = Af as continuous Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by ;

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

dom (AffineMap ((1 / (2 * PI)),1)) = the carrier of R^1
by ;

then Lm52: dom ((AffineMap ((1 / (2 * PI)),1)) | (R^1 E)) = R^1 E
by RELAT_1:91;

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 ;

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 JORDAN16:32;

then reconsider Af1 = Af1 as continuous Function of (R^1 | (R^1 E)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by ;

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

Lm55: dom (ac | (R^1 Q)) = Q
by ;

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 ;

the carrier of (R^1 | (R^1 [.(- 1),1.])) = [.(- 1),1.]
by PRE_TOPC:29;

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

the carrier of (R^1 | ()) =
by PRE_TOPC:29;

then reconsider EE = R^1 E as Subset of (R^1 | ()) by XXREAL_1:35;

Lm57: (R^1 | (R^1 [.(- 1),1.])) | QQ = R^1 | (R^1 Q)
by GOBOARD9:4;

(R^1 | ()) | EE = R^1 | (R^1 E)
by GOBOARD9:4;

then Lm58: c is continuous
by ;

Lm59: for aX1 being Subset of st aX1 = { q where q is Point of () : ( q in the carrier of & 0 <= q `2 ) } holds
Circle2IntervalL | ( | aX1) is continuous
proof end;

Lm60: for aX1 being Subset of st aX1 = { q where q is Point of () : ( q in the carrier of & 0 >= q `2 ) } holds
Circle2IntervalL | ( | aX1) is continuous
proof end;

Lm61: for p being Point of 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 by ;
cluster Circle2IntervalL -> one-to-one onto continuous ;
coherence by ;
end;

Lm63: CircleMap () is open
proof end;

Lm64: CircleMap (R^1 (1 / 2)) is open
by ;

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 end;

theorem :: TOPREALB:45
canceled;

theorem :: TOPREALB:46
CircleMap (R^1 (1 / 2)) is being_homeomorphism
proof end;

theorem :: TOPREALB:47
canceled;

theorem :: TOPREALB:48
canceled;

theorem :: TOPREALB:49
ex F being Subset-Family of () st
( F = {(),(CircleMap .: ].(1 / 2),(3 / 2).[)} & F is Cover of () & F is open & ( for U being Subset of () 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),(() | 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),(() | U) st f = CircleMap | d holds
f is being_homeomorphism ) ) ) ) ) )
proof end;