begin
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 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
;
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
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
theorem Th1:
theorem Th2:
:: 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
theorem
begin
:: deftheorem defines R^1 TOPREALB:def 2 :
for r being real number holds R^1 r = r;
:: deftheorem defines R^1 TOPREALB:def 3 :
for A being Subset of REAL holds R^1 A = A;
:: deftheorem defines R^1 TOPREALB:def 4 :
for f being PartFunc of REAL ,REAL holds R^1 f = f;
theorem Th5:
theorem Th6:
theorem Th7:
theorem Th8:
Lm10:
sin is Function of R^1 ,R^1
Lm11:
cos is Function of R^1 ,R^1
set A = R^1 ].0 ,1.[;
Lm12:
now
let a be non
zero real number ;
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 ;
( 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 A1, A2, TSEP_1:3;
verum
end;
begin
:: 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 );
:: deftheorem defines Tcircle TOPREALB:def 6 :
for n being Nat
for x being Point of (TOP-REAL n)
for r being real number holds Tcircle x,r = (TOP-REAL n) | (Sphere x,r);
theorem Th9:
theorem Th10:
theorem
:: 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:58;
theorem Th12:
theorem Th13:
theorem Th14:
theorem Th15:
theorem
theorem
set TREC = Trectangle p0,p1,p0,p1;
theorem
theorem Th19:
Lm14:
for n being non empty Element of NAT
for r being real positive number
for x being Point of (TOP-REAL n) holds Tunit_circle n, Tcircle x,r are_homeomorphic
theorem
:: 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;
:: 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} );
theorem
canceled;
theorem Th22:
theorem Th23:
theorem Th24:
theorem Th25:
theorem Th26:
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 Th27:
theorem Th28:
theorem Th29:
theorem
theorem
begin
:: 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 number holds b1 . x = |[(cos ((2 * PI ) * x)),(sin ((2 * PI ) * x))]| );
Lm18:
dom CircleMap = REAL
by FUNCT_2:def 1, TOPMETR:24;
theorem Th32:
theorem Th33:
theorem Th34:
Lm19:
CircleMap . (1 / 2) = |[(- 1),0 ]|
theorem Th35:
theorem
theorem
Lm20:
for r being real number holds CircleMap . r = |[((cos * (AffineMap (2 * PI ),0 )) . r),((sin * (AffineMap (2 * PI ),0 )) . r)]|
theorem
Lm21:
for A being Subset of R^1 holds CircleMap | A is Function of (R^1 | A),(Tunit_circle 2)
Lm22:
for r being real number st - 1 <= r & r <= 1 holds
( 0 <= (arccos r) / (2 * PI ) & (arccos r) / (2 * PI ) <= 1 / 2 )
theorem Th39:
Lm23:
CircleMap | [.0 ,1.[ is one-to-one
theorem Th40:
theorem Th41:
:: 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).[
theorem Th42:
definition
func Circle2IntervalR -> Function of
(Topen_unit_circle c[10] ),
(R^1 | (R^1 ].0 ,1.[)) means :
Def13:
for
p being
Point of
(Topen_unit_circle c[10] ) ex
x,
y being
real number 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 number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = (arccos x) / (2 * PI ) ) & ( y <= 0 implies b1 . p = 1 - ((arccos x) / (2 * PI )) ) )
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 number 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 number 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
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 number 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:
for
p being
Point of
(Topen_unit_circle c[-10] ) ex
x,
y being
real number 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 number st
( p = |[x,y]| & ( y >= 0 implies b1 . p = 1 + ((arccos x) / (2 * PI )) ) & ( y <= 0 implies b1 . p = 1 - ((arccos x) / (2 * PI )) ) )
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 number 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 number 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
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 number 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 Th43:
theorem Th44:
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: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 FUNCT_2:def 1, TOPMETR:24;
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.[
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_2:4;
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 ) = 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:24, TOPREALA:29;
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.[
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_2:4;
Lm34:
rng (AffineMap (- 1),1) = REAL
by JORDAN16:32;
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:24, TOPREALA:29;
reconsider ac = R^1 arccos as continuous Function of (R^1 | (R^1 [.(- 1),1.])),(R^1 | (R^1 [.0 ,PI .])) by SIN_COS6:87, SIN_COS6:88;
set c = ac | (R^1 Q);
Lm36:
dom (ac | (R^1 Q)) = Q
by RELAT_1:91, SIN_COS6:88, XXREAL_1:35;
Lm37:
rng (ac | (R^1 Q)) c= E
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_2:4;
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 | (R^1 [.0 ,PI .])) = [.0 ,PI .]
by PRE_TOPC:29;
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 GOBOARD9:4;
(R^1 | (R^1 [.0 ,PI .])) | EE = R^1 | (R^1 E)
by GOBOARD9:4;
then Lm39:
c is continuous
by Lm38, TOPREALA:29;
reconsider p = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:24;
Lm40:
dom p = the carrier of (TOP-REAL 2)
by FUNCT_2:def 1;
Lm41:
p is continuous
by TOPREAL6:83;
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
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
Lm44:
for p being Point of (Topen_unit_circle c[10] ) st p = c[-10] holds
Circle2IntervalR is_continuous_at p
set h1 = REAL --> 1;
reconsider h1 = REAL --> 1 as PartFunc of REAL ,REAL ;
Lm45:
Circle2IntervalR is continuous
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: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 FUNCT_2:def 1, TOPMETR:24;
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).[
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_2:4;
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 Lm35, Lm50, Lm51, TOPMETR:24, TOPREALA:29;
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:24;
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).[
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_2:4;
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 Lm35, Lm53, Lm54, TOPMETR:24, TOPREALA:29;
set c = ac | (R^1 Q);
Lm55:
dom (ac | (R^1 Q)) = Q
by RELAT_1:91, SIN_COS6:88, XXREAL_1:36;
Lm56:
rng (ac | (R^1 Q)) c= E
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_2:4;
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 | (R^1 [.0 ,PI .])) = [.0 ,PI .]
by PRE_TOPC:29;
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 GOBOARD9:4;
(R^1 | (R^1 [.0 ,PI .])) | EE = R^1 | (R^1 E)
by GOBOARD9:4;
then Lm58:
c is continuous
by Lm57, TOPREALA:29;
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
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
Lm61:
for p being Point of (Topen_unit_circle c[-10] ) st p = c[10] holds
Circle2IntervalL is_continuous_at p
Lm62:
Circle2IntervalL is continuous
Lm63:
CircleMap (R^1 0 ) is open
Lm64:
CircleMap (R^1 (1 / 2)) is open
by Lm19, Th44, TOPREALA:35;
theorem
canceled;
theorem
theorem
canceled;
theorem
canceled;
theorem
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 ) ) ) ) ) )