:: TOPREALB semantic presentation
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:18;
Lm3: 1 - 0 <= 1
;
Lm4: (3 / 2) - (1 / 2) <= 1
;
registration
clusterK108(0,1) -> non empty ;
coherence
not ].0,1.[ is empty ;
clusterK105((- 1),1) -> non empty ;
coherence
not [.(- 1),1.] is empty ;
clusterK108((1 / 2),(3 / 2)) -> non empty ;
coherence
not ].(1 / 2),(3 / 2).[ is empty
proof
not ].(1 / 2),((1 / 2) + p1).[ is empty ;
hence not ].(1 / 2),(3 / 2).[ is empty ; ::_thesis: verum
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
let X be non empty TopSpace; ::_thesis: 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
let Y be non empty SubSpace of X; ::_thesis: for Z being non empty SubSpace of Y
for p being Point of Z holds p is Point of X
let Z be non empty SubSpace of Y; ::_thesis: for p being Point of Z holds p is Point of X
let p be Point of Z; ::_thesis: p is Point of X
p is Point of Y by PRE_TOPC:25;
hence p is Point of X by PRE_TOPC:25; ::_thesis: verum
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
let X be TopSpace; ::_thesis: for Y being SubSpace of X
for Z being SubSpace of Y
for A being Subset of Z holds A is Subset of X
let Y be SubSpace of X; ::_thesis: for Z being SubSpace of Y
for A being Subset of Z holds A is Subset of X
let Z be SubSpace of Y; ::_thesis: for A being Subset of Z holds A is Subset of X
let A be Subset of Z; ::_thesis: A is Subset of X
the carrier of Z is Subset of Y by TSEP_1:1;
then A1: A is Subset of Y by XBOOLE_1:1;
the carrier of Y is Subset of X by TSEP_1:1;
hence A is Subset of X by A1, XBOOLE_1:1; ::_thesis: verum
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, r, b being real number holds sin ((a * r) + b) = (sin * (AffineMap (a,b))) . r
proof
let a, r, b be real number ; ::_thesis: sin ((a * r) + b) = (sin * (AffineMap (a,b))) . r
A1: r is Real by XREAL_0:def_1;
thus sin ((a * r) + b) = sin . ((a * r) + b) by SIN_COS:def_17
.= sin . ((AffineMap (a,b)) . r) by FCONT_1:def_4
.= (sin * (AffineMap (a,b))) . r by A1, FUNCT_2:15 ; ::_thesis: verum
end;
theorem Th2: :: TOPREALB:2
for a, r, b being real number holds cos ((a * r) + b) = (cos * (AffineMap (a,b))) . r
proof
let a, r, b be real number ; ::_thesis: cos ((a * r) + b) = (cos * (AffineMap (a,b))) . r
A1: r is Real by XREAL_0:def_1;
thus cos ((a * r) + b) = cos . ((a * r) + b) by SIN_COS:def_19
.= cos . ((AffineMap (a,b)) . r) by FCONT_1:def_4
.= (cos * (AffineMap (a,b))) . r by A1, FUNCT_2:15 ; ::_thesis: verum
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
thus rng (AffineMap (a,b)) = REAL by FCONT_1:55; :: according to FUNCT_2:def_3 ::_thesis: AffineMap (a,b) is one-to-one
thus AffineMap (a,b) is one-to-one by FCONT_1:50; ::_thesis: verum
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
].(a + 0),(b + 0).[ in IntIntervals (a,b) by Lm1;
hence not IntIntervals (a,b) is empty ; ::_thesis: verum
end;
end;
theorem :: TOPREALB:4
for b, a being real number st b - a <= 1 holds
IntIntervals (a,b) is mutually-disjoint
proof
let b, a be real number ; ::_thesis: ( b - a <= 1 implies IntIntervals (a,b) is mutually-disjoint )
assume A1: b - a <= 1 ; ::_thesis: IntIntervals (a,b) is mutually-disjoint
A2: now__::_thesis:_for_k_being_Element_of_NAT_holds_a_+_(k_+_1)_>=_b
let k be Element of NAT ; ::_thesis: a + (k + 1) >= b
A3: (a + 1) + 0 <= (a + 1) + k by XREAL_1:6;
(b - a) + a <= 1 + a by A1, XREAL_1:6;
hence a + (k + 1) >= b by A3, XXREAL_0:2; ::_thesis: verum
end;
let x, y be set ; :: according to TAXONOM2:def_5 ::_thesis: ( not x in IntIntervals (a,b) or not y in IntIntervals (a,b) or x = y or x misses y )
assume x in IntIntervals (a,b) ; ::_thesis: ( not y in IntIntervals (a,b) or x = y or x misses y )
then consider nx being Element of INT such that
A4: x = ].(a + nx),(b + nx).[ ;
assume y in IntIntervals (a,b) ; ::_thesis: ( x = y or x misses y )
then consider ny being Element of INT such that
A5: y = ].(a + ny),(b + ny).[ ;
assume A6: x <> y ; ::_thesis: x misses y
assume x meets y ; ::_thesis: contradiction
then consider z being set such that
A7: z in x and
A8: z in y by XBOOLE_0:3;
reconsider z = z as Real by A4, A7;
A9: a + nx < z by A4, A7, XXREAL_1:4;
A10: z < b + ny by A5, A8, XXREAL_1:4;
A11: a + ny < z by A5, A8, XXREAL_1:4;
A12: z < b + nx by A4, A7, XXREAL_1:4;
percases ( nx = ny or nx < ny or nx > ny ) by XXREAL_0:1;
suppose nx = ny ; ::_thesis: contradiction
hence contradiction by A4, A5, A6; ::_thesis: verum
end;
suppose nx < ny ; ::_thesis: contradiction
then nx + 1 <= ny by INT_1:7;
then reconsider k = ny - (nx + 1) as Element of NAT by INT_1:5;
((a + nx) + 1) + k < b + nx by A12, A11, XXREAL_0:2;
then ((a + nx) + (k + 1)) - nx < (b + nx) - nx by XREAL_1:14;
then a + (k + 1) < b ;
hence contradiction by A2; ::_thesis: verum
end;
suppose nx > ny ; ::_thesis: contradiction
then ny + 1 <= nx by INT_1:7;
then reconsider k = nx - (ny + 1) as Element of NAT by INT_1:5;
((a + ny) + 1) + k < b + ny by A9, A10, XXREAL_0:2;
then ((a + ny) + (k + 1)) - ny < (b + ny) - ny by XREAL_1:14;
then a + (k + 1) < b ;
hence contradiction by A2; ::_thesis: verum
end;
end;
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
IntIntervals (a,b) c= bool the carrier of R^1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in IntIntervals (a,b) or x in bool the carrier of R^1 )
assume x in IntIntervals (a,b) ; ::_thesis: x in bool the carrier of R^1
then ex n being Element of INT st x = ].(a + n),(b + n).[ ;
hence x in bool the carrier of R^1 by TOPMETR:17; ::_thesis: verum
end;
hence IntIntervals (a,b) is Subset-Family of R^1 ; ::_thesis: verum
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
IntIntervals (a,b) is open
proof
let A be Subset of R^1; :: according to TOPS_2:def_1 ::_thesis: ( not A in IntIntervals (a,b) or A is open )
assume A in IntIntervals (a,b) ; ::_thesis: A is open
then ex n being Element of INT st A = ].(a + n),(b + n).[ ;
hence A is open by JORDAN6:35; ::_thesis: verum
end;
hence IntIntervals (a,b) is open Subset-Family of R^1 ; ::_thesis: verum
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 TOPMETR:17, XREAL_0:def_1;
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: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;
clusterR^1 | (R^1 A) -> open ;
coherence
R^1 | (R^1 A) is open
proof
let X be Subset of R^1; :: according to TSEP_1:def_1 ::_thesis: ( not X = the carrier of (R^1 | (R^1 A)) or X is open )
assume X = the carrier of (R^1 | (R^1 A)) ; ::_thesis: X is open
hence X is open by PRE_TOPC:8; ::_thesis: verum
end;
end;
registration
let A be closed Subset of REAL;
clusterR^1 | (R^1 A) -> closed ;
coherence
R^1 | (R^1 A) is closed
proof
let X be Subset of R^1; :: according to BORSUK_1:def_11 ::_thesis: ( not X = the carrier of (R^1 | (R^1 A)) or X is closed )
assume X = the carrier of (R^1 | (R^1 A)) ; ::_thesis: X is closed
hence X is closed by PRE_TOPC:8; ::_thesis: verum
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
A1: the carrier of (R^1 | (R^1 (rng f))) = R^1 (rng f) by PRE_TOPC:8;
the carrier of (R^1 | (R^1 (dom f))) = R^1 (dom f) by PRE_TOPC:8;
hence f is Function of (R^1 | (R^1 (dom f))),(R^1 | (R^1 (rng f))) by A1, FUNCT_2:2; ::_thesis: verum
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
thus rng (R^1 f) = the carrier of (R^1 | (R^1 (rng f))) by PRE_TOPC:8; :: according to FUNCT_2:def_3 ::_thesis: verum
end;
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
[#] R^1 = R^1 ([#] REAL) by TOPMETR:17;
hence R^1 | (R^1 ([#] REAL)) = R^1 by PRE_TOPC:def_5; ::_thesis: verum
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
let f be PartFunc of REAL,REAL; ::_thesis: ( dom f = REAL implies R^1 | (R^1 (dom f)) = R^1 )
assume dom f = REAL ; ::_thesis: R^1 | (R^1 (dom f)) = R^1
then [#] R^1 = R^1 (dom f) by TOPMETR:17;
hence R^1 | (R^1 (dom f)) = R^1 by PRE_TOPC:def_5; ::_thesis: verum
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
let f be Function of REAL,REAL; ::_thesis: f is Function of R^1,(R^1 | (R^1 (rng f)))
REAL = dom f by FUNCT_2:def_1;
then R^1 | (R^1 (dom f)) = R^1 by Th6;
then R^1 f is Function of R^1,(R^1 | (R^1 (rng f))) ;
hence f is Function of R^1,(R^1 | (R^1 (rng f))) ; ::_thesis: verum
end;
theorem Th8: :: TOPREALB:8
for f being Function of REAL,REAL holds f is Function of R^1,R^1
proof
let f be Function of REAL,REAL; ::_thesis: f is Function of R^1,R^1
dom f = REAL by FUNCT_2:def_1;
then reconsider B = rng f as non empty Subset of REAL by RELAT_1:42;
f is Function of R^1,(R^1 | (R^1 B)) by Th7;
hence f is Function of R^1,R^1 by TOPREALA:7; ::_thesis: verum
end;
Lm10: sin is Function of R^1,R^1
proof
A1: sin = R^1 sin ;
R^1 | (R^1 (dom sin)) = R^1 by Th6, SIN_COS:24;
hence sin is Function of R^1,R^1 by A1, COMPTRIG:28, TOPREALA:7; ::_thesis: verum
end;
Lm11: cos is Function of R^1,R^1
proof
A1: cos = R^1 cos ;
R^1 | (R^1 (dom cos)) = R^1 by Th6, SIN_COS:24;
hence cos is Function of R^1,R^1 by A1, COMPTRIG:29, TOPREALA:7; ::_thesis: verum
end;
registration
let f be continuous PartFunc of REAL,REAL;
cluster R^1 f -> continuous ;
coherence
R^1 f is continuous
proof
set g = R^1 f;
percases ( not dom f is empty or dom f is empty ) ;
suppose not dom f is empty ; ::_thesis: R^1 f is continuous
then reconsider A = dom f, B = rng f as non empty Subset of REAL by RELAT_1:42;
reconsider g = R^1 f as Function of (R^1 | (R^1 A)),(R^1 | (R^1 B)) ;
reconsider h = g as Function of (R^1 | (R^1 A)),R^1 by TOPREALA:7;
for x being Point of (R^1 | (R^1 A)) holds h is_continuous_at x
proof
let x be Point of (R^1 | (R^1 A)); ::_thesis: h is_continuous_at x
let G be a_neighborhood of h . x; :: according to TMAP_1:def_2 ::_thesis: ex b1 being a_neighborhood of x st h .: b1 c= G
consider Z being Neighbourhood of f . x such that
A1: Z c= G by TOPREALA:20;
reconsider xx = x as Point of R^1 by PRE_TOPC:25;
the carrier of (R^1 | (R^1 A)) = A by PRE_TOPC:8;
then f is_continuous_in x by FCONT_1:def_2;
then consider N being Neighbourhood of x such that
A2: f .: N c= Z by FCONT_1:5;
consider g being real number such that
A3: 0 < g and
A4: N = ].(x - g),(x + g).[ by RCOMP_1:def_6;
A5: x + 0 < x + g by A3, XREAL_1:6;
reconsider NN = N as open Subset of R^1 by A4, JORDAN6:35, TOPMETR:17;
reconsider M = NN /\ ([#] (R^1 | (R^1 A))) as Subset of (R^1 | (R^1 A)) ;
A6: NN = Int NN by TOPS_1:23;
x - g < x - 0 by A3, XREAL_1:15;
then xx in Int NN by A4, A6, A5, XXREAL_1:4;
then NN is open a_neighborhood of xx by CONNSP_2:def_1;
then reconsider M = M as open a_neighborhood of x by TOPREALA:5;
take M ; ::_thesis: h .: M c= G
h .: M c= h .: NN by RELAT_1:123, XBOOLE_1:17;
then h .: M c= Z by A2, XBOOLE_1:1;
hence h .: M c= G by A1, XBOOLE_1:1; ::_thesis: verum
end;
then h is continuous by TMAP_1:44;
hence R^1 f is continuous by PRE_TOPC:27; ::_thesis: verum
end;
suppose dom f is empty ; ::_thesis: R^1 f is continuous
hence R^1 f is continuous ; ::_thesis: verum
end;
end;
end;
end;
set A = R^1 ].0,1.[;
Lm12: now__::_thesis:_for_a_being_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 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 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 number ;
let b be real number ;
cluster R^1 (AffineMap (a,b)) -> open ;
coherence
R^1 (AffineMap (a,b)) is open
proof
let A be Subset of (R^1 | (R^1 (dom (AffineMap (a,b))))); :: according to T_0TOPSP:def_2 ::_thesis: ( not A is open or (R^1 (AffineMap (a,b))) .: A is open )
A1: b is Real by XREAL_0:def_1;
A2: R^1 = R^1 | (R^1 (dom (AffineMap (a,b)))) by Lm12;
A3: R^1 = R^1 | (R^1 (rng (AffineMap (a,b)))) by Lm12;
a is Real by XREAL_0:def_1;
then R^1 (AffineMap (a,b)) is being_homeomorphism by A1, A2, A3, JORDAN16:20;
hence ( not A is open or (R^1 (AffineMap (a,b))) .: A is open ) by A2, A3, TOPGRP_1:25; ::_thesis: verum
end;
end;
begin
definition
let S be SubSpace of TOP-REAL 2;
attrS 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
let S be SubSpace of TOP-REAL 2; ::_thesis: ( S is being_simple_closed_curve implies ( not S is empty & S is pathwise_connected & S is compact ) )
assume A1: the carrier of S is Simple_closed_curve ; :: according to TOPREALB:def_5 ::_thesis: ( not S is empty & S is pathwise_connected & S is compact )
then reconsider A = the carrier of S as Simple_closed_curve ;
not A is empty ;
hence not the carrier of S is empty ; :: according to STRUCT_0:def_1 ::_thesis: ( S is pathwise_connected & S is compact )
thus S is pathwise_connected by A1, TOPALG_3:10; ::_thesis: S is compact
[#] S = A ;
then [#] S is compact by COMPTS_1:2;
hence S is compact by COMPTS_1:1; ::_thesis: verum
end;
end;
registration
let r be real positive number ;
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
reconsider a = x as Point of (Euclid 2) by TOPREAL3:8;
A1: x = |[(x `1),(x `2)]| by EUCLID:53;
Sphere (x,r) = Sphere (a,r) by TOPREAL9:15
.= circle ((x `1),(x `2),r) by A1, TOPREAL9:49
.= { w where w is Point of (TOP-REAL 2) : |.(w - |[(x `1),(x `2)]|).| = r } by JGRAPH_6:def_5 ;
hence Sphere (x,r) is being_simple_closed_curve by JGRAPH_6:23; ::_thesis: verum
end;
end;
definition
let n be Nat;
let x be Point of (TOP-REAL n);
let r be real number ;
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 number holds Tcircle (x,r) = (TOP-REAL n) | (Sphere (x,r));
registration
let n be non empty Nat;
let x be Point of (TOP-REAL n);
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 (TOP-REAL n) holds the carrier of (Tcircle (x,r)) = Sphere (x,r)
proof
let n be Element of NAT ; ::_thesis: for r being real number
for x being Point of (TOP-REAL n) holds the carrier of (Tcircle (x,r)) = Sphere (x,r)
let r be real number ; ::_thesis: for x being Point of (TOP-REAL n) holds the carrier of (Tcircle (x,r)) = Sphere (x,r)
let x be Point of (TOP-REAL n); ::_thesis: the carrier of (Tcircle (x,r)) = Sphere (x,r)
[#] (Tcircle (x,r)) = Sphere (x,r) by PRE_TOPC:def_5;
hence the carrier of (Tcircle (x,r)) = Sphere (x,r) ; ::_thesis: verum
end;
registration
let n be Nat;
let x be Point of (TOP-REAL n);
let r be empty real number ;
cluster Tcircle (x,r) -> trivial ;
coherence
Tcircle (x,r) is trivial
proof
reconsider e = x as Point of (Euclid n) by TOPREAL3:8;
A1: n in NAT by ORDINAL1:def_12;
then the carrier of (Tcircle (x,r)) = Sphere (x,r) by Th9
.= Sphere (e,r) by A1, TOPREAL9:15
.= {e} by TOPREAL6:54 ;
hence Tcircle (x,r) is trivial ; ::_thesis: verum
end;
end;
theorem Th10: :: TOPREALB:10
for r being real number holds Tcircle ((0. (TOP-REAL 2)),r) is SubSpace of Trectangle ((- r),r,(- r),r)
proof
let r be real number ; ::_thesis: Tcircle ((0. (TOP-REAL 2)),r) is SubSpace of Trectangle ((- r),r,(- r),r)
set C = Tcircle ((0. (TOP-REAL 2)),r);
set T = Trectangle ((- r),r,(- r),r);
the carrier of (Tcircle ((0. (TOP-REAL 2)),r)) c= the carrier of (Trectangle ((- r),r,(- r),r))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (Tcircle ((0. (TOP-REAL 2)),r)) or x in the carrier of (Trectangle ((- r),r,(- r),r)) )
A1: closed_inside_of_rectangle ((- r),r,(- r),r) = { p where p is Point of (TOP-REAL 2) : ( - r <= p `1 & p `1 <= r & - r <= p `2 & p `2 <= r ) } by JGRAPH_6:def_2;
assume A2: x in the carrier of (Tcircle ((0. (TOP-REAL 2)),r)) ; ::_thesis: x in the carrier of (Trectangle ((- r),r,(- r),r))
reconsider x = x as Point of (TOP-REAL 2) by A2, PRE_TOPC:25;
the carrier of (Tcircle ((0. (TOP-REAL 2)),r)) = Sphere ((0. (TOP-REAL 2)),r) by Th9;
then A3: |.x.| = r by A2, TOPREAL9:12;
A4: abs (x `2) <= |.x.| by JGRAPH_1:33;
then A5: - r <= x `2 by A3, ABSVALUE:5;
A6: abs (x `1) <= |.x.| by JGRAPH_1:33;
then A7: x `1 <= r by A3, ABSVALUE:5;
A8: the carrier of (Trectangle ((- r),r,(- r),r)) = closed_inside_of_rectangle ((- r),r,(- r),r) by PRE_TOPC:8;
A9: x `2 <= r by A3, A4, ABSVALUE:5;
- r <= x `1 by A3, A6, ABSVALUE:5;
hence x in the carrier of (Trectangle ((- r),r,(- r),r)) by A1, A8, A7, A5, A9; ::_thesis: verum
end;
hence Tcircle ((0. (TOP-REAL 2)),r) is SubSpace of Trectangle ((- r),r,(- r),r) by TSEP_1:4; ::_thesis: verum
end;
registration
let x be Point of (TOP-REAL 2);
let r be real positive number ;
cluster Tcircle (x,r) -> being_simple_closed_curve ;
coherence
Tcircle (x,r) is being_simple_closed_curve
proof
thus the carrier of (Tcircle (x,r)) is Simple_closed_curve by Th9; :: according to TOPREALB:def_5 ::_thesis: verum
end;
end;
registration
cluster strict TopSpace-like V221() V222() 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
set x = the Point of (TOP-REAL 2);
set r = the real positive number ;
take Tcircle ( the Point of (TOP-REAL 2), the real positive number ) ; ::_thesis: ( Tcircle ( the Point of (TOP-REAL 2), the real positive number ) is being_simple_closed_curve & Tcircle ( the Point of (TOP-REAL 2), the real positive number ) is strict )
thus ( Tcircle ( the Point of (TOP-REAL 2), the real positive number ) is being_simple_closed_curve & Tcircle ( the Point of (TOP-REAL 2), the real positive number ) is strict ) ; ::_thesis: verum
end;
end;
theorem :: TOPREALB:11
for S, T being being_simple_closed_curve SubSpace of TOP-REAL 2 holds S,T are_homeomorphic
proof
let S, T be being_simple_closed_curve SubSpace of TOP-REAL 2; ::_thesis: S,T are_homeomorphic
TopStruct(# the carrier of S, the topology of S #), TopStruct(# the carrier of T, the topology of T #) are_homeomorphic
proof
reconsider A = the carrier of TopStruct(# the carrier of S, the topology of S #) as Simple_closed_curve by Def5;
consider f being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | A) such that
A1: f is being_homeomorphism by TOPREAL2:def_1;
A2: f " is being_homeomorphism by A1, TOPS_2:56;
A3: [#] TopStruct(# the carrier of S, the topology of S #) = A ;
TopStruct(# the carrier of S, the topology of S #) is strict SubSpace of TOP-REAL 2 by TMAP_1:6;
then A4: TopStruct(# the carrier of S, the topology of S #) = (TOP-REAL 2) | A by A3, PRE_TOPC:def_5;
reconsider B = the carrier of TopStruct(# the carrier of T, the topology of T #) as Simple_closed_curve by Def5;
consider g being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | B) such that
A5: g is being_homeomorphism by TOPREAL2:def_1;
A6: [#] TopStruct(# the carrier of T, the topology of T #) = B ;
A7: TopStruct(# the carrier of T, the topology of T #) is strict SubSpace of TOP-REAL 2 by TMAP_1:6;
then reconsider h = g * (f ") as Function of TopStruct(# the carrier of S, the topology of S #),TopStruct(# the carrier of T, the topology of T #) by A4, A6, PRE_TOPC:def_5;
take h ; :: according to T_0TOPSP:def_1 ::_thesis: h is being_homeomorphism
TopStruct(# the carrier of T, the topology of T #) = (TOP-REAL 2) | B by A7, A6, PRE_TOPC:def_5;
hence h is being_homeomorphism by A5, A4, A2, TOPS_2:57; ::_thesis: verum
end;
hence S,T are_homeomorphic by TOPREALA:15; ::_thesis: verum
end;
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 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 (TOP-REAL n) st x is Point of (Tunit_circle n) holds
|.x.| = 1
proof
reconsider j = 1 as real non negative number ;
let n be non empty Element of NAT ; ::_thesis: for x being Point of (TOP-REAL n) st x is Point of (Tunit_circle n) holds
|.x.| = 1
let x be Point of (TOP-REAL n); ::_thesis: ( x is Point of (Tunit_circle n) implies |.x.| = 1 )
assume x is Point of (Tunit_circle n) ; ::_thesis: |.x.| = 1
then x in the carrier of (Tcircle ((0. (TOP-REAL n)),j)) ;
then x in Sphere ((0. (TOP-REAL n)),1) by Th9;
hence |.x.| = 1 by TOPREAL9:12; ::_thesis: verum
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
let x be Point of (TOP-REAL 2); ::_thesis: ( x is Point of (Tunit_circle 2) implies ( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 ) )
assume A1: x is Point of (Tunit_circle 2) ; ::_thesis: ( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 )
consider a, b being Real such that
A2: x = <*a,b*> by EUCLID:51;
A3: b = x `2 by A2, EUCLID:52;
A4: a = x `1 by A2, EUCLID:52;
A5: 1 ^2 = |.x.| ^2 by A1, Th12
.= (a ^2) + (b ^2) by A4, A3, JGRAPH_3:1 ;
0 <= a ^2 by XREAL_1:63;
then - (a ^2) <= - 0 ;
then A6: (b ^2) - 1 <= 0 by A5;
0 <= b ^2 by XREAL_1:63;
then - (b ^2) <= - 0 ;
then (a ^2) - 1 <= 0 by A5;
hence ( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 ) by A4, A3, A6, SQUARE_1:43; ::_thesis: verum
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
let x be Point of (TOP-REAL 2); ::_thesis: ( x is Point of (Tunit_circle 2) & x `1 = 1 implies x `2 = 0 )
assume that
A1: x is Point of (Tunit_circle 2) and
A2: x `1 = 1 ; ::_thesis: x `2 = 0
1 ^2 = |.x.| ^2 by A1, Th12
.= ((x `1) ^2) + ((x `2) ^2) by JGRAPH_3:1 ;
hence x `2 = 0 by A2; ::_thesis: verum
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
let x be Point of (TOP-REAL 2); ::_thesis: ( x is Point of (Tunit_circle 2) & x `1 = - 1 implies x `2 = 0 )
assume that
A1: x is Point of (Tunit_circle 2) and
A2: x `1 = - 1 ; ::_thesis: x `2 = 0
1 ^2 = |.x.| ^2 by A1, Th12
.= ((x `1) ^2) + ((x `2) ^2) by JGRAPH_3:1 ;
hence x `2 = 0 by A2; ::_thesis: verum
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
let x be Point of (TOP-REAL 2); ::_thesis: ( x is Point of (Tunit_circle 2) & x `2 = 1 implies x `1 = 0 )
assume that
A1: x is Point of (Tunit_circle 2) and
A2: x `2 = 1 ; ::_thesis: x `1 = 0
1 ^2 = |.x.| ^2 by A1, Th12
.= ((x `1) ^2) + ((x `2) ^2) by JGRAPH_3:1 ;
hence x `1 = 0 by A2; ::_thesis: verum
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
let x be Point of (TOP-REAL 2); ::_thesis: ( x is Point of (Tunit_circle 2) & x `2 = - 1 implies x `1 = 0 )
assume that
A1: x is Point of (Tunit_circle 2) and
A2: x `2 = - 1 ; ::_thesis: x `1 = 0
1 ^2 = |.x.| ^2 by A1, Th12
.= ((x `1) ^2) + ((x `2) ^2) by JGRAPH_3:1 ;
hence x `1 = 0 by A2; ::_thesis: verum
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 (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
let n be non empty Element of NAT ; ::_thesis: for r being real positive number
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
let r be real positive number ; ::_thesis: 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
let x be Point of (TOP-REAL n); ::_thesis: 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
let f be Function of (Tunit_circle n),(Tcircle (x,r)); ::_thesis: ( ( 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 ) implies f is being_homeomorphism )
assume A1: 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 ; ::_thesis: f is being_homeomorphism
defpred S1[ Point of (TOP-REAL n), set ] means $2 = (r * $1) + x;
set U = Tunit_circle n;
set C = Tcircle (x,r);
A2: for u being Point of (TOP-REAL n) ex y being Point of (TOP-REAL n) st S1[u,y] ;
consider F being Function of (TOP-REAL n),(TOP-REAL n) such that
A3: for x being Point of (TOP-REAL n) holds S1[x,F . x] from FUNCT_2:sch_3(A2);
defpred S2[ Point of (TOP-REAL n), set ] means $2 = (1 / r) * ($1 - x);
A4: for u being Point of (TOP-REAL n) ex y being Point of (TOP-REAL n) st S2[u,y] ;
consider G being Function of (TOP-REAL n),(TOP-REAL n) such that
A5: for a being Point of (TOP-REAL n) holds S2[a,G . a] from FUNCT_2:sch_3(A4);
set f2 = (TOP-REAL n) --> x;
set f1 = id (TOP-REAL n);
dom G = the carrier of (TOP-REAL n) by FUNCT_2:def_1;
then A6: dom (G | (Sphere (x,r))) = Sphere (x,r) by RELAT_1:62;
for p being Point of (TOP-REAL n) holds G . p = ((1 / r) * ((id (TOP-REAL n)) . p)) + ((- (1 / r)) * (((TOP-REAL n) --> x) . p))
proof
let p be Point of (TOP-REAL n); ::_thesis: G . p = ((1 / r) * ((id (TOP-REAL n)) . p)) + ((- (1 / r)) * (((TOP-REAL n) --> x) . p))
thus ((1 / r) * ((id (TOP-REAL n)) . p)) + ((- (1 / r)) * (((TOP-REAL n) --> x) . p)) = ((1 / r) * p) + ((- (1 / r)) * (((TOP-REAL n) --> x) . p)) by FUNCT_1:18
.= ((1 / r) * p) + ((- (1 / r)) * x) by FUNCOP_1:7
.= ((1 / r) * p) - ((1 / r) * x) by EUCLID:40
.= (1 / r) * (p - x) by EUCLID:49
.= G . p by A5 ; ::_thesis: verum
end;
then A7: G is continuous by TOPALG_1:16;
thus dom f = [#] (Tunit_circle n) by FUNCT_2:def_1; :: according to TOPS_2:def_5 ::_thesis: ( rng f = [#] (Tcircle (x,r)) & f is one-to-one & f is continuous & f /" is continuous )
A8: dom f = the carrier of (Tunit_circle n) by FUNCT_2:def_1;
for p being Point of (TOP-REAL n) holds F . p = (r * ((id (TOP-REAL n)) . p)) + (1 * (((TOP-REAL n) --> x) . p))
proof
let p be Point of (TOP-REAL n); ::_thesis: F . p = (r * ((id (TOP-REAL n)) . p)) + (1 * (((TOP-REAL n) --> x) . p))
thus (r * ((id (TOP-REAL n)) . p)) + (1 * (((TOP-REAL n) --> x) . p)) = (r * ((id (TOP-REAL n)) . p)) + (((TOP-REAL n) --> x) . p) by EUCLID:29
.= (r * p) + (((TOP-REAL n) --> x) . p) by FUNCT_1:18
.= (r * p) + x by FUNCOP_1:7
.= F . p by A3 ; ::_thesis: verum
end;
then A9: F is continuous by TOPALG_1:16;
A10: the carrier of (Tcircle (x,r)) = Sphere (x,r) by Th9;
A11: the carrier of (Tunit_circle n) = Sphere ((0. (TOP-REAL n)),1) by Th9;
A12: for a being set st a in dom f holds
f . a = (F | (Sphere ((0. (TOP-REAL n)),1))) . a
proof
let a be set ; ::_thesis: ( a in dom f implies f . a = (F | (Sphere ((0. (TOP-REAL n)),1))) . a )
assume A13: a in dom f ; ::_thesis: f . a = (F | (Sphere ((0. (TOP-REAL n)),1))) . a
reconsider y = a as Point of (TOP-REAL n) by A13, PRE_TOPC:25;
thus f . a = (r * y) + x by A1, A13
.= F . y by A3
.= (F | (Sphere ((0. (TOP-REAL n)),1))) . a by A11, A13, FUNCT_1:49 ; ::_thesis: verum
end;
A14: (1 / r) * r = 1 by XCMPLX_1:87;
thus A15: rng f = [#] (Tcircle (x,r)) ::_thesis: ( f is one-to-one & f is continuous & f /" is continuous )
proof
thus rng f c= [#] (Tcircle (x,r)) ; :: according to XBOOLE_0:def_10 ::_thesis: [#] (Tcircle (x,r)) c= rng f
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in [#] (Tcircle (x,r)) or b in rng f )
assume A16: b in [#] (Tcircle (x,r)) ; ::_thesis: b in rng f
then reconsider c = b as Point of (TOP-REAL n) by PRE_TOPC:25;
set a = (1 / r) * (c - x);
|.(((1 / r) * (c - x)) - (0. (TOP-REAL n))).| = |.((1 / r) * (c - x)).| by RLVECT_1:13
.= (abs (1 / r)) * |.(c - x).| by TOPRNS_1:7
.= (1 / r) * |.(c - x).| by ABSVALUE:def_1
.= (1 / r) * r by A10, A16, TOPREAL9:9 ;
then A17: (1 / r) * (c - x) in Sphere ((0. (TOP-REAL n)),1) by A14;
then f . ((1 / r) * (c - x)) = (r * ((1 / r) * (c - x))) + x by A1, A11
.= ((r * (1 / r)) * (c - x)) + x by EUCLID:30
.= (c - x) + x by A14, EUCLID:29
.= b by EUCLID:48 ;
hence b in rng f by A11, A8, A17, FUNCT_1:def_3; ::_thesis: verum
end;
thus A18: f is one-to-one ::_thesis: ( f is continuous & f /" is continuous )
proof
let a, b be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not a in K121(f) or not b in K121(f) or not f . a = f . b or a = b )
assume that
A19: a in dom f and
A20: b in dom f and
A21: f . a = f . b ; ::_thesis: a = b
reconsider a1 = a, b1 = b as Point of (TOP-REAL n) by A11, A8, A19, A20;
A22: f . b1 = (r * b1) + x by A1, A20;
f . a1 = (r * a1) + x by A1, A19;
then r * a1 = ((r * b1) + x) - x by A21, A22, EUCLID:48;
hence a = b by EUCLID:34, EUCLID:48; ::_thesis: verum
end;
A23: for a being set st a in dom (f ") holds
(f ") . a = (G | (Sphere (x,r))) . a
proof
reconsider ff = f as Function ;
let a be set ; ::_thesis: ( a in dom (f ") implies (f ") . a = (G | (Sphere (x,r))) . a )
assume A24: a in dom (f ") ; ::_thesis: (f ") . a = (G | (Sphere (x,r))) . a
reconsider y = a as Point of (TOP-REAL n) by A24, PRE_TOPC:25;
set e = (1 / r) * (y - x);
A25: f is onto by A15, FUNCT_2:def_3;
|.(((1 / r) * (y - x)) - (0. (TOP-REAL n))).| = |.((1 / r) * (y - x)).| by RLVECT_1:13
.= (abs (1 / r)) * |.(y - x).| by TOPRNS_1:7
.= (1 / r) * |.(y - x).| by ABSVALUE:def_1
.= (1 / r) * r by A10, A24, TOPREAL9:9 ;
then A26: (1 / r) * (y - x) in Sphere ((0. (TOP-REAL n)),1) by A14;
then f . ((1 / r) * (y - x)) = (r * ((1 / r) * (y - x))) + x by A1, A11
.= ((r * (1 / r)) * (y - x)) + x by EUCLID:30
.= (y - x) + x by A14, EUCLID:29
.= y by EUCLID:48 ;
then (ff ") . a = (1 / r) * (y - x) by A11, A8, A18, A26, FUNCT_1:32;
hence (f ") . a = (1 / r) * (y - x) by A25, A18, TOPS_2:def_4
.= G . y by A5
.= (G | (Sphere (x,r))) . a by A10, A24, FUNCT_1:49 ;
::_thesis: verum
end;
dom F = the carrier of (TOP-REAL n) by FUNCT_2:def_1;
then dom (F | (Sphere ((0. (TOP-REAL n)),1))) = Sphere ((0. (TOP-REAL n)),1) by RELAT_1:62;
hence f is continuous by A11, A8, A9, A12, BORSUK_4:44, FUNCT_1:2; ::_thesis: f /" is continuous
dom (f ") = the carrier of (Tcircle (x,r)) by FUNCT_2:def_1;
hence f /" is continuous by A10, A6, A7, A23, BORSUK_4:44, FUNCT_1:2; ::_thesis: verum
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 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
proof
let n be non empty Element of NAT ; ::_thesis: for r being real positive number
for x being Point of (TOP-REAL n) holds Tunit_circle n, Tcircle (x,r) are_homeomorphic
let r be real positive number ; ::_thesis: for x being Point of (TOP-REAL n) holds Tunit_circle n, Tcircle (x,r) are_homeomorphic
let x be Point of (TOP-REAL n); ::_thesis: Tunit_circle n, Tcircle (x,r) are_homeomorphic
set U = Tunit_circle n;
set C = Tcircle (x,r);
defpred S1[ Point of (Tunit_circle n), set ] means ex w being Point of (TOP-REAL n) st
( w = $1 & $2 = (r * w) + x );
A1: r is Real by XREAL_0:def_1;
A2: the carrier of (Tcircle (x,r)) = Sphere (x,r) by Th9;
A3: for u being Point of (Tunit_circle n) ex y being Point of (Tcircle (x,r)) st S1[u,y]
proof
let u be Point of (Tunit_circle n); ::_thesis: ex y being Point of (Tcircle (x,r)) st S1[u,y]
reconsider v = u as Point of (TOP-REAL n) by PRE_TOPC:25;
set y = (r * v) + x;
|.(((r * v) + x) - x).| = |.(r * v).| by EUCLID:48
.= (abs r) * |.v.| by A1, TOPRNS_1:7
.= r * |.v.| by ABSVALUE:def_1
.= r * 1 by Th12 ;
then reconsider y = (r * v) + x as Point of (Tcircle (x,r)) by A2, TOPREAL9:9;
take y ; ::_thesis: S1[u,y]
thus S1[u,y] ; ::_thesis: verum
end;
consider f being Function of (Tunit_circle n),(Tcircle (x,r)) such that
A4: for x being Point of (Tunit_circle n) holds S1[x,f . x] from FUNCT_2:sch_3(A3);
take f ; :: according to T_0TOPSP:def_1 ::_thesis: f is being_homeomorphism
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
proof
let a be Point of (Tunit_circle n); ::_thesis: for b being Point of (TOP-REAL n) st a = b holds
f . a = (r * b) + x
let b be Point of (TOP-REAL n); ::_thesis: ( a = b implies f . a = (r * b) + x )
S1[a,f . a] by A4;
hence ( a = b implies f . a = (r * b) + x ) ; ::_thesis: verum
end;
hence f is being_homeomorphism by Th19; ::_thesis: verum
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 (TOP-REAL n) holds Tcircle (x,r), Tcircle (y,s) are_homeomorphic
proof
let n be non empty Element of NAT ; ::_thesis: for r, s being real positive number
for x, y being Point of (TOP-REAL n) holds Tcircle (x,r), Tcircle (y,s) are_homeomorphic
let r, s be real positive number ; ::_thesis: for x, y being Point of (TOP-REAL n) holds Tcircle (x,r), Tcircle (y,s) are_homeomorphic
let x, y be Point of (TOP-REAL n); ::_thesis: Tcircle (x,r), Tcircle (y,s) are_homeomorphic
A1: Tunit_circle n, Tcircle (y,s) are_homeomorphic by Lm14;
Tcircle (x,r), Tunit_circle n are_homeomorphic by Lm14;
hence Tcircle (x,r), Tcircle (y,s) are_homeomorphic by A1, BORSUK_3:3; ::_thesis: verum
end;
registration
let x be Point of (TOP-REAL 2);
let r be real non negative number ;
cluster Tcircle (x,r) -> pathwise_connected ;
coherence
Tcircle (x,r) is pathwise_connected
proof
percases ( r is positive or not r is positive ) ;
suppose r is positive ; ::_thesis: Tcircle (x,r) is pathwise_connected
then reconsider r = r as real positive number ;
Tcircle (x,r) is pathwise_connected ;
hence Tcircle (x,r) is pathwise_connected ; ::_thesis: verum
end;
suppose not r is positive ; ::_thesis: Tcircle (x,r) is pathwise_connected
then reconsider r = r as real non positive non negative number ;
Tcircle (x,r) is trivial ;
hence Tcircle (x,r) is pathwise_connected ; ::_thesis: verum
end;
end;
end;
end;
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
A1: |[1,0]| `2 = 0 by EUCLID:52;
A2: |[1,0]| `1 = 1 by EUCLID:52;
|.(|[(1 + 0),(0 + 0)]| - |[0,0]|).| = |.((|[1,0]| + |[0,0]|) - |[0,0]|).| by EUCLID:56
.= |.(|[1,0]| + (|[0,0]| - |[0,0]|)).| by EUCLID:45
.= |.(|[1,0]| + (0. (TOP-REAL 2))).| by EUCLID:42
.= |.|[1,0]|.| by EUCLID:27
.= sqrt ((1 ^2) + (0 ^2)) by A2, A1, JGRAPH_1:30
.= 1 by SQUARE_1:22 ;
hence |[1,0]| is Point of (Tunit_circle 2) by Lm13, TOPREAL9:9; ::_thesis: verum
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
A3: |[(- 1),0]| `2 = 0 by EUCLID:52;
A4: |[(- 1),0]| `1 = - 1 by EUCLID:52;
|.(|[((- 1) + 0),(0 + 0)]| - |[0,0]|).| = |.((|[(- 1),0]| + |[0,0]|) - |[0,0]|).| by EUCLID:56
.= |.(|[(- 1),0]| + (|[0,0]| - |[0,0]|)).| by EUCLID:45
.= |.(|[(- 1),0]| + (0. (TOP-REAL 2))).| by EUCLID:42
.= |.|[(- 1),0]|.| by EUCLID:27
.= sqrt (((- 1) ^2) + (0 ^2)) by A4, A3, JGRAPH_1:30
.= sqrt ((1 ^2) + (0 ^2))
.= 1 by SQUARE_1:22 ;
hence |[(- 1),0]| is Point of (Tunit_circle 2) by Lm13, TOPREAL9:9; ::_thesis: verum
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
reconsider A = the carrier of (Tunit_circle 2) \ {p} as Subset of (Tunit_circle 2) ;
take (Tunit_circle 2) | A ; ::_thesis: the carrier of ((Tunit_circle 2) | A) = the carrier of (Tunit_circle 2) \ {p}
thus the carrier of ((Tunit_circle 2) | A) = the carrier of (Tunit_circle 2) \ {p} by PRE_TOPC:8; ::_thesis: verum
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
set X = Topen_unit_circle p;
A1: the carrier of (Topen_unit_circle p) = the carrier of (Tunit_circle 2) \ {p} by Def10;
percases ( p = c[10] or p <> c[10] ) ;
supposeA2: p = c[10] ; ::_thesis: not Topen_unit_circle p is empty
set x = |[0,1]|;
reconsider r = p as Point of (TOP-REAL 2) by PRE_TOPC:25;
A3: |[0,1]| `1 = 0 by EUCLID:52;
A4: |[0,1]| `2 = 1 by EUCLID:52;
|.(|[(0 + 0),(1 + 0)]| - |[0,0]|).| = |.((|[0,1]| + |[0,0]|) - |[0,0]|).| by EUCLID:56
.= |.(|[0,1]| + (|[0,0]| - |[0,0]|)).| by EUCLID:45
.= |.(|[0,1]| + (0. (TOP-REAL 2))).| by EUCLID:42
.= |.|[0,1]|.| by EUCLID:27
.= sqrt ((1 ^2) + (0 ^2)) by A3, A4, JGRAPH_1:30
.= 1 by SQUARE_1:22 ;
then A5: |[0,1]| in the carrier of (Tunit_circle 2) by Lm13;
r `1 = 1 by A2, EUCLID:52;
then not |[0,1]| in {p} by A3, TARSKI:def_1;
hence not the carrier of (Topen_unit_circle p) is empty by A1, A5, XBOOLE_0:def_5; :: according to STRUCT_0:def_1 ::_thesis: verum
end;
suppose p <> c[10] ; ::_thesis: not Topen_unit_circle p is empty
then not c[10] in {p} by TARSKI:def_1;
hence not the carrier of (Topen_unit_circle p) is empty by A1, XBOOLE_0:def_5; :: according to STRUCT_0:def_1 ::_thesis: verum
end;
end;
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
let p be Point of (Tunit_circle 2); ::_thesis: p is not Point of (Topen_unit_circle p)
A1: p in {p} by TARSKI:def_1;
the carrier of (Topen_unit_circle p) = the carrier of (Tunit_circle 2) \ {p} by Def10;
hence p is not Point of (Topen_unit_circle p) by A1, XBOOLE_0:def_5; ::_thesis: verum
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
let p be Point of (Tunit_circle 2); ::_thesis: Topen_unit_circle p = (Tunit_circle 2) | (([#] (Tunit_circle 2)) \ {p})
[#] (Topen_unit_circle p) = ([#] (Tunit_circle 2)) \ {p} by Def10;
hence Topen_unit_circle p = (Tunit_circle 2) | (([#] (Tunit_circle 2)) \ {p}) by PRE_TOPC:def_5; ::_thesis: verum
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
let p, q be Point of (Tunit_circle 2); ::_thesis: ( p <> q implies q is Point of (Topen_unit_circle p) )
assume A1: p <> q ; ::_thesis: q is Point of (Topen_unit_circle p)
the carrier of (Topen_unit_circle p) = the carrier of (Tunit_circle 2) \ {p} by Def10;
hence q is Point of (Topen_unit_circle p) by A1, ZFMISC_1:56; ::_thesis: verum
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
let p be Point of (TOP-REAL 2); ::_thesis: ( p is Point of (Topen_unit_circle c[10]) & p `2 = 0 implies p = c[-10] )
assume that
A1: p is Point of (Topen_unit_circle c[10]) and
A2: p `2 = 0 ; ::_thesis: p = c[-10]
A3: now__::_thesis:_not_p_`1_=_1
assume p `1 = 1 ; ::_thesis: contradiction
then p = c[10] by A2, EUCLID:53;
hence contradiction by A1, Th21; ::_thesis: verum
end;
p is Point of (Tunit_circle 2) by A1, PRE_TOPC:25;
then 1 ^2 = |.p.| ^2 by Th12
.= ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1 ;
then ( p `1 = 1 or p `1 = - 1 ) by A2, SQUARE_1:41;
hence p = c[-10] by A2, A3, EUCLID:53; ::_thesis: verum
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
let p be Point of (TOP-REAL 2); ::_thesis: ( p is Point of (Topen_unit_circle c[-10]) & p `2 = 0 implies p = c[10] )
assume that
A1: p is Point of (Topen_unit_circle c[-10]) and
A2: p `2 = 0 ; ::_thesis: p = c[10]
A3: now__::_thesis:_not_p_`1_=_-_1
assume p `1 = - 1 ; ::_thesis: contradiction
then p = c[-10] by A2, EUCLID:53;
hence contradiction by A1, Th21; ::_thesis: verum
end;
p is Point of (Tunit_circle 2) by A1, PRE_TOPC:25;
then 1 ^2 = |.p.| ^2 by Th12
.= ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1 ;
then ( p `1 = 1 or p `1 = - 1 ) by A2, SQUARE_1:41;
hence p = c[10] by A2, A3, EUCLID:53; ::_thesis: verum
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
let p be Point of (Tunit_circle 2); ::_thesis: 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 )
let x be Point of (TOP-REAL 2); ::_thesis: ( x is Point of (Topen_unit_circle p) implies ( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 ) )
assume x is Point of (Topen_unit_circle p) ; ::_thesis: ( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 )
then A1: x in the carrier of (Topen_unit_circle p) ;
the carrier of (Topen_unit_circle p) is Subset of the carrier of (Tunit_circle 2) by TSEP_1:1;
hence ( - 1 <= x `1 & x `1 <= 1 & - 1 <= x `2 & x `2 <= 1 ) by A1, Th13; ::_thesis: verum
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
let x be Point of (TOP-REAL 2); ::_thesis: ( x is Point of (Topen_unit_circle c[10]) implies ( - 1 <= x `1 & x `1 < 1 ) )
assume A1: x is Point of (Topen_unit_circle c[10]) ; ::_thesis: ( - 1 <= x `1 & x `1 < 1 )
A2: now__::_thesis:_not_x_`1_=_1
A3: the carrier of (Topen_unit_circle c[10]) = the carrier of (Tunit_circle 2) \ {c[10]} by Def10;
then A4: not x in {c[10]} by A1, XBOOLE_0:def_5;
A5: x = |[(x `1),(x `2)]| by EUCLID:53;
assume A6: x `1 = 1 ; ::_thesis: contradiction
x in the carrier of (Tunit_circle 2) by A1, A3, XBOOLE_0:def_5;
then x = c[10] by A6, A5, Th14;
hence contradiction by A4, TARSKI:def_1; ::_thesis: verum
end;
x `1 <= 1 by A1, Th26;
hence ( - 1 <= x `1 & x `1 < 1 ) by A1, A2, Th26, XXREAL_0:1; ::_thesis: verum
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
let x be Point of (TOP-REAL 2); ::_thesis: ( x is Point of (Topen_unit_circle c[-10]) implies ( - 1 < x `1 & x `1 <= 1 ) )
assume A1: x is Point of (Topen_unit_circle c[-10]) ; ::_thesis: ( - 1 < x `1 & x `1 <= 1 )
A2: now__::_thesis:_not_x_`1_=_-_1
A3: the carrier of (Topen_unit_circle c[-10]) = the carrier of (Tunit_circle 2) \ {c[-10]} by Def10;
then A4: not x in {c[-10]} by A1, XBOOLE_0:def_5;
A5: x = |[(x `1),(x `2)]| by EUCLID:53;
assume A6: x `1 = - 1 ; ::_thesis: contradiction
x in the carrier of (Tunit_circle 2) by A1, A3, XBOOLE_0:def_5;
then x = c[-10] by A6, A5, Th15;
hence contradiction by A4, TARSKI:def_1; ::_thesis: verum
end;
- 1 <= x `1 by A1, Th26;
hence ( - 1 < x `1 & x `1 <= 1 ) by A1, A2, Th26, XXREAL_0:1; ::_thesis: verum
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
let A be Subset of (Tunit_circle 2); :: according to TSEP_1:def_1 ::_thesis: ( not A = the carrier of (Topen_unit_circle p) or A is open )
assume A = the carrier of (Topen_unit_circle p) ; ::_thesis: A is open
then A1: A ` = the carrier of (Tunit_circle 2) \ ( the carrier of (Tunit_circle 2) \ {p}) by Def10
.= the carrier of (Tunit_circle 2) /\ {p} by XBOOLE_1:48
.= {p} by ZFMISC_1:46 ;
Tunit_circle 2 is T_2 by TOPMETR:2;
then A ` is closed by A1, PCOMPS_1:7;
hence A is open by TOPS_1:4; ::_thesis: verum
end;
end;
theorem :: TOPREALB:29
for p being Point of (Tunit_circle 2) holds Topen_unit_circle p, I(01) are_homeomorphic
proof
set D = Sphere ((0. (TOP-REAL 2)),p1);
let p be Point of (Tunit_circle 2); ::_thesis: Topen_unit_circle p, I(01) are_homeomorphic
set P = Topen_unit_circle p;
reconsider p2 = p as Point of (TOP-REAL 2) by PRE_TOPC:25;
(Sphere ((0. (TOP-REAL 2)),p1)) \ {p} c= Sphere ((0. (TOP-REAL 2)),p1) by XBOOLE_1:36;
then reconsider A = (Sphere ((0. (TOP-REAL 2)),p1)) \ {p} as Subset of (Tcircle ((0. (TOP-REAL 2)),1)) by Th9;
Topen_unit_circle p = (Tcircle ((0. (TOP-REAL 2)),1)) | A by Lm13, Th22, EUCLID:54
.= (TOP-REAL 2) | ((Sphere ((0. (TOP-REAL 2)),p1)) \ {p2}) by GOBOARD9:2 ;
hence Topen_unit_circle p, I(01) are_homeomorphic by Lm13, BORSUK_4:52, EUCLID:54; ::_thesis: verum
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
set D = Sphere ((0. (TOP-REAL 2)),p1);
let p, q be Point of (Tunit_circle 2); ::_thesis: Topen_unit_circle p, Topen_unit_circle q are_homeomorphic
set P = Topen_unit_circle p;
set Q = Topen_unit_circle q;
reconsider p2 = p, q2 = q as Point of (TOP-REAL 2) by PRE_TOPC:25;
A1: (Sphere ((0. (TOP-REAL 2)),p1)) \ {q} c= Sphere ((0. (TOP-REAL 2)),p1) by XBOOLE_1:36;
(Sphere ((0. (TOP-REAL 2)),p1)) \ {p} c= Sphere ((0. (TOP-REAL 2)),p1) by XBOOLE_1:36;
then reconsider A = (Sphere ((0. (TOP-REAL 2)),p1)) \ {p}, B = (Sphere ((0. (TOP-REAL 2)),p1)) \ {q} as Subset of (Tcircle ((0. (TOP-REAL 2)),1)) by A1, Th9;
A2: Topen_unit_circle q = (Tcircle ((0. (TOP-REAL 2)),1)) | B by Lm13, Th22, EUCLID:54
.= (TOP-REAL 2) | ((Sphere ((0. (TOP-REAL 2)),p1)) \ {q2}) by GOBOARD9:2 ;
Topen_unit_circle p = (Tcircle ((0. (TOP-REAL 2)),1)) | A by Lm13, Th22, EUCLID:54
.= (TOP-REAL 2) | ((Sphere ((0. (TOP-REAL 2)),p1)) \ {p2}) by GOBOARD9:2 ;
hence Topen_unit_circle p, Topen_unit_circle q are_homeomorphic by A2, Lm13, BORSUK_4:53, EUCLID:54; ::_thesis: verum
end;
begin
definition
func CircleMap -> Function of R^1,(Tunit_circle 2) 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,(Tunit_circle 2) st
for x being real number holds b1 . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]|
proof
defpred S1[ real number , set ] means $2 = |[(cos ((2 * PI) * $1)),(sin ((2 * PI) * $1))]|;
A1: for x being Element of R^1 ex y being Element of the carrier of (Tunit_circle 2) st S1[x,y]
proof
let x be Element of R^1; ::_thesis: ex y being Element of the carrier of (Tunit_circle 2) st S1[x,y]
set y = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]|;
|.(|[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| - |[0,0]|).| = |.|[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]|.| by EUCLID:54, RLVECT_1:13
.= sqrt (((|[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| `1) ^2) + ((|[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| `2) ^2)) by JGRAPH_1:30
.= sqrt (((cos ((2 * PI) * x)) ^2) + ((|[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| `2) ^2)) by EUCLID:52
.= sqrt (((cos ((2 * PI) * x)) ^2) + ((sin ((2 * PI) * x)) ^2)) by EUCLID:52
.= 1 by SIN_COS:29, SQUARE_1:18 ;
then |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| is Element of the carrier of (Tunit_circle 2) by Lm13, TOPREAL9:9;
hence ex y being Element of the carrier of (Tunit_circle 2) st S1[x,y] ; ::_thesis: verum
end;
consider f being Function of the carrier of R^1, the carrier of (Tunit_circle 2) such that
A2: for x being Element of R^1 holds S1[x,f . x] from FUNCT_2:sch_3(A1);
take f ; ::_thesis: for x being real number holds f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]|
let x be real number ; ::_thesis: f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]|
x is Point of R^1 by TOPMETR:17, XREAL_0:def_1;
hence f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of R^1,(Tunit_circle 2) 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
let f, g be Function of R^1,(Tunit_circle 2); ::_thesis: ( ( for x being real number holds f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ) & ( for x being real number holds g . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ) implies f = g )
assume that
A3: for x being real number holds f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| and
A4: for x being real number holds g . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| ; ::_thesis: f = g
for x being Point of R^1 holds f . x = g . x
proof
let x be Point of R^1; ::_thesis: f . x = g . x
thus f . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| by A3
.= g . x by A4 ; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
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 number 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 number holds CircleMap . r = CircleMap . (r + i)
proof
let i be Integer; ::_thesis: for r being real number holds CircleMap . r = CircleMap . (r + i)
let r be real number ; ::_thesis: CircleMap . r = CircleMap . (r + i)
defpred S1[ Integer] means CircleMap . r = CircleMap . (r + $1);
A1: for i being Integer st S1[i] holds
( S1[i - 1] & S1[i + 1] )
proof
let i be Integer; ::_thesis: ( S1[i] implies ( S1[i - 1] & S1[i + 1] ) )
assume A2: S1[i] ; ::_thesis: ( S1[i - 1] & S1[i + 1] )
thus CircleMap . (r + (i - 1)) = |[(cos ((2 * PI) * ((r + i) - 1))),(sin ((2 * PI) * ((r + i) - 1)))]| by Def11
.= |[(cos ((2 * PI) * (r + i))),(sin (((2 * PI) * (r + i)) + ((2 * PI) * (- 1))))]| by COMPLEX2:9
.= |[(cos ((2 * PI) * (r + i))),(sin ((2 * PI) * (r + i)))]| by COMPLEX2:8
.= CircleMap . r by A2, Def11 ; ::_thesis: S1[i + 1]
thus CircleMap . (r + (i + 1)) = |[(cos ((2 * PI) * ((r + i) + 1))),(sin ((2 * PI) * ((r + i) + 1)))]| by Def11
.= |[(cos ((2 * PI) * (r + i))),(sin (((2 * PI) * (r + i)) + ((2 * PI) * 1)))]| by COMPLEX2:9
.= |[(cos ((2 * PI) * (r + i))),(sin ((2 * PI) * (r + i)))]| by COMPLEX2:8
.= CircleMap . r by A2, Def11 ; ::_thesis: verum
end;
A3: S1[ 0 ] ;
for i being Integer holds S1[i] from INT_1:sch_4(A3, A1);
hence CircleMap . r = CircleMap . (r + i) ; ::_thesis: verum
end;
theorem Th32: :: TOPREALB:32
for i being Integer holds CircleMap . i = c[10]
proof
let i be Integer; ::_thesis: CircleMap . i = c[10]
thus CircleMap . i = |[(cos (((2 * PI) * i) + 0)),(sin ((2 * PI) * i))]| by Def11
.= |[(cos 0),(sin (((2 * PI) * i) + 0))]| by COMPLEX2:9
.= c[10] by COMPLEX2:8, SIN_COS:31 ; ::_thesis: verum
end;
theorem Th33: :: TOPREALB:33
CircleMap " {c[10]} = INT
proof
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: INT c= CircleMap " {c[10]}
let i be set ; ::_thesis: ( i in CircleMap " {c[10]} implies i in INT )
assume A1: i in CircleMap " {c[10]} ; ::_thesis: i in INT
then reconsider x = i as Real by TOPMETR:17;
CircleMap . i in {c[10]} by A1, FUNCT_2:38;
then CircleMap . i = c[10] by TARSKI:def_1;
then |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| = |[1,0]| by Def11;
then cos ((2 * PI) * x) = 1 by SPPOL_2:1;
hence i in INT by SIN_COS6:44; ::_thesis: verum
end;
let i be set ; :: according to TARSKI:def_3 ::_thesis: ( not i in INT or i in CircleMap " {c[10]} )
assume i in INT ; ::_thesis: i in CircleMap " {c[10]}
then reconsider i = i as Integer ;
CircleMap . i = c[10] by Th32;
then A2: CircleMap . i in {c[10]} by TARSKI:def_1;
i in the carrier of R^1 by TOPMETR:17, XREAL_0:def_1;
hence i in CircleMap " {c[10]} by A2, FUNCT_2:38; ::_thesis: verum
end;
Lm19: CircleMap . (1 / 2) = |[(- 1),0]|
proof
thus CircleMap . (1 / 2) = |[(cos ((2 * PI) * (1 / 2))),(sin ((2 * PI) * (1 / 2)))]| by Def11
.= |[(- 1),0]| by SIN_COS:77 ; ::_thesis: verum
end;
theorem Th34: :: TOPREALB:34
for r being real number st frac r = 1 / 2 holds
CircleMap . r = |[(- 1),0]|
proof
let r be real number ; ::_thesis: ( frac r = 1 / 2 implies CircleMap . r = |[(- 1),0]| )
assume A1: frac r = 1 / 2 ; ::_thesis: CircleMap . r = |[(- 1),0]|
thus CircleMap . r = CircleMap . (r + (- [\r/])) by Th31
.= CircleMap . (r - [\r/])
.= |[(- 1),0]| by A1, Lm19, INT_1:def_8 ; ::_thesis: verum
end;
theorem :: TOPREALB:35
for r being real number st frac r = 1 / 4 holds
CircleMap . r = |[0,1]|
proof
let r be real number ; ::_thesis: ( frac r = 1 / 4 implies CircleMap . r = |[0,1]| )
assume frac r = 1 / 4 ; ::_thesis: CircleMap . r = |[0,1]|
then A1: r - [\r/] = 1 / 4 by INT_1:def_8;
thus CircleMap . r = CircleMap . (r + (- [\r/])) by Th31
.= |[(cos ((2 * PI) * (1 / 4))),(sin ((2 * PI) * (1 / 4)))]| by A1, Def11
.= |[0,1]| by SIN_COS:77 ; ::_thesis: verum
end;
theorem :: TOPREALB:36
for r being real number st frac r = 3 / 4 holds
CircleMap . r = |[0,(- 1)]|
proof
let r be real number ; ::_thesis: ( frac r = 3 / 4 implies CircleMap . r = |[0,(- 1)]| )
assume frac r = 3 / 4 ; ::_thesis: CircleMap . r = |[0,(- 1)]|
then A1: r - [\r/] = 3 / 4 by INT_1:def_8;
thus CircleMap . r = CircleMap . (r + (- [\r/])) by Th31
.= |[(cos ((2 * PI) * (3 / 4))),(sin ((2 * PI) * (3 / 4)))]| by A1, Def11
.= |[0,(- 1)]| by SIN_COS:77 ; ::_thesis: verum
end;
Lm20: for r being real number holds CircleMap . r = |[((cos * (AffineMap ((2 * PI),0))) . r),((sin * (AffineMap ((2 * PI),0))) . r)]|
proof
let r be real number ; ::_thesis: CircleMap . r = |[((cos * (AffineMap ((2 * PI),0))) . r),((sin * (AffineMap ((2 * PI),0))) . r)]|
thus CircleMap . r = |[(cos (((2 * PI) * r) + 0)),(sin ((2 * PI) * r))]| by Def11
.= |[((cos * (AffineMap ((2 * PI),0))) . r),(sin (((2 * PI) * r) + 0))]| by Th2
.= |[((cos * (AffineMap ((2 * PI),0))) . r),((sin * (AffineMap ((2 * PI),0))) . r)]| by Th1 ; ::_thesis: verum
end;
theorem :: TOPREALB:37
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
let r be real number ; ::_thesis: for i, j being Integer holds CircleMap . r = |[((cos * (AffineMap ((2 * PI),((2 * PI) * i)))) . r),((sin * (AffineMap ((2 * PI),((2 * PI) * j)))) . r)]|
let i, j be Integer; ::_thesis: CircleMap . r = |[((cos * (AffineMap ((2 * PI),((2 * PI) * i)))) . r),((sin * (AffineMap ((2 * PI),((2 * PI) * j)))) . r)]|
thus CircleMap . r = |[(cos (((2 * PI) * r) + 0)),(sin ((2 * PI) * r))]| by Def11
.= |[(cos (((2 * PI) * r) + ((2 * PI) * i))),(sin (((2 * PI) * r) + 0))]| by COMPLEX2:9
.= |[(cos (((2 * PI) * r) + ((2 * PI) * i))),(sin (((2 * PI) * r) + ((2 * PI) * j)))]| by COMPLEX2:8
.= |[((cos * (AffineMap ((2 * PI),((2 * PI) * i)))) . r),(sin (((2 * PI) * r) + ((2 * PI) * j)))]| by Th2
.= |[((cos * (AffineMap ((2 * PI),((2 * PI) * i)))) . r),((sin * (AffineMap ((2 * PI),((2 * PI) * j)))) . r)]| by Th1 ; ::_thesis: verum
end;
registration
cluster CircleMap -> continuous ;
coherence
CircleMap is continuous
proof
reconsider l = AffineMap ((2 * PI),0) as Function of R^1,R^1 by Th8;
set sR = R^1 sin;
set cR = R^1 cos;
A1: dom (AffineMap ((2 * PI),0)) = REAL by FUNCT_2:def_1;
reconsider sR = R^1 sin, cR = R^1 cos as Function of R^1,R^1 by Lm10, Lm11;
A2: AffineMap ((2 * PI),0) = R^1 (AffineMap ((2 * PI),0)) ;
reconsider g = CircleMap as Function of R^1,(TOP-REAL 2) by TOPREALA:7;
A3: rng (AffineMap ((2 * PI),0)) = [#] REAL by FCONT_1:55;
set c = cR * l;
set s = sR * l;
A4: R^1 | (R^1 (dom cos)) = R^1 by Th6, SIN_COS:24;
A5: R^1 | (R^1 (dom sin)) = R^1 by Th6, SIN_COS:24;
for p being Point of R^1
for V being Subset of (TOP-REAL 2) st g . p in V & V is open holds
ex W being Subset of R^1 st
( p in W & W is open & g .: W c= V )
proof
let p be Point of R^1; ::_thesis: for V being Subset of (TOP-REAL 2) st g . p in V & V is open holds
ex W being Subset of R^1 st
( p in W & W is open & g .: W c= V )
let V be Subset of (TOP-REAL 2); ::_thesis: ( g . p in V & V is open implies ex W being Subset of R^1 st
( p in W & W is open & g .: W c= V ) )
assume that
A6: g . p in V and
A7: V is open ; ::_thesis: ex W being Subset of R^1 st
( p in W & W is open & g .: W c= V )
reconsider e = g . p as Point of (Euclid 2) by TOPREAL3:8;
V = Int V by A7, TOPS_1:23;
then consider r being real number such that
A8: r > 0 and
A9: Ball (e,r) c= V by A6, GOBOARD6:5;
set B = ].(((g . p) `2) - (r / (sqrt 2))),(((g . p) `2) + (r / (sqrt 2))).[;
set A = ].(((g . p) `1) - (r / (sqrt 2))),(((g . p) `1) + (r / (sqrt 2))).[;
set F = (1,2) --> (].(((g . p) `1) - (r / (sqrt 2))),(((g . p) `1) + (r / (sqrt 2))).[,].(((g . p) `2) - (r / (sqrt 2))),(((g . p) `2) + (r / (sqrt 2))).[);
reconsider A = ].(((g . p) `1) - (r / (sqrt 2))),(((g . p) `1) + (r / (sqrt 2))).[, B = ].(((g . p) `2) - (r / (sqrt 2))),(((g . p) `2) + (r / (sqrt 2))).[ as Subset of R^1 by TOPMETR:17;
A10: B is open by JORDAN6:35;
A11: product ((1,2) --> (].(((g . p) `1) - (r / (sqrt 2))),(((g . p) `1) + (r / (sqrt 2))).[,].(((g . p) `2) - (r / (sqrt 2))),(((g . p) `2) + (r / (sqrt 2))).[)) c= Ball (e,r) by TOPREAL6:41;
A12: cR is continuous by A4, PRE_TOPC:26;
A13: sR is continuous by A5, PRE_TOPC:26;
A14: g . p = |[((cR * l) . p),((sR * l) . p)]| by Lm20;
then (g . p) `2 = (sR * l) . p by EUCLID:52;
then (sR * l) . p in B by A8, SQUARE_1:19, TOPREAL6:15;
then consider Ws being Subset of R^1 such that
A15: p in Ws and
A16: Ws is open and
A17: (sR * l) .: Ws c= B by A2, A1, A3, A10, A13, Th5, JGRAPH_2:10;
A18: A is open by JORDAN6:35;
(g . p) `1 = (cR * l) . p by A14, EUCLID:52;
then (cR * l) . p in A by A8, SQUARE_1:19, TOPREAL6:15;
then consider Wc being Subset of R^1 such that
A19: p in Wc and
A20: Wc is open and
A21: (cR * l) .: Wc c= A by A2, A1, A3, A18, A12, Th5, JGRAPH_2:10;
set W = Ws /\ Wc;
take Ws /\ Wc ; ::_thesis: ( p in Ws /\ Wc & Ws /\ Wc is open & g .: (Ws /\ Wc) c= V )
thus p in Ws /\ Wc by A15, A19, XBOOLE_0:def_4; ::_thesis: ( Ws /\ Wc is open & g .: (Ws /\ Wc) c= V )
thus Ws /\ Wc is open by A16, A20; ::_thesis: g .: (Ws /\ Wc) c= V
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in g .: (Ws /\ Wc) or y in V )
assume y in g .: (Ws /\ Wc) ; ::_thesis: y in V
then consider x being Element of R^1 such that
A22: x in Ws /\ Wc and
A23: y = g . x by FUNCT_2:65;
x in Ws by A22, XBOOLE_0:def_4;
then A24: (sR * l) . x in (sR * l) .: Ws by FUNCT_2:35;
x in Wc by A22, XBOOLE_0:def_4;
then A25: (cR * l) . x in (cR * l) .: Wc by FUNCT_2:35;
|[((cR * l) . x),((sR * l) . x)]| = (1,2) --> (((cR * l) . x),((sR * l) . x)) by TOPREALA:28;
then |[((cR * l) . x),((sR * l) . x)]| in product ((1,2) --> (].(((g . p) `1) - (r / (sqrt 2))),(((g . p) `1) + (r / (sqrt 2))).[,].(((g . p) `2) - (r / (sqrt 2))),(((g . p) `2) + (r / (sqrt 2))).[)) by A17, A21, A24, A25, HILBERT3:11;
then A26: |[((cR * l) . x),((sR * l) . x)]| in Ball (e,r) by A11;
g . x = |[((cR * l) . x),((sR * l) . x)]| by Lm20;
hence y in V by A9, A23, A26; ::_thesis: verum
end;
then g is continuous by JGRAPH_2:10;
hence CircleMap is continuous by PRE_TOPC:27; ::_thesis: verum
end;
end;
Lm21: for A being Subset of R^1 holds CircleMap | A is Function of (R^1 | A),(Tunit_circle 2)
proof
let A be Subset of R^1; ::_thesis: CircleMap | A is Function of (R^1 | A),(Tunit_circle 2)
A1: rng (CircleMap | A) c= the carrier of (Tunit_circle 2) ;
dom (CircleMap | A) = A by Lm18, RELAT_1:62, TOPMETR:17
.= the carrier of (R^1 | A) by PRE_TOPC:8 ;
hence CircleMap | A is Function of (R^1 | A),(Tunit_circle 2) by A1, FUNCT_2:2; ::_thesis: verum
end;
Lm22: for r being real number st - 1 <= r & r <= 1 holds
( 0 <= (arccos r) / (2 * PI) & (arccos r) / (2 * PI) <= 1 / 2 )
proof
let r be real number ; ::_thesis: ( - 1 <= r & r <= 1 implies ( 0 <= (arccos r) / (2 * PI) & (arccos r) / (2 * PI) <= 1 / 2 ) )
assume that
A1: - 1 <= r and
A2: r <= 1 ; ::_thesis: ( 0 <= (arccos r) / (2 * PI) & (arccos r) / (2 * PI) <= 1 / 2 )
arccos r <= PI by A1, A2, SIN_COS6:99;
then A3: (arccos r) / (2 * PI) <= (1 * PI) / (2 * PI) by XREAL_1:72;
0 <= arccos r by A1, A2, SIN_COS6:99;
hence ( 0 <= (arccos r) / (2 * PI) & (arccos r) / (2 * PI) <= 1 / 2 ) by A3, XCMPLX_1:91; ::_thesis: verum
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
let A be Subset of R^1; ::_thesis: for f being Function of (R^1 | A),(Tunit_circle 2) st [.0,1.[ c= A & f = CircleMap | A holds
f is onto
let f be Function of (R^1 | A),(Tunit_circle 2); ::_thesis: ( [.0,1.[ c= A & f = CircleMap | A implies f is onto )
assume that
A1: [.0,1.[ c= A and
A2: f = CircleMap | A ; ::_thesis: f is onto
A3: dom f = A by A2, Lm18, RELAT_1:62, TOPMETR:17;
thus rng f c= the carrier of (Tunit_circle 2) ; :: according to XBOOLE_0:def_10,FUNCT_2:def_3 ::_thesis: the carrier of (Tunit_circle 2) c= rng f
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of (Tunit_circle 2) or y in rng f )
assume A4: y in the carrier of (Tunit_circle 2) ; ::_thesis: y in rng f
then reconsider z = y as Point of (TOP-REAL 2) by PRE_TOPC:25;
set z1 = z `1 ;
set z2 = z `2 ;
A5: z `1 <= 1 by A4, Th13;
set x = (arccos (z `1)) / (2 * PI);
A6: - 1 <= z `1 by A4, Th13;
then A7: 0 <= (arccos (z `1)) / (2 * PI) by A5, Lm22;
(arccos (z `1)) / (2 * PI) <= 1 / 2 by A6, A5, Lm22;
then A8: (arccos (z `1)) / (2 * PI) < 1 by XXREAL_0:2;
A9: ((z `1) ^2) + ((z `2) ^2) = |.z.| ^2 by JGRAPH_1:29;
A10: |.z.| = 1 by A4, Th12;
percases ( z `2 < 0 or z `2 >= 0 ) ;
supposeA11: z `2 < 0 ; ::_thesis: y in rng f
now__::_thesis:_not_(arccos_(z_`1))_/_(2_*_PI)_=_0
assume (arccos (z `1)) / (2 * PI) = 0 ; ::_thesis: contradiction
then arccos (z `1) = 0 ;
then z `1 = 1 by A6, A5, SIN_COS6:96;
hence contradiction by A10, A9, A11; ::_thesis: verum
end;
then A12: 1 - 0 > 1 - ((arccos (z `1)) / (2 * PI)) by A7, XREAL_1:15;
1 - ((arccos (z `1)) / (2 * PI)) > 1 - 1 by A8, XREAL_1:15;
then A13: 1 - ((arccos (z `1)) / (2 * PI)) in [.0,1.[ by A12, XXREAL_1:3;
then f . (1 - ((arccos (z `1)) / (2 * PI))) = CircleMap . ((- ((arccos (z `1)) / (2 * PI))) + 1) by A1, A2, FUNCT_1:49
.= CircleMap . (- ((arccos (z `1)) / (2 * PI))) by Th31
.= |[(cos (- ((2 * PI) * ((arccos (z `1)) / (2 * PI))))),(sin ((2 * PI) * (- ((arccos (z `1)) / (2 * PI)))))]| by Def11
.= |[(cos ((2 * PI) * ((arccos (z `1)) / (2 * PI)))),(sin ((2 * PI) * (- ((arccos (z `1)) / (2 * PI)))))]| by SIN_COS:31
.= |[(cos (arccos (z `1))),(sin (- ((2 * PI) * ((arccos (z `1)) / (2 * PI)))))]| by XCMPLX_1:87
.= |[(cos (arccos (z `1))),(- (sin ((2 * PI) * ((arccos (z `1)) / (2 * PI)))))]| by SIN_COS:31
.= |[(cos (arccos (z `1))),(- (sin (arccos (z `1))))]| by XCMPLX_1:87
.= |[(z `1),(- (sin (arccos (z `1))))]| by A6, A5, SIN_COS6:91
.= |[(z `1),(- (- (z `2)))]| by A10, A9, A11, SIN_COS6:103
.= y by EUCLID:53 ;
hence y in rng f by A1, A3, A13, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA14: z `2 >= 0 ; ::_thesis: y in rng f
A15: (arccos (z `1)) / (2 * PI) in [.0,1.[ by A7, A8, XXREAL_1:3;
then f . ((arccos (z `1)) / (2 * PI)) = CircleMap . ((arccos (z `1)) / (2 * PI)) by A1, A2, FUNCT_1:49
.= |[(cos ((2 * PI) * ((arccos (z `1)) / (2 * PI)))),(sin ((2 * PI) * ((arccos (z `1)) / (2 * PI))))]| by Def11
.= |[(cos (arccos (z `1))),(sin ((2 * PI) * ((arccos (z `1)) / (2 * PI))))]| by XCMPLX_1:87
.= |[(cos (arccos (z `1))),(sin (arccos (z `1)))]| by XCMPLX_1:87
.= |[(z `1),(sin (arccos (z `1)))]| by A6, A5, SIN_COS6:91
.= |[(z `1),(z `2)]| by A10, A9, A14, SIN_COS6:102
.= y by EUCLID:53 ;
hence y in rng f by A1, A3, A15, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
registration
cluster CircleMap -> onto ;
coherence
CircleMap is onto
proof
A1: R^1 | ([#] R^1) = R^1 by TSEP_1:3;
CircleMap | REAL = CircleMap by Lm18, RELAT_1:69;
hence CircleMap is onto by A1, Th38, TOPMETR:17; ::_thesis: verum
end;
end;
Lm23: CircleMap | [.0,1.[ is one-to-one
proof
A1: sin | [.(PI / 2),((3 / 2) * PI).] is one-to-one ;
let x1, y1 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in K121((CircleMap | [.0,1.[)) or not y1 in K121((CircleMap | [.0,1.[)) or not (CircleMap | [.0,1.[) . x1 = (CircleMap | [.0,1.[) . y1 or x1 = y1 )
set f = CircleMap | [.0,1.[;
A2: [.0,(PI / 2).] c= [.(- (PI / 2)),(PI / 2).] by XXREAL_1:34;
A3: dom (CircleMap | [.0,1.[) = [.0,1.[ by Lm18, RELAT_1:62;
assume A4: x1 in dom (CircleMap | [.0,1.[) ; ::_thesis: ( not y1 in K121((CircleMap | [.0,1.[)) or not (CircleMap | [.0,1.[) . x1 = (CircleMap | [.0,1.[) . y1 or x1 = y1 )
then reconsider x = x1 as Real by A3;
A5: (CircleMap | [.0,1.[) . x = CircleMap . x by A3, A4, FUNCT_1:49
.= |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x))]| by Def11 ;
assume A6: y1 in dom (CircleMap | [.0,1.[) ; ::_thesis: ( not (CircleMap | [.0,1.[) . x1 = (CircleMap | [.0,1.[) . y1 or x1 = y1 )
then reconsider y = y1 as Real by A3;
assume A7: (CircleMap | [.0,1.[) . x1 = (CircleMap | [.0,1.[) . y1 ; ::_thesis: x1 = y1
A8: (CircleMap | [.0,1.[) . y = CircleMap . y by A3, A6, FUNCT_1:49
.= |[(cos ((2 * PI) * y)),(sin ((2 * PI) * y))]| by Def11 ;
then A9: cos ((2 * PI) * x) = cos ((2 * PI) * y) by A7, A5, SPPOL_2:1;
A10: cos ((2 * PI) * y) = cos . ((2 * PI) * y) by SIN_COS:def_19;
A11: cos ((2 * PI) * x) = cos . ((2 * PI) * x) by SIN_COS:def_19;
A12: sin ((2 * PI) * x) = sin ((2 * PI) * y) by A7, A5, A8, SPPOL_2:1;
A13: sin ((2 * PI) * y) = sin . ((2 * PI) * y) by SIN_COS:def_17;
A14: sin ((2 * PI) * x) = sin . ((2 * PI) * x) by SIN_COS:def_17;
percases ( ( 0 <= x & x <= 1 / 4 ) or ( 1 / 4 < x & x <= 1 / 2 ) or ( 1 / 2 < x & x <= 3 / 4 ) or ( 3 / 4 < x & x < 1 ) ) by A3, A4, XXREAL_1:3;
supposeA15: ( 0 <= x & x <= 1 / 4 ) ; ::_thesis: x1 = y1
A16: [.0,(PI / 2).] c= [.0,PI.] by Lm5, XXREAL_1:34;
(2 * PI) * x <= (2 * PI) * (1 / 4) by A15, XREAL_1:64;
then A17: (2 * PI) * x in [.0,(((2 * PI) * 1) / 4).] by A15, XXREAL_1:1;
percases ( ( 0 <= y & y <= 1 / 4 ) or ( 1 / 4 < y & y < 3 / 4 ) or ( 3 / 4 <= y & y < 1 ) ) by A3, A6, XXREAL_1:3;
supposeA18: ( 0 <= y & y <= 1 / 4 ) ; ::_thesis: x1 = y1
then (2 * PI) * y <= (2 * PI) * (1 / 4) by XREAL_1:64;
then A19: (2 * PI) * y in [.0,(((2 * PI) * 1) / 4).] by A18, XXREAL_1:1;
set g = sin | [.0,(PI / 2).];
A20: dom (sin | [.0,(PI / 2).]) = [.0,(PI / 2).] by RELAT_1:62, SIN_COS:24;
(sin | [.0,(PI / 2).]) . ((2 * PI) * x) = sin . ((2 * PI) * x) by A17, FUNCT_1:49
.= sin . ((2 * PI) * y) by A12, A14, SIN_COS:def_17
.= (sin | [.0,(PI / 2).]) . ((2 * PI) * y) by A19, FUNCT_1:49 ;
then (2 * PI) * x = (2 * PI) * y by A17, A19, A20, FUNCT_1:def_4;
then x = ((2 * PI) * y) / (2 * PI) by XCMPLX_1:89;
hence x1 = y1 by XCMPLX_1:89; ::_thesis: verum
end;
supposeA21: ( 1 / 4 < y & y < 3 / 4 ) ; ::_thesis: x1 = y1
then A22: (2 * PI) * y < (2 * PI) * (3 / 4) by XREAL_1:68;
(2 * PI) * (1 / 4) < (2 * PI) * y by A21, XREAL_1:68;
then (2 * PI) * y in ].(PI / 2),((3 / 2) * PI).[ by A22, XXREAL_1:4;
hence x1 = y1 by A9, A11, A10, A2, A17, COMPTRIG:12, COMPTRIG:13; ::_thesis: verum
end;
supposeA23: ( 3 / 4 <= y & y < 1 ) ; ::_thesis: x1 = y1
then A24: (2 * PI) * y < (2 * PI) * 1 by XREAL_1:68;
A25: [.((3 / 2) * PI),(2 * PI).[ c= ].PI,(2 * PI).[ by Lm6, XXREAL_1:48;
(2 * PI) * (3 / 4) <= (2 * PI) * y by A23, XREAL_1:64;
then (2 * PI) * y in [.((3 / 2) * PI),(2 * PI).[ by A24, XXREAL_1:3;
hence x1 = y1 by A12, A14, A13, A17, A16, A25, COMPTRIG:8, COMPTRIG:9; ::_thesis: verum
end;
end;
end;
supposeA26: ( 1 / 4 < x & x <= 1 / 2 ) ; ::_thesis: x1 = y1
then A27: (2 * PI) * x <= (2 * PI) * (1 / 2) by XREAL_1:64;
(2 * PI) * (1 / 4) < (2 * PI) * x by A26, XREAL_1:68;
then A28: (2 * PI) * x in ].(PI / 2),((2 * PI) * (1 / 2)).] by A27, XXREAL_1:2;
A29: ].(PI / 2),PI.] c= ].(PI / 2),((3 / 2) * PI).[ by Lm6, XXREAL_1:49;
A30: ].(PI / 2),PI.] c= [.0,PI.] by XXREAL_1:36;
percases ( ( 0 <= y & y <= 1 / 4 ) or ( 1 / 4 < y & y <= 1 / 2 ) or ( 1 / 2 < y & y < 1 ) ) by A3, A6, XXREAL_1:3;
supposeA31: ( 0 <= y & y <= 1 / 4 ) ; ::_thesis: x1 = y1
then (2 * PI) * y <= (2 * PI) * (1 / 4) by XREAL_1:64;
then (2 * PI) * y in [.0,((2 * PI) * (1 / 4)).] by A31, XXREAL_1:1;
hence x1 = y1 by A9, A11, A10, A2, A28, A29, COMPTRIG:12, COMPTRIG:13; ::_thesis: verum
end;
supposeA32: ( 1 / 4 < y & y <= 1 / 2 ) ; ::_thesis: x1 = y1
then A33: (2 * PI) * y <= (2 * PI) * (1 / 2) by XREAL_1:64;
(2 * PI) * (1 / 4) < (2 * PI) * y by A32, XREAL_1:68;
then A34: (2 * PI) * y in ].((2 * PI) * (1 / 4)),((2 * PI) * (1 / 2)).] by A33, XXREAL_1:2;
set g = sin | ].(PI / 2),PI.];
A35: dom (sin | ].(PI / 2),PI.]) = ].(PI / 2),PI.] by RELAT_1:62, SIN_COS:24;
A36: sin | ].(PI / 2),PI.] is one-to-one by A1, Lm6, SIN_COS6:2, XXREAL_1:36;
(sin | ].(PI / 2),PI.]) . ((2 * PI) * x) = sin . ((2 * PI) * x) by A28, FUNCT_1:49
.= sin . ((2 * PI) * y) by A12, A14, SIN_COS:def_17
.= (sin | ].(PI / 2),PI.]) . ((2 * PI) * y) by A34, FUNCT_1:49 ;
then (2 * PI) * x = (2 * PI) * y by A28, A34, A36, A35, FUNCT_1:def_4;
then x = ((2 * PI) * y) / (2 * PI) by XCMPLX_1:89;
hence x1 = y1 by XCMPLX_1:89; ::_thesis: verum
end;
supposeA37: ( 1 / 2 < y & y < 1 ) ; ::_thesis: x1 = y1
then A38: (2 * PI) * y < (2 * PI) * 1 by XREAL_1:68;
(2 * PI) * (1 / 2) < (2 * PI) * y by A37, XREAL_1:68;
then (2 * PI) * y in ].PI,(2 * PI).[ by A38, XXREAL_1:4;
hence x1 = y1 by A12, A14, A13, A28, A30, COMPTRIG:8, COMPTRIG:9; ::_thesis: verum
end;
end;
end;
supposeA39: ( 1 / 2 < x & x <= 3 / 4 ) ; ::_thesis: x1 = y1
then A40: (2 * PI) * x <= (2 * PI) * (3 / 4) by XREAL_1:64;
(2 * PI) * (1 / 2) < (2 * PI) * x by A39, XREAL_1:68;
then A41: (2 * PI) * x in ].PI,((2 * PI) * (3 / 4)).] by A40, XXREAL_1:2;
A42: ].PI,((3 / 2) * PI).] c= [.(PI / 2),((3 / 2) * PI).] by Lm5, XXREAL_1:36;
A43: ].PI,((3 / 2) * PI).] c= ].PI,(2 * PI).[ by Lm7, XXREAL_1:49;
percases ( ( 0 <= y & y <= 1 / 2 ) or ( 1 / 2 < y & y <= 3 / 4 ) or ( 3 / 4 < y & y < 1 ) ) by A3, A6, XXREAL_1:3;
supposeA44: ( 0 <= y & y <= 1 / 2 ) ; ::_thesis: x1 = y1
then (2 * PI) * y <= (2 * PI) * (1 / 2) by XREAL_1:64;
then (2 * PI) * y in [.0,PI.] by A44, XXREAL_1:1;
hence x1 = y1 by A12, A14, A13, A41, A43, COMPTRIG:8, COMPTRIG:9; ::_thesis: verum
end;
supposeA45: ( 1 / 2 < y & y <= 3 / 4 ) ; ::_thesis: x1 = y1
then A46: (2 * PI) * y <= (2 * PI) * (3 / 4) by XREAL_1:64;
(2 * PI) * (1 / 2) < (2 * PI) * y by A45, XREAL_1:68;
then A47: (2 * PI) * y in ].PI,((2 * PI) * (3 / 4)).] by A46, XXREAL_1:2;
set g = sin | ].PI,((3 / 2) * PI).];
A48: dom (sin | ].PI,((3 / 2) * PI).]) = ].PI,((3 / 2) * PI).] by RELAT_1:62, SIN_COS:24;
A49: sin | ].PI,((3 / 2) * PI).] is one-to-one by A1, Lm5, SIN_COS6:2, XXREAL_1:36;
(sin | ].PI,((3 / 2) * PI).]) . ((2 * PI) * x) = sin . ((2 * PI) * x) by A41, FUNCT_1:49
.= sin . ((2 * PI) * y) by A12, A14, SIN_COS:def_17
.= (sin | ].PI,((3 / 2) * PI).]) . ((2 * PI) * y) by A47, FUNCT_1:49 ;
then (2 * PI) * x = (2 * PI) * y by A41, A47, A49, A48, FUNCT_1:def_4;
then x = ((2 * PI) * y) / (2 * PI) by XCMPLX_1:89;
hence x1 = y1 by XCMPLX_1:89; ::_thesis: verum
end;
supposeA50: ( 3 / 4 < y & y < 1 ) ; ::_thesis: x1 = y1
then A51: (2 * PI) * y < (2 * PI) * 1 by XREAL_1:68;
(2 * PI) * (3 / 4) < (2 * PI) * y by A50, XREAL_1:68;
then (2 * PI) * y in ].((3 / 2) * PI),(2 * PI).[ by A51, XXREAL_1:4;
hence x1 = y1 by A9, A11, A10, A41, A42, COMPTRIG:14, COMPTRIG:15; ::_thesis: verum
end;
end;
end;
supposeA52: ( 3 / 4 < x & x < 1 ) ; ::_thesis: x1 = y1
then A53: (2 * PI) * x < (2 * PI) * 1 by XREAL_1:68;
(2 * PI) * (3 / 4) < (2 * PI) * x by A52, XREAL_1:68;
then A54: (2 * PI) * x in ].((3 / 2) * PI),(2 * PI).[ by A53, XXREAL_1:4;
A55: ].((3 / 2) * PI),(2 * PI).[ c= ].PI,(2 * PI).[ by Lm6, XXREAL_1:46;
percases ( ( 0 <= y & y <= 1 / 2 ) or ( 1 / 2 < y & y <= 3 / 4 ) or ( 3 / 4 < y & y < 1 ) ) by A3, A6, XXREAL_1:3;
supposeA56: ( 0 <= y & y <= 1 / 2 ) ; ::_thesis: x1 = y1
then (2 * PI) * y <= (2 * PI) * (1 / 2) by XREAL_1:64;
then (2 * PI) * y in [.0,PI.] by A56, XXREAL_1:1;
hence x1 = y1 by A12, A14, A13, A54, A55, COMPTRIG:8, COMPTRIG:9; ::_thesis: verum
end;
supposeA57: ( 1 / 2 < y & y <= 3 / 4 ) ; ::_thesis: x1 = y1
then A58: (2 * PI) * y <= (2 * PI) * (3 / 4) by XREAL_1:64;
A59: ].PI,((3 / 2) * PI).] c= [.(PI / 2),((3 / 2) * PI).] by Lm5, XXREAL_1:36;
(2 * PI) * (1 / 2) < (2 * PI) * y by A57, XREAL_1:68;
then (2 * PI) * y in ].PI,((3 / 2) * PI).] by A58, XXREAL_1:2;
hence x1 = y1 by A9, A11, A10, A54, A59, COMPTRIG:14, COMPTRIG:15; ::_thesis: verum
end;
supposeA60: ( 3 / 4 < y & y < 1 ) ; ::_thesis: x1 = y1
then A61: (2 * PI) * y < (2 * PI) * 1 by XREAL_1:68;
(2 * PI) * (3 / 4) < (2 * PI) * y by A60, XREAL_1:68;
then A62: (2 * PI) * y in ].((3 / 2) * PI),(2 * PI).[ by A61, XXREAL_1:4;
set g = sin | ].((3 / 2) * PI),(2 * PI).[;
A63: dom (sin | ].((3 / 2) * PI),(2 * PI).[) = ].((3 / 2) * PI),(2 * PI).[ by RELAT_1:62, SIN_COS:24;
(sin | ].((3 / 2) * PI),(2 * PI).[) . ((2 * PI) * x) = sin . ((2 * PI) * x) by A54, FUNCT_1:49
.= sin . ((2 * PI) * y) by A12, A14, SIN_COS:def_17
.= (sin | ].((3 / 2) * PI),(2 * PI).[) . ((2 * PI) * y) by A62, FUNCT_1:49 ;
then (2 * PI) * x = (2 * PI) * y by A54, A62, A63, FUNCT_1:def_4;
then x = ((2 * PI) * y) / (2 * PI) by XCMPLX_1:89;
hence x1 = y1 by XCMPLX_1:89; ::_thesis: verum
end;
end;
end;
end;
end;
registration
let r be real number ;
clusterK69(CircleMap,[.r,(r + 1).[) -> one-to-one ;
coherence
CircleMap | [.r,(r + 1).[ is one-to-one
proof
let x, y be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x in K121((CircleMap | [.r,(r + 1).[)) or not y in K121((CircleMap | [.r,(r + 1).[)) or not (CircleMap | [.r,(r + 1).[) . x = (CircleMap | [.r,(r + 1).[) . y or x = y )
set g = CircleMap | [.0,1.[;
set f = CircleMap | [.r,(r + 1).[;
assume that
A1: x in dom (CircleMap | [.r,(r + 1).[) and
A2: y in dom (CircleMap | [.r,(r + 1).[) and
A3: (CircleMap | [.r,(r + 1).[) . x = (CircleMap | [.r,(r + 1).[) . y ; ::_thesis: x = y
A4: dom (CircleMap | [.r,(r + 1).[) = [.r,(r + 1).[ by Lm18, RELAT_1:62;
then reconsider x = x, y = y as Real by A1, A2;
A5: dom (CircleMap | [.0,1.[) = [.0,1.[ by Lm18, RELAT_1:62;
A6: r <= y by A4, A2, XXREAL_1:3;
A7: x < r + 1 by A4, A1, XXREAL_1:3;
set x1 = frac x;
A8: frac x = x - [\x/] by INT_1:def_8;
A9: frac x < 1 by INT_1:43;
0 <= frac x by INT_1:43;
then A10: frac x in [.0,1.[ by A9, XXREAL_1:3;
set y1 = frac y;
A11: frac y = y - [\y/] by INT_1:def_8;
A12: frac y < 1 by INT_1:43;
0 <= frac y by INT_1:43;
then A13: frac y in [.0,1.[ by A12, XXREAL_1:3;
A14: (CircleMap | [.r,(r + 1).[) . y = CircleMap . y by A2, FUNCT_1:47
.= CircleMap . (y + (- [\y/])) by Th31
.= (CircleMap | [.0,1.[) . (frac y) by A5, A11, A13, FUNCT_1:47 ;
(CircleMap | [.r,(r + 1).[) . x = CircleMap . x by A1, FUNCT_1:47
.= CircleMap . (x + (- [\x/])) by Th31
.= (CircleMap | [.0,1.[) . (frac x) by A5, A8, A10, FUNCT_1:47 ;
then A15: frac x = frac y by A5, A3, A10, A13, A14, Lm23, FUNCT_1:def_4;
A16: y < r + 1 by A4, A2, XXREAL_1:3;
r <= x by A4, A1, XXREAL_1:3;
hence x = y by A7, A6, A16, A15, INT_1:72; ::_thesis: verum
end;
end;
registration
let r be real number ;
clusterK69(CircleMap,].r,(r + 1).[) -> one-to-one ;
coherence
CircleMap | ].r,(r + 1).[ is one-to-one
proof
CircleMap | [.r,(r + 1).[ is one-to-one ;
hence CircleMap | ].r,(r + 1).[ is one-to-one by SIN_COS6:2, XXREAL_1:45; ::_thesis: verum
end;
end;
theorem Th39: :: TOPREALB:39
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
let b, a be real number ; ::_thesis: ( b - a <= 1 implies for d being set st d in IntIntervals (a,b) holds
CircleMap | d is one-to-one )
assume A1: b - a <= 1 ; ::_thesis: for d being set st d in IntIntervals (a,b) holds
CircleMap | d is one-to-one
let d be set ; ::_thesis: ( d in IntIntervals (a,b) implies CircleMap | d is one-to-one )
assume d in IntIntervals (a,b) ; ::_thesis: CircleMap | d is one-to-one
then consider n being Element of INT such that
A2: d = ].(a + n),(b + n).[ ;
A3: CircleMap | [.(a + n),((a + n) + 1).[ is one-to-one ;
(b - a) + (a + n) <= 1 + (a + n) by A1, XREAL_1:6;
hence CircleMap | d is one-to-one by A2, A3, SIN_COS6:2, XXREAL_1:45; ::_thesis: verum
end;
theorem Th40: :: TOPREALB:40
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
let a, b be real number ; ::_thesis: for d being set st d in IntIntervals (a,b) holds
CircleMap .: d = CircleMap .: (union (IntIntervals (a,b)))
set D = IntIntervals (a,b);
let d be set ; ::_thesis: ( d in IntIntervals (a,b) implies CircleMap .: d = CircleMap .: (union (IntIntervals (a,b))) )
assume A1: d in IntIntervals (a,b) ; ::_thesis: CircleMap .: d = CircleMap .: (union (IntIntervals (a,b)))
hence CircleMap .: d c= CircleMap .: (union (IntIntervals (a,b))) by RELAT_1:123, ZFMISC_1:74; :: according to XBOOLE_0:def_10 ::_thesis: CircleMap .: (union (IntIntervals (a,b))) c= CircleMap .: d
consider i being Element of INT such that
A2: d = ].(a + i),(b + i).[ by A1;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in CircleMap .: (union (IntIntervals (a,b))) or y in CircleMap .: d )
assume y in CircleMap .: (union (IntIntervals (a,b))) ; ::_thesis: y in CircleMap .: d
then consider x being Element of R^1 such that
A3: x in union (IntIntervals (a,b)) and
A4: y = CircleMap . x by FUNCT_2:65;
consider Z being set such that
A5: x in Z and
A6: Z in IntIntervals (a,b) by A3, TARSKI:def_4;
consider n being Element of INT such that
A7: Z = ].(a + n),(b + n).[ by A6;
x < b + n by A5, A7, XXREAL_1:4;
then x + i < (b + n) + i by XREAL_1:6;
then A8: (x + i) - n < ((b + n) + i) - n by XREAL_1:9;
set k = (x + i) - n;
A9: CircleMap . ((x + i) - n) = CircleMap . (x + (i - n))
.= y by A4, Th31 ;
A10: (x + i) - n in the carrier of R^1 by TOPMETR:17, XREAL_0:def_1;
a + n < x by A5, A7, XXREAL_1:4;
then (a + n) + i < x + i by XREAL_1:6;
then ((a + n) + i) - n < (x + i) - n by XREAL_1:9;
then (x + i) - n in d by A2, A8, XXREAL_1:4;
hence y in CircleMap .: d by A10, A9, FUNCT_2:35; ::_thesis: verum
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
set B = [.r,(r + 1).[;
set A = ].r,(r + 1).[;
set X = Topen_unit_circle (CircleMap . r);
set f = CircleMap | ].r,(r + 1).[;
set g = CircleMap | [.r,(r + 1).[;
A1: ].r,(r + 1).[ c= [.r,(r + 1).[ by XXREAL_1:45;
A2: dom (CircleMap | ].r,(r + 1).[) = ].r,(r + 1).[ by Lm18, RELAT_1:62;
A3: the carrier of (Topen_unit_circle (CircleMap . r)) = the carrier of (Tunit_circle 2) \ {(CircleMap . r)} by Def10;
A4: rng (CircleMap | ].r,(r + 1).[) c= the carrier of (Topen_unit_circle (CircleMap . r))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (CircleMap | ].r,(r + 1).[) or y in the carrier of (Topen_unit_circle (CircleMap . r)) )
assume A5: y in rng (CircleMap | ].r,(r + 1).[) ; ::_thesis: y in the carrier of (Topen_unit_circle (CircleMap . r))
now__::_thesis:_not_y_=_CircleMap_._r
A6: dom (CircleMap | [.r,(r + 1).[) = [.r,(r + 1).[ by Lm18, RELAT_1:62;
assume A7: y = CircleMap . r ; ::_thesis: contradiction
r + 0 < r + 1 by XREAL_1:8;
then A8: r in [.r,(r + 1).[ by XXREAL_1:3;
consider x being set such that
A9: x in dom (CircleMap | ].r,(r + 1).[) and
A10: (CircleMap | ].r,(r + 1).[) . x = y by A5, FUNCT_1:def_3;
(CircleMap | [.r,(r + 1).[) . x = CircleMap . x by A1, A2, A9, FUNCT_1:49
.= CircleMap . r by A2, A7, A9, A10, FUNCT_1:49
.= (CircleMap | [.r,(r + 1).[) . r by A8, FUNCT_1:49 ;
then x = r by A1, A2, A9, A8, A6, FUNCT_1:def_4;
hence contradiction by A2, A9, XXREAL_1:4; ::_thesis: verum
end;
then not y in {(CircleMap . r)} by TARSKI:def_1;
hence y in the carrier of (Topen_unit_circle (CircleMap . r)) by A3, A5, XBOOLE_0:def_5; ::_thesis: verum
end;
the carrier of (R^1 | (R^1 ].r,(r + 1).[)) = ].r,(r + 1).[ by PRE_TOPC:8;
hence CircleMap | ].r,(r + 1).[ is Function of (R^1 | (R^1 ].r,(r + 1).[)),(Topen_unit_circle (CircleMap . r)) by A2, A4, FUNCT_2:2; ::_thesis: verum
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
let a, r be real number ; ::_thesis: rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) = ].r,(r + 1).[
set F = AffineMap (1,(- a));
set f = (AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[;
dom (AffineMap (1,(- a))) = REAL by FUNCT_2:def_1;
then A1: dom ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) = ].(r + a),((r + a) + 1).[ by RELAT_1:62;
thus rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) = ].r,(r + 1).[ ::_thesis: verum
proof
thus rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) c= ].r,(r + 1).[ :: according to XBOOLE_0:def_10 ::_thesis: ].r,(r + 1).[ c= rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) or y in ].r,(r + 1).[ )
assume y in rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) ; ::_thesis: y in ].r,(r + 1).[
then consider x being set such that
A2: x in dom ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) and
A3: ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) . x = y by FUNCT_1:def_3;
reconsider x = x as Real by A2;
r + a < x by A1, A2, XXREAL_1:4;
then A4: (r + a) - a < x - a by XREAL_1:9;
x < (r + a) + 1 by A1, A2, XXREAL_1:4;
then A5: x - a < ((r + a) + 1) - a by XREAL_1:9;
y = (AffineMap (1,(- a))) . x by A2, A3, FUNCT_1:47
.= (1 * x) + (- a) by FCONT_1:def_4 ;
hence y in ].r,(r + 1).[ by A4, A5, XXREAL_1:4; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in ].r,(r + 1).[ or y in rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) )
assume A6: y in ].r,(r + 1).[ ; ::_thesis: y in rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[)
then reconsider y = y as Real ;
y < r + 1 by A6, XXREAL_1:4;
then A7: y + a < (r + 1) + a by XREAL_1:6;
r < y by A6, XXREAL_1:4;
then r + a < y + a by XREAL_1:6;
then A8: y + a in ].(r + a),((r + a) + 1).[ by A7, XXREAL_1:4;
then ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) . (y + a) = (AffineMap (1,(- a))) . (y + a) by FUNCT_1:49
.= (1 * (y + a)) + (- a) by FCONT_1:def_4
.= y ;
hence y in rng ((AffineMap (1,(- a))) | ].(r + a),((r + a) + 1).[) by A1, A8, FUNCT_1:def_3; ::_thesis: verum
end;
end;
theorem Th41: :: TOPREALB:41
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
let i be Integer; ::_thesis: for a being real number holds CircleMap (R^1 (a + i)) = (CircleMap (R^1 a)) * ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[)
let a be real number ; ::_thesis: CircleMap (R^1 (a + i)) = (CircleMap (R^1 a)) * ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[)
set W = ].a,(a + 1).[;
set Q = ].(a + i),((a + i) + 1).[;
set h = CircleMap (R^1 (a + i));
set g = CircleMap (R^1 a);
set F = AffineMap (1,(- i));
set f = (AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[;
A1: dom (CircleMap (R^1 (a + i))) = ].(a + i),((a + i) + 1).[ by Lm18, RELAT_1:62;
dom (AffineMap (1,(- i))) = REAL by FUNCT_2:def_1;
then A2: dom ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[) = ].(a + i),((a + i) + 1).[ by RELAT_1:62;
A3: for x being set st x in dom (CircleMap (R^1 (a + i))) holds
(CircleMap (R^1 (a + i))) . x = ((CircleMap (R^1 a)) * ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[)) . x
proof
let x be set ; ::_thesis: ( x in dom (CircleMap (R^1 (a + i))) implies (CircleMap (R^1 (a + i))) . x = ((CircleMap (R^1 a)) * ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[)) . x )
assume A4: x in dom (CircleMap (R^1 (a + i))) ; ::_thesis: (CircleMap (R^1 (a + i))) . x = ((CircleMap (R^1 a)) * ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[)) . x
then reconsider y = x as Real by A1;
y < (a + i) + 1 by A1, A4, XXREAL_1:4;
then A5: y - i < ((a + i) + 1) - i by XREAL_1:9;
a + i < y by A1, A4, XXREAL_1:4;
then (a + i) - i < y - i by XREAL_1:9;
then A6: y - i in ].a,(a + 1).[ by A5, XXREAL_1:4;
thus ((CircleMap (R^1 a)) * ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[)) . x = (CircleMap (R^1 a)) . (((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[) . x) by A1, A2, A4, FUNCT_1:13
.= (CircleMap (R^1 a)) . ((AffineMap (1,(- i))) . x) by A1, A4, FUNCT_1:49
.= (CircleMap (R^1 a)) . ((1 * y) + (- i)) by FCONT_1:def_4
.= CircleMap . (y + (- i)) by A6, FUNCT_1:49
.= CircleMap . y by Th31
.= (CircleMap (R^1 (a + i))) . x by A1, A4, FUNCT_1:49 ; ::_thesis: verum
end;
A7: rng ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[) = ].a,(a + 1).[ by Lm24;
dom (CircleMap (R^1 a)) = ].a,(a + 1).[ by Lm18, RELAT_1:62;
then dom ((CircleMap (R^1 a)) * ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[)) = dom ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[) by A7, RELAT_1:27;
hence CircleMap (R^1 (a + i)) = (CircleMap (R^1 a)) * ((AffineMap (1,(- i))) | ].(a + i),((a + i) + 1).[) by A2, A3, Lm18, FUNCT_1:2, RELAT_1:62; ::_thesis: verum
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
thus CircleMap r is one-to-one ; ::_thesis: ( CircleMap r is onto & CircleMap r is continuous )
thus CircleMap r is onto ::_thesis: CircleMap r is continuous
proof
set TOUC = Topen_unit_circle (CircleMap . r);
set A = ].r,(r + 1).[;
set f = CircleMap | ].r,(r + 1).[;
set X = the carrier of (Topen_unit_circle (CircleMap . r));
thus rng (CircleMap r) c= the carrier of (Topen_unit_circle (CircleMap . r)) ; :: according to XBOOLE_0:def_10,FUNCT_2:def_3 ::_thesis: the carrier of (Topen_unit_circle (CircleMap . r)) c= rng (CircleMap r)
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of (Topen_unit_circle (CircleMap . r)) or y in rng (CircleMap r) )
A1: [\r/] <= r by INT_1:def_6;
A2: dom (CircleMap | ].r,(r + 1).[) = ].r,(r + 1).[ by Lm18, RELAT_1:62;
assume A3: y in the carrier of (Topen_unit_circle (CircleMap . r)) ; ::_thesis: y in rng (CircleMap r)
then reconsider z = y as Point of (TOP-REAL 2) by Lm8;
set z1 = z `1 ;
set z2 = z `2 ;
A4: z `1 <= 1 by A3, Th26;
set x = (arccos (z `1)) / (2 * PI);
A5: - 1 <= z `1 by A3, Th26;
then A6: 0 <= (arccos (z `1)) / (2 * PI) by A4, Lm22;
(arccos (z `1)) / (2 * PI) <= 1 / 2 by A5, A4, Lm22;
then A7: (arccos (z `1)) / (2 * PI) < 1 by XXREAL_0:2;
then A8: ((arccos (z `1)) / (2 * PI)) - ((arccos (z `1)) / (2 * PI)) < 1 - ((arccos (z `1)) / (2 * PI)) by XREAL_1:14;
A9: ((z `1) ^2) + ((z `2) ^2) = |.z.| ^2 by JGRAPH_1:29;
z is Point of (Tunit_circle 2) by A3, PRE_TOPC:25;
then A10: |.z.| = 1 by Th12;
percases ( z `2 < 0 or z `2 >= 0 ) ;
supposeA11: z `2 < 0 ; ::_thesis: y in rng (CircleMap r)
A12: CircleMap . (- ((arccos (z `1)) / (2 * PI))) = |[(cos (- ((2 * PI) * ((arccos (z `1)) / (2 * PI))))),(sin ((2 * PI) * (- ((arccos (z `1)) / (2 * PI)))))]| by Def11
.= |[(cos ((2 * PI) * ((arccos (z `1)) / (2 * PI)))),(sin ((2 * PI) * (- ((arccos (z `1)) / (2 * PI)))))]| by SIN_COS:31
.= |[(cos (arccos (z `1))),(sin (- ((2 * PI) * ((arccos (z `1)) / (2 * PI)))))]| by XCMPLX_1:87
.= |[(cos (arccos (z `1))),(- (sin ((2 * PI) * ((arccos (z `1)) / (2 * PI)))))]| by SIN_COS:31
.= |[(cos (arccos (z `1))),(- (sin (arccos (z `1))))]| by XCMPLX_1:87
.= |[(z `1),(- (sin (arccos (z `1))))]| by A5, A4, SIN_COS6:91
.= |[(z `1),(- (- (z `2)))]| by A10, A9, A11, SIN_COS6:103
.= y by EUCLID:53 ;
percases ( (1 - ((arccos (z `1)) / (2 * PI))) + [\r/] in ].r,(r + 1).[ or not (1 - ((arccos (z `1)) / (2 * PI))) + [\r/] in ].r,(r + 1).[ ) ;
supposeA13: (1 - ((arccos (z `1)) / (2 * PI))) + [\r/] in ].r,(r + 1).[ ; ::_thesis: y in rng (CircleMap r)
then (CircleMap | ].r,(r + 1).[) . ((1 - ((arccos (z `1)) / (2 * PI))) + [\r/]) = CircleMap . ((- ((arccos (z `1)) / (2 * PI))) + ([\r/] + 1)) by FUNCT_1:49
.= CircleMap . (- ((arccos (z `1)) / (2 * PI))) by Th31 ;
hence y in rng (CircleMap r) by A2, A12, A13, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA14: not (1 - ((arccos (z `1)) / (2 * PI))) + [\r/] in ].r,(r + 1).[ ; ::_thesis: y in rng (CircleMap r)
now__::_thesis:_not_(arccos_(z_`1))_/_(2_*_PI)_=_0
assume (arccos (z `1)) / (2 * PI) = 0 ; ::_thesis: contradiction
then arccos (z `1) = 0 ;
then z `1 = 1 by A5, A4, SIN_COS6:96;
hence contradiction by A10, A9, A11; ::_thesis: verum
end;
then [\r/] - ((arccos (z `1)) / (2 * PI)) < r - 0 by A1, A6, XREAL_1:15;
then ((- ((arccos (z `1)) / (2 * PI))) + [\r/]) + 1 < r + 1 by XREAL_1:6;
then A15: r >= (1 - ((arccos (z `1)) / (2 * PI))) + [\r/] by A14, XXREAL_1:4;
([\r/] + 1) + 0 < ([\r/] + 1) + (1 - ((arccos (z `1)) / (2 * PI))) by A8, XREAL_1:6;
then A16: r < (2 - ((arccos (z `1)) / (2 * PI))) + [\r/] by INT_1:29, XXREAL_0:2;
now__::_thesis:_not_((-_((arccos_(z_`1))_/_(2_*_PI)))_+_[\r/])_+_1_=_r
assume ((- ((arccos (z `1)) / (2 * PI))) + [\r/]) + 1 = r ; ::_thesis: contradiction
then CircleMap . r = CircleMap . ((- ((arccos (z `1)) / (2 * PI))) + ([\r/] + 1))
.= CircleMap . (- ((arccos (z `1)) / (2 * PI))) by Th31 ;
hence contradiction by A3, A12, Th21; ::_thesis: verum
end;
then (1 - ((arccos (z `1)) / (2 * PI))) + [\r/] < r by A15, XXREAL_0:1;
then ((1 - ((arccos (z `1)) / (2 * PI))) + [\r/]) + 1 < r + 1 by XREAL_1:6;
then A17: ((- ((arccos (z `1)) / (2 * PI))) + [\r/]) + 2 in ].r,(r + 1).[ by A16, XXREAL_1:4;
then (CircleMap | ].r,(r + 1).[) . ((- ((arccos (z `1)) / (2 * PI))) + ([\r/] + 2)) = CircleMap . ((- ((arccos (z `1)) / (2 * PI))) + ([\r/] + 2)) by FUNCT_1:49
.= CircleMap . (- ((arccos (z `1)) / (2 * PI))) by Th31 ;
hence y in rng (CircleMap r) by A2, A12, A17, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
supposeA18: z `2 >= 0 ; ::_thesis: y in rng (CircleMap r)
A19: CircleMap . ((arccos (z `1)) / (2 * PI)) = |[(cos ((2 * PI) * ((arccos (z `1)) / (2 * PI)))),(sin ((2 * PI) * ((arccos (z `1)) / (2 * PI))))]| by Def11
.= |[(cos (arccos (z `1))),(sin ((2 * PI) * ((arccos (z `1)) / (2 * PI))))]| by XCMPLX_1:87
.= |[(cos (arccos (z `1))),(sin (arccos (z `1)))]| by XCMPLX_1:87
.= |[(z `1),(sin (arccos (z `1)))]| by A5, A4, SIN_COS6:91
.= |[(z `1),(z `2)]| by A10, A9, A18, SIN_COS6:102
.= y by EUCLID:53 ;
percases ( ((arccos (z `1)) / (2 * PI)) + [\r/] in ].r,(r + 1).[ or not ((arccos (z `1)) / (2 * PI)) + [\r/] in ].r,(r + 1).[ ) ;
supposeA20: ((arccos (z `1)) / (2 * PI)) + [\r/] in ].r,(r + 1).[ ; ::_thesis: y in rng (CircleMap r)
then (CircleMap | ].r,(r + 1).[) . (((arccos (z `1)) / (2 * PI)) + [\r/]) = CircleMap . (((arccos (z `1)) / (2 * PI)) + [\r/]) by FUNCT_1:49
.= CircleMap . ((arccos (z `1)) / (2 * PI)) by Th31 ;
hence y in rng (CircleMap r) by A2, A19, A20, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA21: not ((arccos (z `1)) / (2 * PI)) + [\r/] in ].r,(r + 1).[ ; ::_thesis: y in rng (CircleMap r)
0 + ([\r/] + 1) <= ((arccos (z `1)) / (2 * PI)) + ([\r/] + 1) by A6, XREAL_1:6;
then A22: r < (((arccos (z `1)) / (2 * PI)) + [\r/]) + 1 by INT_1:29, XXREAL_0:2;
A23: now__::_thesis:_not_(((arccos_(z_`1))_/_(2_*_PI))_+_[\r/])_+_1_=_r_+_1
assume (((arccos (z `1)) / (2 * PI)) + [\r/]) + 1 = r + 1 ; ::_thesis: contradiction
then CircleMap . r = CircleMap . ((arccos (z `1)) / (2 * PI)) by Th31;
hence contradiction by A3, A19, Th21; ::_thesis: verum
end;
((arccos (z `1)) / (2 * PI)) + [\r/] < 1 + r by A1, A7, XREAL_1:8;
then ((arccos (z `1)) / (2 * PI)) + [\r/] <= r by A21, XXREAL_1:4;
then (((arccos (z `1)) / (2 * PI)) + [\r/]) + 1 <= r + 1 by XREAL_1:6;
then (((arccos (z `1)) / (2 * PI)) + [\r/]) + 1 < r + 1 by A23, XXREAL_0:1;
then A24: (((arccos (z `1)) / (2 * PI)) + [\r/]) + 1 in ].r,(r + 1).[ by A22, XXREAL_1:4;
then (CircleMap | ].r,(r + 1).[) . ((((arccos (z `1)) / (2 * PI)) + [\r/]) + 1) = CircleMap . (((arccos (z `1)) / (2 * PI)) + ([\r/] + 1)) by FUNCT_1:49
.= CircleMap . ((arccos (z `1)) / (2 * PI)) by Th31 ;
hence y in rng (CircleMap r) by A2, A19, A24, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
end;
end;
Topen_unit_circle (CircleMap . r) = (Tunit_circle 2) | (([#] (Tunit_circle 2)) \ {(CircleMap . r)}) by Th22;
hence CircleMap r is continuous by TOPREALA:8; ::_thesis: verum
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 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)) ) )
proof
defpred S1[ set , set ] means ex x, y being real number st
( $1 = |[x,y]| & ( y >= 0 implies $2 = (arccos x) / (2 * PI) ) & ( y <= 0 implies $2 = 1 - ((arccos x) / (2 * PI)) ) );
reconsider A = R^1 ].0,1.[ as non empty Subset of R^1 ;
A1: the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) = A by PRE_TOPC:8;
A2: for x being Element of the carrier of (Topen_unit_circle c[10]) ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
proof
let x be Element of the carrier of (Topen_unit_circle c[10]); ::_thesis: ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
A3: the carrier of (Topen_unit_circle c[10]) = the carrier of (Tunit_circle 2) \ {c[10]} by Def10;
then A4: x in the carrier of (Tunit_circle 2) by XBOOLE_0:def_5;
A5: the carrier of (Tunit_circle 2) is Subset of (TOP-REAL 2) by TSEP_1:1;
then consider a, b being Real such that
A6: x = <*a,b*> by A4, EUCLID:51;
reconsider x1 = x as Point of (TOP-REAL 2) by A4, A5;
A7: b = x1 `2 by A6, EUCLID:52;
set k = arccos a;
A8: a = x1 `1 by A6, EUCLID:52;
then A9: - 1 <= a by Th26;
A10: 1 ^2 = |.x1.| ^2 by A4, Th12
.= (a ^2) + (b ^2) by A8, A7, JGRAPH_3:1 ;
A11: a <= 1 by A8, Th26;
then A12: 0 <= arccos a by A9, SIN_COS6:99;
A13: (arccos a) / (2 * PI) <= 1 / 2 by A9, A11, Lm22;
A14: not x in {c[10]} by A3, XBOOLE_0:def_5;
A15: now__::_thesis:_not_arccos_a_=_0
assume A16: arccos a = 0 ; ::_thesis: contradiction
then 1 - 1 = (1 + (b ^2)) - 1 by A9, A11, A10, SIN_COS6:96;
then A17: b = 0 ;
a = 1 by A9, A11, A16, SIN_COS6:96;
hence contradiction by A6, A14, A17, TARSKI:def_1; ::_thesis: verum
end;
A18: arccos a <= PI by A9, A11, SIN_COS6:99;
A19: 0 <= (arccos a) / (2 * PI) by A9, A11, Lm22;
percases ( b = 0 or b > 0 or b < 0 ) ;
supposeA20: b = 0 ; ::_thesis: ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
set y = (arccos a) / (2 * PI);
(arccos a) / (2 * PI) < 1 by A13, XXREAL_0:2;
then reconsider y = (arccos a) / (2 * PI) as Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) by A1, A19, A15, XXREAL_1:4;
take y ; ::_thesis: S1[x,y]
take a ; ::_thesis: ex y being real number st
( x = |[a,y]| & ( y >= 0 implies y = (arccos a) / (2 * PI) ) & ( y <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
take b ; ::_thesis: ( x = |[a,b]| & ( b >= 0 implies y = (arccos a) / (2 * PI) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus x = |[a,b]| by A6; ::_thesis: ( ( b >= 0 implies y = (arccos a) / (2 * PI) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus ( b >= 0 implies y = (arccos a) / (2 * PI) ) ; ::_thesis: ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) )
assume b <= 0 ; ::_thesis: y = 1 - ((arccos a) / (2 * PI))
A21: a <> 1 by A6, A14, A20, TARSKI:def_1;
hence y = (1 * PI) / (2 * PI) by A10, A20, SIN_COS6:93, SQUARE_1:41
.= 1 - (1 / 2) by XCMPLX_1:91
.= 1 - ((1 * PI) / (2 * PI)) by XCMPLX_1:91
.= 1 - ((arccos a) / (2 * PI)) by A10, A20, A21, SIN_COS6:93, SQUARE_1:41 ;
::_thesis: verum
end;
supposeA22: b > 0 ; ::_thesis: ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
set y = (arccos a) / (2 * PI);
(arccos a) / (2 * PI) < 1 by A13, XXREAL_0:2;
then reconsider y = (arccos a) / (2 * PI) as Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) by A1, A19, A15, XXREAL_1:4;
take y ; ::_thesis: S1[x,y]
take a ; ::_thesis: ex y being real number st
( x = |[a,y]| & ( y >= 0 implies y = (arccos a) / (2 * PI) ) & ( y <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
take b ; ::_thesis: ( x = |[a,b]| & ( b >= 0 implies y = (arccos a) / (2 * PI) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus ( x = |[a,b]| & ( b >= 0 implies y = (arccos a) / (2 * PI) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) ) by A6, A22; ::_thesis: verum
end;
supposeA23: b < 0 ; ::_thesis: ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
set y = 1 - ((arccos a) / (2 * PI));
A24: ((2 * PI) - (arccos a)) / (2 * PI) = ((2 * PI) / (2 * PI)) - ((arccos a) / (2 * PI)) by XCMPLX_1:120
.= 1 - ((arccos a) / (2 * PI)) by XCMPLX_1:60 ;
(2 * PI) - (arccos a) < (2 * PI) - 0 by A12, A15, XREAL_1:15;
then 1 - ((arccos a) / (2 * PI)) < (2 * PI) / (2 * PI) by A24, XREAL_1:74;
then A25: 1 - ((arccos a) / (2 * PI)) < 1 by XCMPLX_1:60;
1 * (arccos a) < 2 * PI by A18, XREAL_1:98;
then (arccos a) - (arccos a) < (2 * PI) - (arccos a) by XREAL_1:14;
then reconsider y = 1 - ((arccos a) / (2 * PI)) as Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) by A1, A24, A25, XXREAL_1:4;
take y ; ::_thesis: S1[x,y]
take a ; ::_thesis: ex y being real number st
( x = |[a,y]| & ( y >= 0 implies y = (arccos a) / (2 * PI) ) & ( y <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
take b ; ::_thesis: ( x = |[a,b]| & ( b >= 0 implies y = (arccos a) / (2 * PI) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus ( x = |[a,b]| & ( b >= 0 implies y = (arccos a) / (2 * PI) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) ) by A6, A23; ::_thesis: verum
end;
end;
end;
ex G being Function of (Topen_unit_circle c[10]),(R^1 | (R^1 ].0,(0 + p1).[)) st
for p being Point of (Topen_unit_circle c[10]) holds S1[p,G . p] from FUNCT_2:sch_3(A2);
hence 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)) ) ) ; ::_thesis: verum
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 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
proof
let f, g be Function of (Topen_unit_circle c[10]),(R^1 | (R^1 ].0,1.[)); ::_thesis: ( ( for p being Point of (Topen_unit_circle c[10]) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies f . p = (arccos x) / (2 * PI) ) & ( y <= 0 implies f . 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 g . p = (arccos x) / (2 * PI) ) & ( y <= 0 implies g . p = 1 - ((arccos x) / (2 * PI)) ) ) ) implies f = g )
assume that
A26: for p being Point of (Topen_unit_circle c[10]) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies f . p = (arccos x) / (2 * PI) ) & ( y <= 0 implies f . p = 1 - ((arccos x) / (2 * PI)) ) ) and
A27: for p being Point of (Topen_unit_circle c[10]) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies g . p = (arccos x) / (2 * PI) ) & ( y <= 0 implies g . p = 1 - ((arccos x) / (2 * PI)) ) ) ; ::_thesis: f = g
now__::_thesis:_for_p_being_Point_of_(Topen_unit_circle_c[10])_holds_f_._p_=_g_._p
let p be Point of (Topen_unit_circle c[10]); ::_thesis: f . p = g . p
A28: ex x2, y2 being real number st
( p = |[x2,y2]| & ( y2 >= 0 implies g . p = (arccos x2) / (2 * PI) ) & ( y2 <= 0 implies g . p = 1 - ((arccos x2) / (2 * PI)) ) ) by A27;
ex x1, y1 being real number st
( p = |[x1,y1]| & ( y1 >= 0 implies f . p = (arccos x1) / (2 * PI) ) & ( y1 <= 0 implies f . p = 1 - ((arccos x1) / (2 * PI)) ) ) by A26;
hence f . p = g . p by A28, SPPOL_2:1; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
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 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: :: TOPREALB:def 14
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)) ) )
proof
defpred S1[ set , set ] means ex x, y being real number st
( $1 = |[x,y]| & ( y >= 0 implies $2 = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies $2 = 1 - ((arccos x) / (2 * PI)) ) );
reconsider A1 = R^1 ].(1 / 2),((1 / 2) + p1).[ as non empty Subset of R^1 ;
A1: the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) = A1 by PRE_TOPC:8;
A2: for x being Element of the carrier of (Topen_unit_circle c[-10]) ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S1[x,y]
proof
let x be Element of the carrier of (Topen_unit_circle c[-10]); ::_thesis: ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S1[x,y]
A3: the carrier of (Topen_unit_circle c[-10]) = the carrier of (Tunit_circle 2) \ {c[-10]} by Def10;
then A4: x in the carrier of (Tunit_circle 2) by XBOOLE_0:def_5;
A5: not x in {c[-10]} by A3, XBOOLE_0:def_5;
A6: the carrier of (Tunit_circle 2) is Subset of (TOP-REAL 2) by TSEP_1:1;
then consider a, b being Real such that
A7: x = <*a,b*> by A4, EUCLID:51;
reconsider x1 = x as Point of (TOP-REAL 2) by A4, A6;
A8: b = x1 `2 by A7, EUCLID:52;
set k = arccos a;
A9: a = x1 `1 by A7, EUCLID:52;
then A10: - 1 <= a by Th26;
A11: a <= 1 by A9, Th26;
then A12: (arccos a) / (2 * PI) <= 1 / 2 by A10, Lm22;
A13: 1 ^2 = |.x1.| ^2 by A4, Th12
.= (a ^2) + (b ^2) by A9, A8, JGRAPH_3:1 ;
A14: now__::_thesis:_not_arccos_a_=_PI
assume A15: arccos a = PI ; ::_thesis: contradiction
then 1 - 1 = (((- 1) ^2) + (b ^2)) - 1 by A10, A11, A13, SIN_COS6:98
.= ((1 ^2) + (b ^2)) - 1 ;
then A16: b = 0 ;
a = - 1 by A10, A11, A15, SIN_COS6:98;
hence contradiction by A7, A5, A16, TARSKI:def_1; ::_thesis: verum
end;
A17: now__::_thesis:_not_(arccos_a)_/_(2_*_PI)_=_1_/_2
assume (arccos a) / (2 * PI) = 1 / 2 ; ::_thesis: contradiction
then ((arccos a) / (2 * PI)) * 2 = (1 / 2) * 2 ;
then (arccos a) / PI = 1 by XCMPLX_1:92;
hence contradiction by A14, XCMPLX_1:58; ::_thesis: verum
end;
A18: 0 <= (arccos a) / (2 * PI) by A10, A11, Lm22;
A19: now__::_thesis:_for_y_being_real_number_st_y_=_1_+_((arccos_a)_/_(2_*_PI))_holds_
y_is_Element_of_the_carrier_of_(R^1_|_(R^1_].(1_/_2),((1_/_2)_+_p1).[))
let y be real number ; ::_thesis: ( y = 1 + ((arccos a) / (2 * PI)) implies y is Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) )
assume A20: y = 1 + ((arccos a) / (2 * PI)) ; ::_thesis: y is Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[))
then A21: y <> 1 + (1 / 2) by A17;
1 + 0 <= y by A18, A20, XREAL_1:6;
then A22: 1 / 2 < y by XXREAL_0:2;
y <= 1 + (1 / 2) by A12, A20, XREAL_1:6;
then y < 3 / 2 by A21, XXREAL_0:1;
hence y is Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by A1, A22, XXREAL_1:4; ::_thesis: verum
end;
percases ( b = 0 or b > 0 or b < 0 ) ;
supposeA23: b = 0 ; ::_thesis: ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S1[x,y]
reconsider y = 1 + ((arccos a) / (2 * PI)) as Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by A19;
take y ; ::_thesis: S1[x,y]
take a ; ::_thesis: ex y being real number st
( x = |[a,y]| & ( y >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( y <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
take b ; ::_thesis: ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus x = |[a,b]| by A7; ::_thesis: ( ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
a <> - 1 by A7, A5, A23, TARSKI:def_1;
then a = 1 by A13, A23, SQUARE_1:41;
hence ( ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) ) by SIN_COS6:95; ::_thesis: verum
end;
supposeA24: b > 0 ; ::_thesis: ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S1[x,y]
reconsider y = 1 + ((arccos a) / (2 * PI)) as Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by A19;
take y ; ::_thesis: S1[x,y]
take a ; ::_thesis: ex y being real number st
( x = |[a,y]| & ( y >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( y <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
take b ; ::_thesis: ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) ) by A7, A24; ::_thesis: verum
end;
supposeA25: b < 0 ; ::_thesis: ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S1[x,y]
set y = 1 - ((arccos a) / (2 * PI));
A26: 1 - ((arccos a) / (2 * PI)) <> 1 / 2 by A17;
1 - ((arccos a) / (2 * PI)) >= 1 - (1 / 2) by A12, XREAL_1:13;
then A27: 1 / 2 < 1 - ((arccos a) / (2 * PI)) by A26, XXREAL_0:1;
1 - 0 >= 1 - ((arccos a) / (2 * PI)) by A18, XREAL_1:13;
then 1 - ((arccos a) / (2 * PI)) < 3 / 2 by XXREAL_0:2;
then reconsider y = 1 - ((arccos a) / (2 * PI)) as Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by A1, A27, XXREAL_1:4;
take y ; ::_thesis: S1[x,y]
take a ; ::_thesis: ex y being real number st
( x = |[a,y]| & ( y >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( y <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
take b ; ::_thesis: ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) ) by A7, A25; ::_thesis: verum
end;
end;
end;
ex G being Function of (Topen_unit_circle c[-10]),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st
for p being Point of (Topen_unit_circle c[-10]) holds S1[p,G . p] from FUNCT_2:sch_3(A2);
hence 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)) ) ) ; ::_thesis: verum
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 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
proof
let f, g be Function of (Topen_unit_circle c[-10]),(R^1 | (R^1 ].(1 / 2),(3 / 2).[)); ::_thesis: ( ( for p being Point of (Topen_unit_circle c[-10]) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies f . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies f . 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 g . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies g . p = 1 - ((arccos x) / (2 * PI)) ) ) ) implies f = g )
assume that
A28: for p being Point of (Topen_unit_circle c[-10]) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies f . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies f . p = 1 - ((arccos x) / (2 * PI)) ) ) and
A29: for p being Point of (Topen_unit_circle c[-10]) ex x, y being real number st
( p = |[x,y]| & ( y >= 0 implies g . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies g . p = 1 - ((arccos x) / (2 * PI)) ) ) ; ::_thesis: f = g
now__::_thesis:_for_p_being_Point_of_(Topen_unit_circle_c[-10])_holds_f_._p_=_g_._p
let p be Point of (Topen_unit_circle c[-10]); ::_thesis: f . p = g . p
A30: ex x2, y2 being real number st
( p = |[x2,y2]| & ( y2 >= 0 implies g . p = 1 + ((arccos x2) / (2 * PI)) ) & ( y2 <= 0 implies g . p = 1 - ((arccos x2) / (2 * PI)) ) ) by A29;
ex x1, y1 being real number st
( p = |[x1,y1]| & ( y1 >= 0 implies f . p = 1 + ((arccos x1) / (2 * PI)) ) & ( y1 <= 0 implies f . p = 1 - ((arccos x1) / (2 * PI)) ) ) by A28;
hence f . p = g . p by A30, SPPOL_2:1; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
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 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 Th42: :: TOPREALB:42
(CircleMap (R^1 0)) " = Circle2IntervalR
proof
reconsider A = R^1 ].0,1.[ as non empty Subset of R^1 ;
set f = CircleMap (R^1 0);
set Y = the carrier of (R^1 | A);
reconsider f = CircleMap (R^1 0) as Function of (R^1 | A),(Topen_unit_circle c[10]) by Th32;
reconsider r0 = 0 as Point of R^1 by TOPMETR:17;
set F = AffineMap ((2 * PI),0);
A1: dom (id the carrier of (R^1 | A)) = the carrier of (R^1 | A) by RELAT_1:45;
CircleMap . r0 = c[10] by Th32;
then A2: rng f = the carrier of (Topen_unit_circle c[10]) by FUNCT_2:def_3;
A3: the carrier of (R^1 | A) = A by PRE_TOPC:8;
A4: now__::_thesis:_for_a_being_set_st_a_in_dom_(Circle2IntervalR_*_f)_holds_
(Circle2IntervalR_*_f)_._a_=_(id_the_carrier_of_(R^1_|_A))_._a
let a be set ; ::_thesis: ( a in dom (Circle2IntervalR * f) implies (Circle2IntervalR * f) . a = (id the carrier of (R^1 | A)) . a )
assume A5: a in dom (Circle2IntervalR * f) ; ::_thesis: (Circle2IntervalR * f) . a = (id the carrier of (R^1 | A)) . a
then reconsider b = a as Point of (R^1 | A) ;
reconsider c = b as Real by XREAL_0:def_1;
consider x, y being real number such that
A6: f . b = |[x,y]| and
A7: ( y >= 0 implies Circle2IntervalR . (f . b) = (arccos x) / (2 * PI) ) and
A8: ( y <= 0 implies Circle2IntervalR . (f . b) = 1 - ((arccos x) / (2 * PI)) ) by Def13;
A9: f . b = CircleMap . b by A3, FUNCT_1:49
.= |[((cos * (AffineMap ((2 * PI),0))) . c),((sin * (AffineMap ((2 * PI),0))) . c)]| by Lm20 ;
then y = (sin * (AffineMap ((2 * PI),0))) . c by A6, SPPOL_2:1;
then A10: y = sin . ((AffineMap ((2 * PI),0)) . c) by FUNCT_2:15;
x = (cos * (AffineMap ((2 * PI),0))) . c by A6, A9, SPPOL_2:1;
then x = cos . ((AffineMap ((2 * PI),0)) . c) by FUNCT_2:15;
then A11: x = cos ((AffineMap ((2 * PI),0)) . c) by SIN_COS:def_19;
A12: c < 1 by A3, XXREAL_1:4;
A13: (AffineMap ((2 * PI),0)) . c = ((2 * PI) * c) + 0 by FCONT_1:def_4;
then A14: ((AffineMap ((2 * PI),0)) . c) / ((2 * PI) * 1) = c / 1 by XCMPLX_1:91;
A15: (AffineMap ((2 * PI),0)) . (1 - c) = ((2 * PI) * (1 - c)) + 0 by FCONT_1:def_4;
then A16: ((AffineMap ((2 * PI),0)) . (1 - c)) / ((2 * PI) * 1) = (1 - c) / 1 by XCMPLX_1:91;
A17: now__::_thesis:_Circle2IntervalR_._(f_._b)_=_b
percases ( y >= 0 or y < 0 ) ;
supposeA18: y >= 0 ; ::_thesis: Circle2IntervalR . (f . b) = b
then not (AffineMap ((2 * PI),0)) . c in ].PI,(2 * PI).[ by A10, COMPTRIG:9;
then ( (AffineMap ((2 * PI),0)) . c <= PI or (AffineMap ((2 * PI),0)) . c >= 2 * PI ) by XXREAL_1:4;
then ( ((AffineMap ((2 * PI),0)) . c) / (2 * PI) <= (1 * PI) / (2 * PI) or ((AffineMap ((2 * PI),0)) . c) / (2 * PI) >= (2 * PI) / (2 * PI) ) by XREAL_1:72;
then c <= 1 / 2 by A14, A12, XCMPLX_1:60, XCMPLX_1:91;
then A19: (2 * PI) * c <= (2 * PI) * (1 / 2) by XREAL_1:64;
0 <= c by A3, XXREAL_1:4;
hence Circle2IntervalR . (f . b) = ((AffineMap ((2 * PI),0)) . c) / (2 * PI) by A7, A11, A13, A18, A19, SIN_COS6:92
.= b by A13, XCMPLX_1:89 ;
::_thesis: verum
end;
supposeA20: y < 0 ; ::_thesis: Circle2IntervalR . (f . b) = b
then not (AffineMap ((2 * PI),0)) . c in [.0,PI.] by A10, COMPTRIG:8;
then ( (AffineMap ((2 * PI),0)) . c < 0 or (AffineMap ((2 * PI),0)) . c > PI ) by XXREAL_1:1;
then ( ((AffineMap ((2 * PI),0)) . c) / (2 * PI) < 0 / (2 * PI) or ((AffineMap ((2 * PI),0)) . c) / (2 * PI) > (1 * PI) / (2 * PI) ) by XREAL_1:74;
then ( c < 0 or c > 1 / 2 ) by A14, XCMPLX_1:91;
then 1 - c < 1 - (1 / 2) by A3, XREAL_1:15, XXREAL_1:4;
then A21: (2 * PI) * (1 - c) <= (2 * PI) * (1 / 2) by XREAL_1:64;
A22: 1 - c > 1 - 1 by A12, XREAL_1:15;
cos . ((AffineMap ((2 * PI),0)) . (1 - c)) = cos ((- ((2 * PI) * c)) + ((2 * PI) * 1)) by A15, SIN_COS:def_19
.= cos (- ((2 * PI) * c)) by COMPLEX2:9
.= cos ((2 * PI) * c) by SIN_COS:31 ;
then arccos x = arccos (cos ((AffineMap ((2 * PI),0)) . (1 - c))) by A11, A13, SIN_COS:def_19
.= (AffineMap ((2 * PI),0)) . (1 - c) by A15, A22, A21, SIN_COS6:92 ;
hence Circle2IntervalR . (f . b) = b by A8, A16, A20; ::_thesis: verum
end;
end;
end;
thus (Circle2IntervalR * f) . a = Circle2IntervalR . (f . b) by A5, FUNCT_1:12
.= (id the carrier of (R^1 | A)) . a by A17, FUNCT_1:18 ; ::_thesis: verum
end;
dom (Circle2IntervalR * f) = the carrier of (R^1 | A) by FUNCT_2:def_1;
then Circle2IntervalR = f " by A2, A1, A4, FUNCT_1:2, FUNCT_2:30;
hence (CircleMap (R^1 0)) " = Circle2IntervalR by TOPS_2:def_4; ::_thesis: verum
end;
theorem Th43: :: TOPREALB:43
(CircleMap (R^1 (1 / 2))) " = Circle2IntervalL
proof
reconsider A1 = R^1 ].(1 / 2),((1 / 2) + p1).[ as non empty Subset of R^1 ;
set f = CircleMap (R^1 (1 / 2));
set Y = the carrier of (R^1 | A1);
reconsider f = CircleMap (R^1 (1 / 2)) as Function of (R^1 | A1),(Topen_unit_circle c[-10]) by Lm19;
set G = AffineMap ((2 * PI),0);
A1: dom (id the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[))) = the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by RELAT_1:45;
A2: rng f = the carrier of (Topen_unit_circle c[-10]) by Lm19, FUNCT_2:def_3;
A3: the carrier of (R^1 | A1) = A1 by PRE_TOPC:8;
A4: now__::_thesis:_for_a_being_set_st_a_in_dom_(Circle2IntervalL_*_f)_holds_
(Circle2IntervalL_*_f)_._a_=_(id_the_carrier_of_(R^1_|_(R^1_].(1_/_2),((1_/_2)_+_p1).[)))_._a
let a be set ; ::_thesis: ( a in dom (Circle2IntervalL * f) implies (Circle2IntervalL * f) . a = (id the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[))) . a )
assume A5: a in dom (Circle2IntervalL * f) ; ::_thesis: (Circle2IntervalL * f) . a = (id the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[))) . a
then reconsider b = a as Point of (R^1 | A1) ;
reconsider c = b as Real by XREAL_0:def_1;
consider x, y being real number such that
A6: f . b = |[x,y]| and
A7: ( y >= 0 implies Circle2IntervalL . (f . b) = 1 + ((arccos x) / (2 * PI)) ) and
A8: ( y <= 0 implies Circle2IntervalL . (f . b) = 1 - ((arccos x) / (2 * PI)) ) by Def14;
A9: f . b = CircleMap . b by A3, FUNCT_1:49
.= |[(cos ((2 * PI) * c)),(sin ((2 * PI) * c))]| by Def11 ;
then A10: y = sin ((2 * PI) * c) by A6, SPPOL_2:1;
A11: 1 / 2 < c by A3, XXREAL_1:4;
then (2 * PI) * (1 / 2) < (2 * PI) * c by XREAL_1:68;
then A12: PI + ((2 * PI) * 0) < (2 * PI) * c ;
A13: c < 3 / 2 by A3, XXREAL_1:4;
then c - 1 < (3 / 2) - 1 by XREAL_1:9;
then A14: (2 * PI) * (c - 1) <= (2 * PI) * (1 / 2) by XREAL_1:64;
(2 * PI) * c <= (2 * PI) * ((1 / 2) + 1) by A13, XREAL_1:64;
then A15: (2 * PI) * c <= PI + ((2 * PI) * 1) ;
A16: (AffineMap ((2 * PI),0)) . (1 - c) = ((2 * PI) * (1 - c)) + 0 by FCONT_1:def_4;
then A17: ((AffineMap ((2 * PI),0)) . (1 - c)) / ((2 * PI) * 1) = (1 - c) / 1 by XCMPLX_1:91;
A18: x = cos ((2 * PI) * c) by A6, A9, SPPOL_2:1
.= cos (((2 * PI) * c) + ((2 * PI) * (- 1))) by COMPLEX2:9
.= cos ((2 * PI) * (c - 1)) ;
A19: now__::_thesis:_Circle2IntervalL_._(f_._b)_=_b
percases ( c >= 1 or c < 1 ) ;
supposeA20: c >= 1 ; ::_thesis: Circle2IntervalL . (f . b) = b
then A21: 1 - 1 <= c - 1 by XREAL_1:9;
(2 * PI) * c >= (2 * PI) * 1 by A20, XREAL_1:64;
hence Circle2IntervalL . (f . b) = 1 + (((2 * PI) * (c - 1)) / (2 * PI)) by A7, A18, A10, A14, A15, A21, SIN_COS6:16, SIN_COS6:92
.= 1 + (c - 1) by XCMPLX_1:89
.= b ;
::_thesis: verum
end;
supposeA22: c < 1 ; ::_thesis: Circle2IntervalL . (f . b) = b
then (2 * PI) * c < (2 * PI) * 1 by XREAL_1:68;
then A23: (2 * PI) * c < (2 * PI) + ((2 * PI) * 0) ;
1 - c < 1 - (1 / 2) by A11, XREAL_1:15;
then A24: (2 * PI) * (1 - c) <= (2 * PI) * (1 / 2) by XREAL_1:64;
A25: 1 - 1 <= 1 - c by A22, XREAL_1:15;
arccos x = arccos (cos ((2 * PI) * c)) by A6, A9, SPPOL_2:1
.= arccos (cos (- ((2 * PI) * c))) by SIN_COS:31
.= arccos (cos (((2 * PI) * (- c)) + ((2 * PI) * 1))) by COMPLEX2:9
.= (AffineMap ((2 * PI),0)) . (1 - c) by A16, A25, A24, SIN_COS6:92 ;
hence Circle2IntervalL . (f . b) = b by A8, A10, A17, A12, A23, SIN_COS6:12; ::_thesis: verum
end;
end;
end;
thus (Circle2IntervalL * f) . a = Circle2IntervalL . (f . b) by A5, FUNCT_1:12
.= (id the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[))) . a by A19, FUNCT_1:18 ; ::_thesis: verum
end;
dom (Circle2IntervalL * f) = the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by FUNCT_2:def_1;
then Circle2IntervalL = f " by A2, A1, A4, FUNCT_1:2, FUNCT_2:30;
hence (CircleMap (R^1 (1 / 2))) " = Circle2IntervalL by TOPS_2:def_4; ::_thesis: verum
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
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((AffineMap ((1 / (2 * PI)),0)) | (R^1 E)) or y in ].0,1.[ )
assume y in rng ((AffineMap ((1 / (2 * PI)),0)) | (R^1 E)) ; ::_thesis: y in ].0,1.[
then consider x being set such that
A1: x in dom ((AffineMap ((1 / (2 * PI)),0)) | (R^1 E)) and
A2: ((AffineMap ((1 / (2 * PI)),0)) | (R^1 E)) . x = y by FUNCT_1:def_3;
reconsider x = x as Real by A1;
A3: y = (AffineMap ((1 / (2 * PI)),0)) . x by A1, A2, Lm28, FUNCT_1:49
.= ((1 / (2 * PI)) * x) + 0 by FCONT_1:def_4
.= x / (2 * PI) by XCMPLX_1:99 ;
then reconsider y = y as Real ;
x <= PI by A1, Lm28, XXREAL_1:2;
then y <= (1 * PI) / (2 * PI) by A3, XREAL_1:72;
then y <= 1 / 2 by XCMPLX_1:91;
then A4: y < 1 by XXREAL_0:2;
0 < x by A1, Lm28, XXREAL_1:2;
hence y in ].0,1.[ by A3, A4, XXREAL_1:4; ::_thesis: verum
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
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((R^1 (AffineMap ((- 1),1))) | (R^1 ].0,1.[)) or y in ].0,1.[ )
assume y in rng ((R^1 (AffineMap ((- 1),1))) | (R^1 ].0,1.[)) ; ::_thesis: y in ].0,1.[
then consider x being set such that
A1: x in dom ((R^1 (AffineMap ((- 1),1))) | (R^1 ].0,1.[)) and
A2: ((R^1 (AffineMap ((- 1),1))) | (R^1 ].0,1.[)) . x = y by FUNCT_1:def_3;
reconsider x = x as Real by A1, Lm33;
0 < x by A1, Lm33, XXREAL_1:4;
then A3: 1 - x < 1 - 0 by XREAL_1:15;
x < 1 by A1, Lm33, XXREAL_1:4;
then A4: 1 - 1 < 1 - x by XREAL_1:15;
y = (AffineMap ((- 1),1)) . x by A1, A2, Lm33, FUNCT_1:49
.= ((- 1) * x) + 1 by FCONT_1:def_4 ;
hence y in ].0,1.[ by A4, A3, XXREAL_1:4; ::_thesis: verum
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
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (ac | (R^1 Q)) or y in E )
assume A1: y in rng (ac | (R^1 Q)) ; ::_thesis: y in E
then consider x being set such that
A2: x in dom (ac | (R^1 Q)) and
A3: (ac | (R^1 Q)) . x = y by FUNCT_1:def_3;
reconsider x = x as Real by A2, Lm36;
A4: - 1 <= x by A2, Lm36, XXREAL_1:3;
A5: x < 1 by A2, Lm36, XXREAL_1:3;
A6: rng (ac | (R^1 Q)) c= rng ac by RELAT_1:70;
then y in [.0,PI.] by A1, SIN_COS6:85;
then reconsider y = y as Real ;
A7: y <= PI by A1, A6, SIN_COS6:85, XXREAL_1:1;
y = arccos . x by A2, A3, Lm36, FUNCT_1:49
.= arccos x by SIN_COS6:def_4 ;
then A8: y <> 0 by A4, A5, SIN_COS6:96;
0 <= y by A1, A6, SIN_COS6:85, XXREAL_1:1;
hence y in E by A7, A8, XXREAL_1:2; ::_thesis: verum
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
reconsider c1 = c[-10] as Point of (TOP-REAL 2) ;
let aX1 be Subset of (Topen_unit_circle c[10]); ::_thesis: ( aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) } implies Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is continuous )
assume A1: aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) } ; ::_thesis: Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is continuous
A2: c1 `2 = 0 by EUCLID:52;
c[-10] is Point of (Topen_unit_circle c[10]) by Lm15, Th23;
then c[-10] in aX1 by A1, A2;
then reconsider aX1 = aX1 as non empty Subset of (Topen_unit_circle c[10]) ;
set X1 = (Topen_unit_circle c[10]) | aX1;
A3: the carrier of (Tunit_circle 2) is Subset of (TOP-REAL 2) by TSEP_1:1;
[#] ((Topen_unit_circle c[10]) | aX1) is Subset of (Tunit_circle 2) by Lm9;
then reconsider B = [#] ((Topen_unit_circle c[10]) | aX1) as non empty Subset of (TOP-REAL 2) by A3, XBOOLE_1:1;
set f = p | B;
A4: dom (p | B) = B by Lm40, RELAT_1:62;
A5: aX1 = the carrier of ((Topen_unit_circle c[10]) | aX1) by PRE_TOPC:8;
A6: rng (p | B) c= Q
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (p | B) or y in Q )
assume y in rng (p | B) ; ::_thesis: y in Q
then consider x being set such that
A7: x in dom (p | B) and
A8: (p | B) . x = y by FUNCT_1:def_3;
consider q being Point of (TOP-REAL 2) such that
A9: q = x and
A10: q in the carrier of (Topen_unit_circle c[10]) and
0 <= q `2 by A1, A5, A4, A7;
A11: - 1 <= q `1 by A10, Th27;
A12: q `1 < 1 by A10, Th27;
y = p . x by A4, A7, A8, FUNCT_1:49
.= q `1 by A9, PSCOMP_1:def_5 ;
hence y in Q by A11, A12, XXREAL_1:3; ::_thesis: verum
end;
the carrier of ((TOP-REAL 2) | B) = B by PRE_TOPC:8;
then reconsider f = p | B as Function of ((TOP-REAL 2) | B),(R^1 | (R^1 Q)) by A4, A6, Lm25, FUNCT_2:2;
(Topen_unit_circle c[10]) | aX1 is SubSpace of Tunit_circle 2 by TSEP_1:7;
then (Topen_unit_circle c[10]) | aX1 is SubSpace of TOP-REAL 2 by TSEP_1:7;
then A13: (TOP-REAL 2) | B = (Topen_unit_circle c[10]) | aX1 by PRE_TOPC:def_5;
A14: for a being Point of ((Topen_unit_circle c[10]) | aX1) holds (Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1)) . a = (Af * (c * f)) . a
proof
let a be Point of ((Topen_unit_circle c[10]) | aX1); ::_thesis: (Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1)) . a = (Af * (c * f)) . a
reconsider b = a as Point of (Topen_unit_circle c[10]) by PRE_TOPC:25;
consider x, y being real number such that
A15: b = |[x,y]| and
A16: ( y >= 0 implies Circle2IntervalR . b = (arccos x) / (2 * PI) ) and
( y <= 0 implies Circle2IntervalR . b = 1 - ((arccos x) / (2 * PI)) ) by Def13;
A17: |[x,y]| `1 < 1 by A15, Th27;
A18: |[x,y]| `1 = x by EUCLID:52;
- 1 <= |[x,y]| `1 by A15, Th26;
then A19: x in Q by A18, A17, XXREAL_1:3;
then arccos . x = c . x by FUNCT_1:49;
then A20: arccos . x in rng c by A19, Lm36, FUNCT_1:def_3;
a in aX1 by A5;
then ex q being Point of (TOP-REAL 2) st
( a = q & q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) by A1;
hence (Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1)) . a = (arccos x) / (2 * PI) by A15, A16, EUCLID:52, FUNCT_1:49
.= (arccos . x) / (2 * PI) by SIN_COS6:def_4
.= ((1 / (2 * PI)) * (arccos . x)) + 0 by XCMPLX_1:99
.= (AffineMap ((1 / (2 * PI)),0)) . (arccos . x) by FCONT_1:def_4
.= Af . (arccos . x) by A20, Lm37, FUNCT_1:49
.= Af . (c . x) by A19, FUNCT_1:49
.= Af . (c . (|[x,y]| `1)) by EUCLID:52
.= Af . (c . (p . a)) by A15, PSCOMP_1:def_5
.= Af . (c . (f . a)) by FUNCT_1:49
.= Af . ((c * f) . a) by A13, FUNCT_2:15
.= (Af * (c * f)) . a by A13, FUNCT_2:15 ;
::_thesis: verum
end;
f is continuous by Lm41, TOPREALA:8;
hence Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is continuous by A13, A14, Lm39, FUNCT_2:63; ::_thesis: verum
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
reconsider c1 = c[-10] as Point of (TOP-REAL 2) ;
let aX1 be Subset of (Topen_unit_circle c[10]); ::_thesis: ( aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) } implies Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is continuous )
assume A1: aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) } ; ::_thesis: Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is continuous
A2: c1 `2 = 0 by EUCLID:52;
c[-10] is Point of (Topen_unit_circle c[10]) by Lm15, Th23;
then c[-10] in aX1 by A1, A2;
then reconsider aX1 = aX1 as non empty Subset of (Topen_unit_circle c[10]) ;
set X1 = (Topen_unit_circle c[10]) | aX1;
A3: the carrier of (Tunit_circle 2) is Subset of (TOP-REAL 2) by TSEP_1:1;
[#] ((Topen_unit_circle c[10]) | aX1) is Subset of (Tunit_circle 2) by Lm9;
then reconsider B = [#] ((Topen_unit_circle c[10]) | aX1) as non empty Subset of (TOP-REAL 2) by A3, XBOOLE_1:1;
set f = p | B;
A4: dom (p | B) = B by Lm40, RELAT_1:62;
A5: aX1 = the carrier of ((Topen_unit_circle c[10]) | aX1) by PRE_TOPC:8;
A6: rng (p | B) c= Q
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (p | B) or y in Q )
assume y in rng (p | B) ; ::_thesis: y in Q
then consider x being set such that
A7: x in dom (p | B) and
A8: (p | B) . x = y by FUNCT_1:def_3;
consider q being Point of (TOP-REAL 2) such that
A9: q = x and
A10: q in the carrier of (Topen_unit_circle c[10]) and
0 >= q `2 by A1, A5, A4, A7;
A11: - 1 <= q `1 by A10, Th27;
A12: q `1 < 1 by A10, Th27;
y = p . x by A4, A7, A8, FUNCT_1:49
.= q `1 by A9, PSCOMP_1:def_5 ;
hence y in Q by A11, A12, XXREAL_1:3; ::_thesis: verum
end;
A13: the carrier of ((TOP-REAL 2) | B) = B by PRE_TOPC:8;
then reconsider f = p | B as Function of ((TOP-REAL 2) | B),(R^1 | (R^1 Q)) by A4, A6, Lm25, FUNCT_2:2;
(Topen_unit_circle c[10]) | aX1 is SubSpace of Tunit_circle 2 by TSEP_1:7;
then (Topen_unit_circle c[10]) | aX1 is SubSpace of TOP-REAL 2 by TSEP_1:7;
then A14: (TOP-REAL 2) | B = (Topen_unit_circle c[10]) | aX1 by PRE_TOPC:def_5;
A15: for a being Point of ((Topen_unit_circle c[10]) | aX1) holds (Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1)) . a = (L * (Af * (c * f))) . a
proof
let a be Point of ((Topen_unit_circle c[10]) | aX1); ::_thesis: (Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1)) . a = (L * (Af * (c * f))) . a
reconsider b = a as Point of (Topen_unit_circle c[10]) by PRE_TOPC:25;
consider x, y being real number such that
A16: b = |[x,y]| and
( y >= 0 implies Circle2IntervalR . b = (arccos x) / (2 * PI) ) and
A17: ( y <= 0 implies Circle2IntervalR . b = 1 - ((arccos x) / (2 * PI)) ) by Def13;
A18: |[x,y]| `1 < 1 by A16, Th27;
dom (Af * (c * f)) = the carrier of ((TOP-REAL 2) | B) by FUNCT_2:def_1;
then A19: (Af * (c * f)) . a in rng (Af * (c * f)) by A13, FUNCT_1:def_3;
then (Af * (c * f)) . a in ].0,1.[ by Lm27;
then reconsider r = (Af * (c * f)) . a as Real ;
a in aX1 by A5;
then A20: ex q being Point of (TOP-REAL 2) st
( a = q & q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) by A1;
A21: |[x,y]| `1 = x by EUCLID:52;
- 1 <= |[x,y]| `1 by A16, Th26;
then A22: x in Q by A21, A18, XXREAL_1:3;
then arccos . x = c . x by FUNCT_1:49;
then A23: arccos . x in rng c by A22, Lm36, FUNCT_1:def_3;
(arccos x) / (2 * PI) = (arccos . x) / (2 * PI) by SIN_COS6:def_4
.= ((1 / (2 * PI)) * (arccos . x)) + 0 by XCMPLX_1:99
.= (AffineMap ((1 / (2 * PI)),0)) . (arccos . x) by FCONT_1:def_4
.= Af . (arccos . x) by A23, Lm37, FUNCT_1:49
.= Af . (c . x) by A22, FUNCT_1:49
.= Af . (c . (|[x,y]| `1)) by EUCLID:52
.= Af . (c . (p . a)) by A16, PSCOMP_1:def_5
.= Af . (c . (f . a)) by FUNCT_1:49
.= Af . ((c * f) . a) by A14, FUNCT_2:15
.= (Af * (c * f)) . a by A14, FUNCT_2:15 ;
hence (Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1)) . a = ((- 1) * r) + 1 by A16, A17, A20, EUCLID:52, FUNCT_1:49
.= (R^1 (AffineMap ((- 1),1))) . r by FCONT_1:def_4
.= L . r by A19, Lm27, FUNCT_1:49
.= (L * (Af * (c * f))) . a by A14, FUNCT_2:15 ;
::_thesis: verum
end;
f is continuous by Lm41, TOPREALA:8;
hence Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is continuous by A14, A15, Lm39, FUNCT_2:63; ::_thesis: verum
end;
Lm44: for p being Point of (Topen_unit_circle c[10]) st p = c[-10] holds
Circle2IntervalR is_continuous_at p
proof
reconsider c1 = c[-10] as Point of (TOP-REAL 2) ;
set aX2 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) } ;
set aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) } ;
A1: { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) } c= the carrier of (Topen_unit_circle c[10])
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) } or x in the carrier of (Topen_unit_circle c[10]) )
assume x in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) } ; ::_thesis: x in the carrier of (Topen_unit_circle c[10])
then ex q being Point of (TOP-REAL 2) st
( x = q & q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) ;
hence x in the carrier of (Topen_unit_circle c[10]) ; ::_thesis: verum
end;
A2: { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) } c= the carrier of (Topen_unit_circle c[10])
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) } or x in the carrier of (Topen_unit_circle c[10]) )
assume x in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) } ; ::_thesis: x in the carrier of (Topen_unit_circle c[10])
then ex q being Point of (TOP-REAL 2) st
( x = q & q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) ;
hence x in the carrier of (Topen_unit_circle c[10]) ; ::_thesis: verum
end;
A3: Topen_unit_circle c[10] is SubSpace of Topen_unit_circle c[10] by TSEP_1:2;
A4: c1 `2 = 0 by EUCLID:52;
A5: c[-10] is Point of (Topen_unit_circle c[10]) by Lm15, Th23;
then A6: c[-10] in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) } by A4;
A7: c[-10] in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) } by A4, A5;
then reconsider aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 <= q `2 ) } , aX2 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[10]) & 0 >= q `2 ) } as non empty Subset of (Topen_unit_circle c[10]) by A1, A2, A6;
set X1 = (Topen_unit_circle c[10]) | aX1;
let p be Point of (Topen_unit_circle c[10]); ::_thesis: ( p = c[-10] implies Circle2IntervalR is_continuous_at p )
assume A8: p = c[-10] ; ::_thesis: Circle2IntervalR is_continuous_at p
reconsider x1 = p as Point of ((Topen_unit_circle c[10]) | aX1) by A8, A6, PRE_TOPC:8;
set X2 = (Topen_unit_circle c[10]) | aX2;
reconsider x2 = p as Point of ((Topen_unit_circle c[10]) | aX2) by A8, A7, PRE_TOPC:8;
A9: the carrier of ((Topen_unit_circle c[10]) | aX2) = aX2 by PRE_TOPC:8;
the carrier of (Topen_unit_circle c[10]) c= aX1 \/ aX2
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of (Topen_unit_circle c[10]) or a in aX1 \/ aX2 )
assume A10: a in the carrier of (Topen_unit_circle c[10]) ; ::_thesis: a in aX1 \/ aX2
then reconsider a = a as Point of (TOP-REAL 2) by Lm8;
( 0 >= a `2 or 0 <= a `2 ) ;
then ( a in aX1 or a in aX2 ) by A10;
hence a in aX1 \/ aX2 by XBOOLE_0:def_3; ::_thesis: verum
end;
then A11: the carrier of (Topen_unit_circle c[10]) = aX1 \/ aX2 by XBOOLE_0:def_10;
Circle2IntervalR | ((Topen_unit_circle c[10]) | aX2) is continuous by Lm43;
then A12: Circle2IntervalR | ((Topen_unit_circle c[10]) | aX2) is_continuous_at x2 by TMAP_1:44;
Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is continuous by Lm42;
then A13: Circle2IntervalR | ((Topen_unit_circle c[10]) | aX1) is_continuous_at x1 by TMAP_1:44;
the carrier of ((Topen_unit_circle c[10]) | aX1) = aX1 by PRE_TOPC:8;
then Topen_unit_circle c[10] = ((Topen_unit_circle c[10]) | aX1) union ((Topen_unit_circle c[10]) | aX2) by A9, A3, A11, TSEP_1:def_2;
hence Circle2IntervalR is_continuous_at p by A13, A12, TMAP_1:113; ::_thesis: verum
end;
set h1 = REAL --> 1;
reconsider h1 = REAL --> 1 as PartFunc of REAL,REAL ;
Lm45: Circle2IntervalR is continuous
proof
set h = (1 / (2 * PI)) (#) arccos;
set K = [.(- 1),1.];
set J = [.p0,0.[;
set I = ].0,p1.];
set Z = R^1 | (R^1 ].0,(0 + p1).[);
for p being Point of (Topen_unit_circle c[10]) holds Circle2IntervalR is_continuous_at p
proof
Tcircle ((0. (TOP-REAL 2)),1) is SubSpace of Trectangle (p0,p1,p0,p1) by Th10;
then A1: Topen_unit_circle c[10] is SubSpace of Trectangle (p0,p1,p0,p1) by TSEP_1:7;
let p be Point of (Topen_unit_circle c[10]); ::_thesis: Circle2IntervalR is_continuous_at p
A2: [.(- 1),1.] = [#] (Closed-Interval-TSpace ((- 1),1)) by TOPMETR:18;
reconsider q = p as Point of (TOP-REAL 2) by Lm8;
A3: the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) = ].0,(0 + p1).[ by PRE_TOPC:8;
consider x, y being real number such that
A4: p = |[x,y]| and
A5: ( y >= 0 implies Circle2IntervalR . p = (arccos x) / (2 * PI) ) and
A6: ( y <= 0 implies Circle2IntervalR . p = 1 - ((arccos x) / (2 * PI)) ) by Def13;
A7: y = q `2 by A4, EUCLID:52;
A8: x = q `1 by A4, EUCLID:52;
then A9: x <= 1 by Th26;
- 1 <= x by A8, Th26;
then A10: x in [.(- 1),1.] by A9, XXREAL_1:1;
then A11: (((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) . x = ((1 / (2 * PI)) (#) arccos) . x by FUNCT_1:49;
dom ((1 / (2 * PI)) (#) arccos) = dom arccos by VALUED_1:def_5
.= [.(- 1),1.] by SIN_COS6:86 ;
then x in dom (((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) by A10, RELAT_1:57;
then A12: ((1 / (2 * PI)) (#) arccos) | [.(- 1),1.] is_continuous_in x by FCONT_1:def_2;
A13: dom ((1 / (2 * PI)) (#) arccos) = dom arccos by VALUED_1:def_5;
then A14: ((1 / (2 * PI)) (#) arccos) . x = (arccos . x) * (1 / (2 * PI)) by A10, SIN_COS6:86, VALUED_1:def_5
.= (1 * (arccos . x)) / (2 * PI) by XCMPLX_1:74 ;
percases ( y = 0 or y < 0 or y > 0 ) ;
suppose y = 0 ; ::_thesis: Circle2IntervalR is_continuous_at p
hence Circle2IntervalR is_continuous_at p by A7, Lm44, Th24; ::_thesis: verum
end;
supposeA15: y < 0 ; ::_thesis: Circle2IntervalR is_continuous_at p
for V being Subset of (R^1 | (R^1 ].0,(0 + p1).[)) st V is open & Circle2IntervalR . p in V holds
ex W being Subset of (Topen_unit_circle c[10]) st
( W is open & p in W & Circle2IntervalR .: W c= V )
proof
set hh = h1 - ((1 / (2 * PI)) (#) arccos);
let V be Subset of (R^1 | (R^1 ].0,(0 + p1).[)); ::_thesis: ( V is open & Circle2IntervalR . p in V implies ex W being Subset of (Topen_unit_circle c[10]) st
( W is open & p in W & Circle2IntervalR .: W c= V ) )
assume that
A16: V is open and
A17: Circle2IntervalR . p in V ; ::_thesis: ex W being Subset of (Topen_unit_circle c[10]) st
( W is open & p in W & Circle2IntervalR .: W c= V )
reconsider V1 = V as Subset of REAL by A3, XBOOLE_1:1;
reconsider V2 = V1 as Subset of R^1 by TOPMETR:17;
V2 is open by A16, TSEP_1:17;
then reconsider V1 = V1 as open Subset of REAL by BORSUK_5:39;
consider N1 being Neighbourhood of Circle2IntervalR . p such that
A18: N1 c= V1 by A17, RCOMP_1:18;
A19: ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . x = (h1 - ((1 / (2 * PI)) (#) arccos)) . x by A10, FUNCT_1:49;
dom (h1 - ((1 / (2 * PI)) (#) arccos)) = (dom h1) /\ (dom ((1 / (2 * PI)) (#) arccos)) by VALUED_1:12;
then A20: dom (h1 - ((1 / (2 * PI)) (#) arccos)) = REAL /\ (dom ((1 / (2 * PI)) (#) arccos)) by FUNCOP_1:13
.= [.(- 1),1.] by A13, SIN_COS6:86, XBOOLE_1:28 ;
then A21: dom ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) = [.(- 1),1.] by RELAT_1:62;
A22: Circle2IntervalR . p = 1 - ((arccos . x) / (2 * PI)) by A6, A15, SIN_COS6:def_4;
A23: p = (1,2) --> (x,y) by A4, TOPREALA:28;
x in dom ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) by A10, A20, RELAT_1:57;
then A24: (h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.] is_continuous_in x by FCONT_1:def_2;
(h1 - ((1 / (2 * PI)) (#) arccos)) . x = (h1 . x) - (((1 / (2 * PI)) (#) arccos) . x) by A10, A20, VALUED_1:13
.= 1 - ((1 * (arccos . x)) / (2 * PI)) by A10, A14, FUNCOP_1:7 ;
then consider N being Neighbourhood of x such that
A25: ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N c= N1 by A22, A19, A24, FCONT_1:5;
set N3 = N /\ [.(- 1),1.];
A26: N /\ [.(- 1),1.] c= [.(- 1),1.] by XBOOLE_1:17;
reconsider N3 = N /\ [.(- 1),1.], J = [.p0,0.[ as Subset of (Closed-Interval-TSpace ((- 1),1)) by Lm2, XBOOLE_1:17, XXREAL_1:35;
set W = (product ((1,2) --> (N3,J))) /\ the carrier of (Topen_unit_circle c[10]);
reconsider W = (product ((1,2) --> (N3,J))) /\ the carrier of (Topen_unit_circle c[10]) as Subset of (Topen_unit_circle c[10]) by XBOOLE_1:17;
take W ; ::_thesis: ( W is open & p in W & Circle2IntervalR .: W c= V )
reconsider KK = product ((1,2) --> (N3,J)) as Subset of (Trectangle (p0,p1,p0,p1)) by TOPREALA:38;
reconsider I3 = J as open Subset of (Closed-Interval-TSpace ((- 1),1)) by TOPREALA:26;
A27: ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N3 c= ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N by RELAT_1:123, XBOOLE_1:17;
R^1 N = N ;
then reconsider M3 = N3 as open Subset of (Closed-Interval-TSpace ((- 1),1)) by A2, TOPS_2:24;
KK = product ((1,2) --> (M3,I3)) ;
then KK is open by TOPREALA:39;
hence W is open by A1, Lm16, TOPS_2:24; ::_thesis: ( p in W & Circle2IntervalR .: W c= V )
x in N by RCOMP_1:16;
then A28: x in N3 by A10, XBOOLE_0:def_4;
y >= - 1 by A7, Th26;
then y in J by A15, XXREAL_1:3;
then p in product ((1,2) --> (N3,J)) by A23, A28, HILBERT3:11;
hence p in W by XBOOLE_0:def_4; ::_thesis: Circle2IntervalR .: W c= V
let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in Circle2IntervalR .: W or m in V )
assume m in Circle2IntervalR .: W ; ::_thesis: m in V
then consider c being Element of (Topen_unit_circle c[10]) such that
A29: c in W and
A30: m = Circle2IntervalR . c by FUNCT_2:65;
A31: c in product ((1,2) --> (N3,J)) by A29, XBOOLE_0:def_4;
then A32: c . 1 in N3 by TOPREALA:3;
consider c1, c2 being real number such that
A33: c = |[c1,c2]| and
( c2 >= 0 implies Circle2IntervalR . c = (arccos c1) / (2 * PI) ) and
A34: ( c2 <= 0 implies Circle2IntervalR . c = 1 - ((arccos c1) / (2 * PI)) ) by Def13;
c . 2 in J by A31, TOPREALA:3;
then c2 in J by A33, TOPREALA:29;
then A35: 1 - ((1 * (arccos c1)) * (1 / (2 * PI))) = m by A30, A34, XCMPLX_1:74, XXREAL_1:3;
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . (c . 1) = (h1 - ((1 / (2 * PI)) (#) arccos)) . (c . 1) by A26, A32, FUNCT_1:49
.= (h1 . (c . 1)) - (((1 / (2 * PI)) (#) arccos) . (c . 1)) by A20, A26, A32, VALUED_1:13
.= 1 - (((1 / (2 * PI)) (#) arccos) . (c . 1)) by A32, FUNCOP_1:7
.= 1 - ((arccos . (c . 1)) * (1 / (2 * PI))) by A13, A26, A32, SIN_COS6:86, VALUED_1:def_5
.= 1 - ((arccos . c1) * (1 / (2 * PI))) by A33, TOPREALA:29
.= 1 - ((arccos c1) * (1 / (2 * PI))) by SIN_COS6:def_4 ;
then m in ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N3 by A26, A32, A21, A35, FUNCT_1:def_6;
then m in ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N by A27;
then m in N1 by A25;
hence m in V by A18; ::_thesis: verum
end;
hence Circle2IntervalR is_continuous_at p by TMAP_1:43; ::_thesis: verum
end;
supposeA36: y > 0 ; ::_thesis: Circle2IntervalR is_continuous_at p
for V being Subset of (R^1 | (R^1 ].0,(0 + p1).[)) st V is open & Circle2IntervalR . p in V holds
ex W being Subset of (Topen_unit_circle c[10]) st
( W is open & p in W & Circle2IntervalR .: W c= V )
proof
let V be Subset of (R^1 | (R^1 ].0,(0 + p1).[)); ::_thesis: ( V is open & Circle2IntervalR . p in V implies ex W being Subset of (Topen_unit_circle c[10]) st
( W is open & p in W & Circle2IntervalR .: W c= V ) )
assume that
A37: V is open and
A38: Circle2IntervalR . p in V ; ::_thesis: ex W being Subset of (Topen_unit_circle c[10]) st
( W is open & p in W & Circle2IntervalR .: W c= V )
reconsider V1 = V as Subset of REAL by A3, XBOOLE_1:1;
reconsider V2 = V1 as Subset of R^1 by TOPMETR:17;
V2 is open by A37, TSEP_1:17;
then reconsider V1 = V1 as open Subset of REAL by BORSUK_5:39;
consider N1 being Neighbourhood of Circle2IntervalR . p such that
A39: N1 c= V1 by A38, RCOMP_1:18;
Circle2IntervalR . p = (arccos . x) / (2 * PI) by A5, A36, SIN_COS6:def_4;
then consider N being Neighbourhood of x such that
A40: (((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) .: N c= N1 by A11, A14, A12, FCONT_1:5;
set N3 = N /\ [.(- 1),1.];
A41: N /\ [.(- 1),1.] c= [.(- 1),1.] by XBOOLE_1:17;
reconsider N3 = N /\ [.(- 1),1.], I = ].0,p1.] as Subset of (Closed-Interval-TSpace ((- 1),1)) by Lm2, XBOOLE_1:17, XXREAL_1:36;
set W = (product ((1,2) --> (N3,I))) /\ the carrier of (Topen_unit_circle c[10]);
reconsider W = (product ((1,2) --> (N3,I))) /\ the carrier of (Topen_unit_circle c[10]) as Subset of (Topen_unit_circle c[10]) by XBOOLE_1:17;
take W ; ::_thesis: ( W is open & p in W & Circle2IntervalR .: W c= V )
reconsider KK = product ((1,2) --> (N3,I)) as Subset of (Trectangle (p0,p1,p0,p1)) by TOPREALA:38;
reconsider I3 = I as open Subset of (Closed-Interval-TSpace ((- 1),1)) by TOPREALA:25;
A42: (((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) .: N3 c= (((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) .: N by RELAT_1:123, XBOOLE_1:17;
R^1 N = N ;
then reconsider M3 = N3 as open Subset of (Closed-Interval-TSpace ((- 1),1)) by A2, TOPS_2:24;
KK = product ((1,2) --> (M3,I3)) ;
then KK is open by TOPREALA:39;
hence W is open by A1, Lm16, TOPS_2:24; ::_thesis: ( p in W & Circle2IntervalR .: W c= V )
x in N by RCOMP_1:16;
then A43: x in N3 by A10, XBOOLE_0:def_4;
A44: dom (((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) = [.(- 1),1.] by A13, RELAT_1:62, SIN_COS6:86;
A45: p = (1,2) --> (x,y) by A4, TOPREALA:28;
y <= 1 by A7, Th26;
then y in I by A36, XXREAL_1:2;
then p in product ((1,2) --> (N3,I)) by A45, A43, HILBERT3:11;
hence p in W by XBOOLE_0:def_4; ::_thesis: Circle2IntervalR .: W c= V
let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in Circle2IntervalR .: W or m in V )
assume m in Circle2IntervalR .: W ; ::_thesis: m in V
then consider c being Element of (Topen_unit_circle c[10]) such that
A46: c in W and
A47: m = Circle2IntervalR . c by FUNCT_2:65;
A48: c in product ((1,2) --> (N3,I)) by A46, XBOOLE_0:def_4;
then A49: c . 1 in N3 by TOPREALA:3;
consider c1, c2 being real number such that
A50: c = |[c1,c2]| and
A51: ( c2 >= 0 implies Circle2IntervalR . c = (arccos c1) / (2 * PI) ) and
( c2 <= 0 implies Circle2IntervalR . c = 1 - ((arccos c1) / (2 * PI)) ) by Def13;
c . 2 in I by A48, TOPREALA:3;
then c2 in I by A50, TOPREALA:29;
then A52: (1 * (arccos c1)) * (1 / (2 * PI)) = m by A47, A51, XCMPLX_1:74, XXREAL_1:2;
(((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) . (c . 1) = ((1 / (2 * PI)) (#) arccos) . (c . 1) by A41, A49, FUNCT_1:49
.= (arccos . (c . 1)) * (1 / (2 * PI)) by A13, A41, A49, SIN_COS6:86, VALUED_1:def_5
.= (arccos . c1) * (1 / (2 * PI)) by A50, TOPREALA:29
.= (arccos c1) * (1 / (2 * PI)) by SIN_COS6:def_4 ;
then m in (((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) .: N3 by A41, A49, A44, A52, FUNCT_1:def_6;
then m in (((1 / (2 * PI)) (#) arccos) | [.(- 1),1.]) .: N by A42;
then m in N1 by A40;
hence m in V by A39; ::_thesis: verum
end;
hence Circle2IntervalR is_continuous_at p by TMAP_1:43; ::_thesis: verum
end;
end;
end;
hence Circle2IntervalR is continuous by TMAP_1:44; ::_thesis: verum
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
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((AffineMap ((- (1 / (2 * PI))),1)) | (R^1 E)) or y in ].(1 / 2),((1 / 2) + p1).[ )
assume y in rng ((AffineMap ((- (1 / (2 * PI))),1)) | (R^1 E)) ; ::_thesis: y in ].(1 / 2),((1 / 2) + p1).[
then consider x being set such that
A1: x in dom ((AffineMap ((- (1 / (2 * PI))),1)) | (R^1 E)) and
A2: ((AffineMap ((- (1 / (2 * PI))),1)) | (R^1 E)) . x = y by FUNCT_1:def_3;
reconsider x = x as Real by A1;
A3: y = (AffineMap ((- (1 / (2 * PI))),1)) . x by A1, A2, Lm49, FUNCT_1:49
.= ((- (1 / (2 * PI))) * x) + 1 by FCONT_1:def_4
.= (- ((1 / (2 * PI)) * x)) + 1
.= (- (x / (2 * PI))) + 1 by XCMPLX_1:99 ;
then reconsider y = y as Real ;
x < PI by A1, Lm49, XXREAL_1:3;
then x / (2 * PI) < (1 * PI) / (2 * PI) by XREAL_1:74;
then x / (2 * PI) < 1 / 2 by XCMPLX_1:91;
then - (x / (2 * PI)) > - (1 / 2) by XREAL_1:24;
then A4: (- (x / (2 * PI))) + 1 > (- (1 / 2)) + 1 by XREAL_1:6;
0 <= x by A1, Lm49, XXREAL_1:3;
then 0 + 1 >= (- (x / (2 * PI))) + 1 by XREAL_1:6;
then y < 3 / 2 by A3, XXREAL_0:2;
hence y in ].(1 / 2),((1 / 2) + p1).[ by A3, A4, XXREAL_1:4; ::_thesis: verum
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
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((AffineMap ((1 / (2 * PI)),1)) | (R^1 E)) or y in ].(1 / 2),((1 / 2) + p1).[ )
assume y in rng ((AffineMap ((1 / (2 * PI)),1)) | (R^1 E)) ; ::_thesis: y in ].(1 / 2),((1 / 2) + p1).[
then consider x being set such that
A1: x in dom ((AffineMap ((1 / (2 * PI)),1)) | (R^1 E)) and
A2: ((AffineMap ((1 / (2 * PI)),1)) | (R^1 E)) . x = y by FUNCT_1:def_3;
reconsider x = x as Real by A1;
A3: y = (AffineMap ((1 / (2 * PI)),1)) . x by A1, A2, Lm52, FUNCT_1:49
.= ((1 / (2 * PI)) * x) + 1 by FCONT_1:def_4
.= (x / (2 * PI)) + 1 by XCMPLX_1:99 ;
then reconsider y = y as Real ;
x < PI by A1, Lm52, XXREAL_1:3;
then x / (2 * PI) < (1 * PI) / (2 * PI) by XREAL_1:74;
then x / (2 * PI) < 1 / 2 by XCMPLX_1:91;
then A4: (x / (2 * PI)) + 1 < (1 / 2) + 1 by XREAL_1:6;
0 <= x by A1, Lm52, XXREAL_1:3;
then 0 + 1 <= (x / (2 * PI)) + 1 by XREAL_1:6;
then 1 / 2 < y by A3, XXREAL_0:2;
hence y in ].(1 / 2),((1 / 2) + p1).[ by A3, A4, XXREAL_1:4; ::_thesis: verum
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
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (ac | (R^1 Q)) or y in E )
assume A1: y in rng (ac | (R^1 Q)) ; ::_thesis: y in E
then consider x being set such that
A2: x in dom (ac | (R^1 Q)) and
A3: (ac | (R^1 Q)) . x = y by FUNCT_1:def_3;
A4: rng (ac | (R^1 Q)) c= rng ac by RELAT_1:70;
then y in [.0,PI.] by A1, SIN_COS6:85;
then reconsider y = y as Real ;
A5: 0 <= y by A1, A4, SIN_COS6:85, XXREAL_1:1;
A6: y <= PI by A1, A4, SIN_COS6:85, XXREAL_1:1;
reconsider x = x as Real by A2, Lm55;
A7: - 1 < x by A2, Lm55, XXREAL_1:2;
A8: x <= 1 by A2, Lm55, XXREAL_1:2;
y = arccos . x by A2, A3, Lm55, FUNCT_1:49
.= arccos x by SIN_COS6:def_4 ;
then y < PI by A6, A7, A8, SIN_COS6:98, XXREAL_0:1;
hence y in E by A5, XXREAL_1:3; ::_thesis: verum
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
reconsider c1 = c[10] as Point of (TOP-REAL 2) ;
let aX1 be Subset of (Topen_unit_circle c[-10]); ::_thesis: ( aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) } implies Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is continuous )
assume A1: aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) } ; ::_thesis: Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is continuous
A2: c1 `2 = 0 by EUCLID:52;
c[10] is Point of (Topen_unit_circle c[-10]) by Lm15, Th23;
then c[10] in aX1 by A1, A2;
then reconsider aX1 = aX1 as non empty Subset of (Topen_unit_circle c[-10]) ;
set X1 = (Topen_unit_circle c[-10]) | aX1;
A3: the carrier of (Tunit_circle 2) is Subset of (TOP-REAL 2) by TSEP_1:1;
[#] ((Topen_unit_circle c[-10]) | aX1) is Subset of (Tunit_circle 2) by Lm9;
then reconsider B = [#] ((Topen_unit_circle c[-10]) | aX1) as non empty Subset of (TOP-REAL 2) by A3, XBOOLE_1:1;
set f = p | B;
A4: dom (p | B) = B by Lm40, RELAT_1:62;
A5: aX1 = the carrier of ((Topen_unit_circle c[-10]) | aX1) by PRE_TOPC:8;
A6: rng (p | B) c= Q
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (p | B) or y in Q )
assume y in rng (p | B) ; ::_thesis: y in Q
then consider x being set such that
A7: x in dom (p | B) and
A8: (p | B) . x = y by FUNCT_1:def_3;
consider q being Point of (TOP-REAL 2) such that
A9: q = x and
A10: q in the carrier of (Topen_unit_circle c[-10]) and
0 <= q `2 by A1, A5, A4, A7;
A11: - 1 < q `1 by A10, Th28;
A12: q `1 <= 1 by A10, Th28;
y = p . x by A4, A7, A8, FUNCT_1:49
.= q `1 by A9, PSCOMP_1:def_5 ;
hence y in Q by A11, A12, XXREAL_1:2; ::_thesis: verum
end;
the carrier of ((TOP-REAL 2) | B) = B by PRE_TOPC:8;
then reconsider f = p | B as Function of ((TOP-REAL 2) | B),(R^1 | (R^1 Q)) by A4, A6, Lm46, FUNCT_2:2;
(Topen_unit_circle c[-10]) | aX1 is SubSpace of Tunit_circle 2 by TSEP_1:7;
then (Topen_unit_circle c[-10]) | aX1 is SubSpace of TOP-REAL 2 by TSEP_1:7;
then A13: (TOP-REAL 2) | B = (Topen_unit_circle c[-10]) | aX1 by PRE_TOPC:def_5;
A14: for a being Point of ((Topen_unit_circle c[-10]) | aX1) holds (Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1)) . a = (Af1 * (c * f)) . a
proof
let a be Point of ((Topen_unit_circle c[-10]) | aX1); ::_thesis: (Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1)) . a = (Af1 * (c * f)) . a
reconsider b = a as Point of (Topen_unit_circle c[-10]) by PRE_TOPC:25;
consider x, y being real number such that
A15: b = |[x,y]| and
A16: ( y >= 0 implies Circle2IntervalL . b = 1 + ((arccos x) / (2 * PI)) ) and
( y <= 0 implies Circle2IntervalL . b = 1 - ((arccos x) / (2 * PI)) ) by Def14;
A17: |[x,y]| `1 <= 1 by A15, Th28;
A18: |[x,y]| `1 = x by EUCLID:52;
- 1 < |[x,y]| `1 by A15, Th28;
then A19: x in Q by A18, A17, XXREAL_1:2;
then arccos . x = c . x by FUNCT_1:49;
then A20: arccos . x in rng c by A19, Lm55, FUNCT_1:def_3;
a in aX1 by A5;
then ex q being Point of (TOP-REAL 2) st
( a = q & q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) by A1;
hence (Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1)) . a = 1 + ((arccos x) / (2 * PI)) by A15, A16, EUCLID:52, FUNCT_1:49
.= 1 + ((arccos . x) / (2 * PI)) by SIN_COS6:def_4
.= 1 + ((1 / (2 * PI)) * (arccos . x)) by XCMPLX_1:99
.= (AffineMap ((1 / (2 * PI)),1)) . (arccos . x) by FCONT_1:def_4
.= Af1 . (arccos . x) by A20, Lm56, FUNCT_1:49
.= Af1 . (c . x) by A19, FUNCT_1:49
.= Af1 . (c . (|[x,y]| `1)) by EUCLID:52
.= Af1 . (c . (p . a)) by A15, PSCOMP_1:def_5
.= Af1 . (c . (f . a)) by FUNCT_1:49
.= Af1 . ((c * f) . a) by A13, FUNCT_2:15
.= (Af1 * (c * f)) . a by A13, FUNCT_2:15 ;
::_thesis: verum
end;
f is continuous by Lm41, TOPREALA:8;
hence Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is continuous by A13, A14, Lm58, FUNCT_2:63; ::_thesis: verum
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
reconsider c1 = c[10] as Point of (TOP-REAL 2) ;
let aX1 be Subset of (Topen_unit_circle c[-10]); ::_thesis: ( aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) } implies Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is continuous )
assume A1: aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) } ; ::_thesis: Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is continuous
A2: c1 `2 = 0 by EUCLID:52;
c[10] is Point of (Topen_unit_circle c[-10]) by Lm15, Th23;
then c[10] in aX1 by A1, A2;
then reconsider aX1 = aX1 as non empty Subset of (Topen_unit_circle c[-10]) ;
set X1 = (Topen_unit_circle c[-10]) | aX1;
A3: the carrier of (Tunit_circle 2) is Subset of (TOP-REAL 2) by TSEP_1:1;
[#] ((Topen_unit_circle c[-10]) | aX1) is Subset of (Tunit_circle 2) by Lm9;
then reconsider B = [#] ((Topen_unit_circle c[-10]) | aX1) as non empty Subset of (TOP-REAL 2) by A3, XBOOLE_1:1;
set f = p | B;
A4: dom (p | B) = B by Lm40, RELAT_1:62;
A5: aX1 = the carrier of ((Topen_unit_circle c[-10]) | aX1) by PRE_TOPC:8;
A6: rng (p | B) c= Q
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (p | B) or y in Q )
assume y in rng (p | B) ; ::_thesis: y in Q
then consider x being set such that
A7: x in dom (p | B) and
A8: (p | B) . x = y by FUNCT_1:def_3;
consider q being Point of (TOP-REAL 2) such that
A9: q = x and
A10: q in the carrier of (Topen_unit_circle c[-10]) and
0 >= q `2 by A1, A5, A4, A7;
A11: - 1 < q `1 by A10, Th28;
A12: q `1 <= 1 by A10, Th28;
y = p . x by A4, A7, A8, FUNCT_1:49
.= q `1 by A9, PSCOMP_1:def_5 ;
hence y in Q by A11, A12, XXREAL_1:2; ::_thesis: verum
end;
the carrier of ((TOP-REAL 2) | B) = B by PRE_TOPC:8;
then reconsider f = p | B as Function of ((TOP-REAL 2) | B),(R^1 | (R^1 Q)) by A4, A6, Lm46, FUNCT_2:2;
(Topen_unit_circle c[-10]) | aX1 is SubSpace of Tunit_circle 2 by TSEP_1:7;
then (Topen_unit_circle c[-10]) | aX1 is SubSpace of TOP-REAL 2 by TSEP_1:7;
then A13: (TOP-REAL 2) | B = (Topen_unit_circle c[-10]) | aX1 by PRE_TOPC:def_5;
A14: for a being Point of ((Topen_unit_circle c[-10]) | aX1) holds (Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1)) . a = (Af * (c * f)) . a
proof
let a be Point of ((Topen_unit_circle c[-10]) | aX1); ::_thesis: (Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1)) . a = (Af * (c * f)) . a
reconsider b = a as Point of (Topen_unit_circle c[-10]) by PRE_TOPC:25;
consider x, y being real number such that
A15: b = |[x,y]| and
( y >= 0 implies Circle2IntervalL . b = 1 + ((arccos x) / (2 * PI)) ) and
A16: ( y <= 0 implies Circle2IntervalL . b = 1 - ((arccos x) / (2 * PI)) ) by Def14;
A17: |[x,y]| `1 <= 1 by A15, Th28;
A18: |[x,y]| `1 = x by EUCLID:52;
- 1 < |[x,y]| `1 by A15, Th28;
then A19: x in Q by A18, A17, XXREAL_1:2;
then arccos . x = c . x by FUNCT_1:49;
then A20: arccos . x in rng c by A19, Lm55, FUNCT_1:def_3;
a in aX1 by A5;
then ex q being Point of (TOP-REAL 2) st
( a = q & q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) by A1;
hence (Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1)) . a = 1 - ((arccos x) / (2 * PI)) by A15, A16, EUCLID:52, FUNCT_1:49
.= 1 - ((arccos . x) / (2 * PI)) by SIN_COS6:def_4
.= 1 - ((1 / (2 * PI)) * (arccos . x)) by XCMPLX_1:99
.= ((- (1 / (2 * PI))) * (arccos . x)) + 1
.= (AffineMap ((- (1 / (2 * PI))),1)) . (arccos . x) by FCONT_1:def_4
.= Af . (arccos . x) by A20, Lm56, FUNCT_1:49
.= Af . (c . x) by A19, FUNCT_1:49
.= Af . (c . (|[x,y]| `1)) by EUCLID:52
.= Af . (c . (p . a)) by A15, PSCOMP_1:def_5
.= Af . (c . (f . a)) by FUNCT_1:49
.= Af . ((c * f) . a) by A13, FUNCT_2:15
.= (Af * (c * f)) . a by A13, FUNCT_2:15 ;
::_thesis: verum
end;
f is continuous by Lm41, TOPREALA:8;
hence Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is continuous by A13, A14, Lm58, FUNCT_2:63; ::_thesis: verum
end;
Lm61: for p being Point of (Topen_unit_circle c[-10]) st p = c[10] holds
Circle2IntervalL is_continuous_at p
proof
reconsider c1 = c[10] as Point of (TOP-REAL 2) ;
set aX2 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) } ;
set aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) } ;
A1: { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) } c= the carrier of (Topen_unit_circle c[-10])
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) } or x in the carrier of (Topen_unit_circle c[-10]) )
assume x in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) } ; ::_thesis: x in the carrier of (Topen_unit_circle c[-10])
then ex q being Point of (TOP-REAL 2) st
( x = q & q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) ;
hence x in the carrier of (Topen_unit_circle c[-10]) ; ::_thesis: verum
end;
A2: { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) } c= the carrier of (Topen_unit_circle c[-10])
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) } or x in the carrier of (Topen_unit_circle c[-10]) )
assume x in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) } ; ::_thesis: x in the carrier of (Topen_unit_circle c[-10])
then ex q being Point of (TOP-REAL 2) st
( x = q & q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) ;
hence x in the carrier of (Topen_unit_circle c[-10]) ; ::_thesis: verum
end;
A3: Topen_unit_circle c[-10] is SubSpace of Topen_unit_circle c[-10] by TSEP_1:2;
A4: c1 `2 = 0 by EUCLID:52;
A5: c[10] is Point of (Topen_unit_circle c[-10]) by Lm15, Th23;
then A6: c[10] in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) } by A4;
A7: c[10] in { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) } by A4, A5;
then reconsider aX1 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 <= q `2 ) } , aX2 = { q where q is Point of (TOP-REAL 2) : ( q in the carrier of (Topen_unit_circle c[-10]) & 0 >= q `2 ) } as non empty Subset of (Topen_unit_circle c[-10]) by A1, A2, A6;
set X1 = (Topen_unit_circle c[-10]) | aX1;
let p be Point of (Topen_unit_circle c[-10]); ::_thesis: ( p = c[10] implies Circle2IntervalL is_continuous_at p )
assume A8: p = c[10] ; ::_thesis: Circle2IntervalL is_continuous_at p
reconsider x1 = p as Point of ((Topen_unit_circle c[-10]) | aX1) by A8, A6, PRE_TOPC:8;
set X2 = (Topen_unit_circle c[-10]) | aX2;
reconsider x2 = p as Point of ((Topen_unit_circle c[-10]) | aX2) by A8, A7, PRE_TOPC:8;
A9: the carrier of ((Topen_unit_circle c[-10]) | aX2) = aX2 by PRE_TOPC:8;
the carrier of (Topen_unit_circle c[-10]) c= aX1 \/ aX2
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of (Topen_unit_circle c[-10]) or a in aX1 \/ aX2 )
assume A10: a in the carrier of (Topen_unit_circle c[-10]) ; ::_thesis: a in aX1 \/ aX2
then reconsider a = a as Point of (TOP-REAL 2) by Lm8;
( 0 >= a `2 or 0 <= a `2 ) ;
then ( a in aX1 or a in aX2 ) by A10;
hence a in aX1 \/ aX2 by XBOOLE_0:def_3; ::_thesis: verum
end;
then A11: the carrier of (Topen_unit_circle c[-10]) = aX1 \/ aX2 by XBOOLE_0:def_10;
Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX2) is continuous by Lm60;
then A12: Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX2) is_continuous_at x2 by TMAP_1:44;
Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is continuous by Lm59;
then A13: Circle2IntervalL | ((Topen_unit_circle c[-10]) | aX1) is_continuous_at x1 by TMAP_1:44;
the carrier of ((Topen_unit_circle c[-10]) | aX1) = aX1 by PRE_TOPC:8;
then Topen_unit_circle c[-10] = ((Topen_unit_circle c[-10]) | aX1) union ((Topen_unit_circle c[-10]) | aX2) by A9, A3, A11, TSEP_1:def_2;
hence Circle2IntervalL is_continuous_at p by A13, A12, TMAP_1:113; ::_thesis: verum
end;
Lm62: Circle2IntervalL is continuous
proof
set h = (1 / (2 * PI)) (#) arccos;
set K = [.(- 1),1.];
set J = [.p0,0.[;
set I = ].0,p1.];
set Z = R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[);
for p being Point of (Topen_unit_circle c[-10]) holds Circle2IntervalL is_continuous_at p
proof
Tcircle ((0. (TOP-REAL 2)),1) is SubSpace of Trectangle (p0,p1,p0,p1) by Th10;
then A1: Topen_unit_circle c[-10] is SubSpace of Trectangle (p0,p1,p0,p1) by TSEP_1:7;
let p be Point of (Topen_unit_circle c[-10]); ::_thesis: Circle2IntervalL is_continuous_at p
A2: [.(- 1),1.] = [#] (Closed-Interval-TSpace ((- 1),1)) by TOPMETR:18;
reconsider q = p as Point of (TOP-REAL 2) by Lm8;
A3: the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) = ].(1 / 2),((1 / 2) + p1).[ by PRE_TOPC:8;
consider x, y being real number such that
A4: p = |[x,y]| and
A5: ( y >= 0 implies Circle2IntervalL . p = 1 + ((arccos x) / (2 * PI)) ) and
A6: ( y <= 0 implies Circle2IntervalL . p = 1 - ((arccos x) / (2 * PI)) ) by Def14;
A7: y = q `2 by A4, EUCLID:52;
A8: x = q `1 by A4, EUCLID:52;
then A9: x <= 1 by Th26;
- 1 <= x by A8, Th26;
then A10: x in [.(- 1),1.] by A9, XXREAL_1:1;
A11: dom ((1 / (2 * PI)) (#) arccos) = dom arccos by VALUED_1:def_5;
then A12: ((1 / (2 * PI)) (#) arccos) . x = (arccos . x) * (1 / (2 * PI)) by A10, SIN_COS6:86, VALUED_1:def_5
.= (1 * (arccos . x)) / (2 * PI) by XCMPLX_1:74 ;
percases ( y = 0 or y < 0 or y > 0 ) ;
suppose y = 0 ; ::_thesis: Circle2IntervalL is_continuous_at p
hence Circle2IntervalL is_continuous_at p by A7, Lm61, Th25; ::_thesis: verum
end;
supposeA13: y < 0 ; ::_thesis: Circle2IntervalL is_continuous_at p
for V being Subset of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st V is open & Circle2IntervalL . p in V holds
ex W being Subset of (Topen_unit_circle c[-10]) st
( W is open & p in W & Circle2IntervalL .: W c= V )
proof
set hh = h1 - ((1 / (2 * PI)) (#) arccos);
let V be Subset of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)); ::_thesis: ( V is open & Circle2IntervalL . p in V implies ex W being Subset of (Topen_unit_circle c[-10]) st
( W is open & p in W & Circle2IntervalL .: W c= V ) )
assume that
A14: V is open and
A15: Circle2IntervalL . p in V ; ::_thesis: ex W being Subset of (Topen_unit_circle c[-10]) st
( W is open & p in W & Circle2IntervalL .: W c= V )
reconsider V1 = V as Subset of REAL by A3, XBOOLE_1:1;
reconsider V2 = V1 as Subset of R^1 by TOPMETR:17;
V2 is open by A14, TSEP_1:17;
then reconsider V1 = V1 as open Subset of REAL by BORSUK_5:39;
consider N1 being Neighbourhood of Circle2IntervalL . p such that
A16: N1 c= V1 by A15, RCOMP_1:18;
A17: ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . x = (h1 - ((1 / (2 * PI)) (#) arccos)) . x by A10, FUNCT_1:49;
dom (h1 - ((1 / (2 * PI)) (#) arccos)) = (dom h1) /\ (dom ((1 / (2 * PI)) (#) arccos)) by VALUED_1:12;
then A18: dom (h1 - ((1 / (2 * PI)) (#) arccos)) = REAL /\ (dom ((1 / (2 * PI)) (#) arccos)) by FUNCOP_1:13
.= [.(- 1),1.] by A11, SIN_COS6:86, XBOOLE_1:28 ;
then A19: dom ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) = [.(- 1),1.] by RELAT_1:62;
A20: Circle2IntervalL . p = 1 - ((arccos . x) / (2 * PI)) by A6, A13, SIN_COS6:def_4;
A21: p = (1,2) --> (x,y) by A4, TOPREALA:28;
x in dom ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) by A10, A18, RELAT_1:57;
then A22: (h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.] is_continuous_in x by FCONT_1:def_2;
(h1 - ((1 / (2 * PI)) (#) arccos)) . x = (h1 . x) - (((1 / (2 * PI)) (#) arccos) . x) by A10, A18, VALUED_1:13
.= 1 - ((1 * (arccos . x)) / (2 * PI)) by A10, A12, FUNCOP_1:7 ;
then consider N being Neighbourhood of x such that
A23: ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N c= N1 by A20, A17, A22, FCONT_1:5;
set N3 = N /\ [.(- 1),1.];
A24: N /\ [.(- 1),1.] c= [.(- 1),1.] by XBOOLE_1:17;
reconsider N3 = N /\ [.(- 1),1.], J = [.p0,0.[ as Subset of (Closed-Interval-TSpace ((- 1),1)) by Lm2, XBOOLE_1:17, XXREAL_1:35;
set W = (product ((1,2) --> (N3,J))) /\ the carrier of (Topen_unit_circle c[-10]);
reconsider W = (product ((1,2) --> (N3,J))) /\ the carrier of (Topen_unit_circle c[-10]) as Subset of (Topen_unit_circle c[-10]) by XBOOLE_1:17;
take W ; ::_thesis: ( W is open & p in W & Circle2IntervalL .: W c= V )
reconsider KK = product ((1,2) --> (N3,J)) as Subset of (Trectangle (p0,p1,p0,p1)) by TOPREALA:38;
reconsider I3 = J as open Subset of (Closed-Interval-TSpace ((- 1),1)) by TOPREALA:26;
A25: ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N3 c= ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N by RELAT_1:123, XBOOLE_1:17;
R^1 N = N ;
then reconsider M3 = N3 as open Subset of (Closed-Interval-TSpace ((- 1),1)) by A2, TOPS_2:24;
KK = product ((1,2) --> (M3,I3)) ;
then KK is open by TOPREALA:39;
hence W is open by A1, Lm17, TOPS_2:24; ::_thesis: ( p in W & Circle2IntervalL .: W c= V )
x in N by RCOMP_1:16;
then A26: x in N3 by A10, XBOOLE_0:def_4;
y >= - 1 by A7, Th26;
then y in J by A13, XXREAL_1:3;
then p in product ((1,2) --> (N3,J)) by A21, A26, HILBERT3:11;
hence p in W by XBOOLE_0:def_4; ::_thesis: Circle2IntervalL .: W c= V
let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in Circle2IntervalL .: W or m in V )
assume m in Circle2IntervalL .: W ; ::_thesis: m in V
then consider c being Element of (Topen_unit_circle c[-10]) such that
A27: c in W and
A28: m = Circle2IntervalL . c by FUNCT_2:65;
A29: c in product ((1,2) --> (N3,J)) by A27, XBOOLE_0:def_4;
then A30: c . 1 in N3 by TOPREALA:3;
consider c1, c2 being real number such that
A31: c = |[c1,c2]| and
( c2 >= 0 implies Circle2IntervalL . c = 1 + ((arccos c1) / (2 * PI)) ) and
A32: ( c2 <= 0 implies Circle2IntervalL . c = 1 - ((arccos c1) / (2 * PI)) ) by Def14;
c . 2 in J by A29, TOPREALA:3;
then c2 in J by A31, TOPREALA:29;
then A33: 1 - ((1 * (arccos c1)) * (1 / (2 * PI))) = m by A28, A32, XCMPLX_1:74, XXREAL_1:3;
((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . (c . 1) = (h1 - ((1 / (2 * PI)) (#) arccos)) . (c . 1) by A24, A30, FUNCT_1:49
.= (h1 . (c . 1)) - (((1 / (2 * PI)) (#) arccos) . (c . 1)) by A18, A24, A30, VALUED_1:13
.= 1 - (((1 / (2 * PI)) (#) arccos) . (c . 1)) by A30, FUNCOP_1:7
.= 1 - ((arccos . (c . 1)) * (1 / (2 * PI))) by A11, A24, A30, SIN_COS6:86, VALUED_1:def_5
.= 1 - ((arccos . c1) * (1 / (2 * PI))) by A31, TOPREALA:29
.= 1 - ((arccos c1) * (1 / (2 * PI))) by SIN_COS6:def_4 ;
then m in ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N3 by A24, A30, A19, A33, FUNCT_1:def_6;
then m in ((h1 - ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N by A25;
then m in N1 by A23;
hence m in V by A16; ::_thesis: verum
end;
hence Circle2IntervalL is_continuous_at p by TMAP_1:43; ::_thesis: verum
end;
supposeA34: y > 0 ; ::_thesis: Circle2IntervalL is_continuous_at p
for V being Subset of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st V is open & Circle2IntervalL . p in V holds
ex W being Subset of (Topen_unit_circle c[-10]) st
( W is open & p in W & Circle2IntervalL .: W c= V )
proof
set hh = h1 + ((1 / (2 * PI)) (#) arccos);
let V be Subset of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)); ::_thesis: ( V is open & Circle2IntervalL . p in V implies ex W being Subset of (Topen_unit_circle c[-10]) st
( W is open & p in W & Circle2IntervalL .: W c= V ) )
assume that
A35: V is open and
A36: Circle2IntervalL . p in V ; ::_thesis: ex W being Subset of (Topen_unit_circle c[-10]) st
( W is open & p in W & Circle2IntervalL .: W c= V )
reconsider V1 = V as Subset of REAL by A3, XBOOLE_1:1;
reconsider V2 = V1 as Subset of R^1 by TOPMETR:17;
V2 is open by A35, TSEP_1:17;
then reconsider V1 = V1 as open Subset of REAL by BORSUK_5:39;
consider N1 being Neighbourhood of Circle2IntervalL . p such that
A37: N1 c= V1 by A36, RCOMP_1:18;
A38: ((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . x = (h1 + ((1 / (2 * PI)) (#) arccos)) . x by A10, FUNCT_1:49;
dom (h1 + ((1 / (2 * PI)) (#) arccos)) = (dom h1) /\ (dom ((1 / (2 * PI)) (#) arccos)) by VALUED_1:def_1;
then A39: dom (h1 + ((1 / (2 * PI)) (#) arccos)) = REAL /\ (dom ((1 / (2 * PI)) (#) arccos)) by FUNCOP_1:13
.= [.(- 1),1.] by A11, SIN_COS6:86, XBOOLE_1:28 ;
then A40: dom ((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) = [.(- 1),1.] by RELAT_1:62;
A41: Circle2IntervalL . p = 1 + ((arccos . x) / (2 * PI)) by A5, A34, SIN_COS6:def_4;
A42: p = (1,2) --> (x,y) by A4, TOPREALA:28;
x in dom ((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) by A10, A39, RELAT_1:57;
then A43: (h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.] is_continuous_in x by FCONT_1:def_2;
(h1 + ((1 / (2 * PI)) (#) arccos)) . x = (h1 . x) + (((1 / (2 * PI)) (#) arccos) . x) by A10, A39, VALUED_1:def_1
.= 1 + ((1 * (arccos . x)) / (2 * PI)) by A10, A12, FUNCOP_1:7 ;
then consider N being Neighbourhood of x such that
A44: ((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N c= N1 by A41, A38, A43, FCONT_1:5;
set N3 = N /\ [.(- 1),1.];
A45: N /\ [.(- 1),1.] c= [.(- 1),1.] by XBOOLE_1:17;
reconsider N3 = N /\ [.(- 1),1.], I = ].0,p1.] as Subset of (Closed-Interval-TSpace ((- 1),1)) by Lm2, XBOOLE_1:17, XXREAL_1:36;
set W = (product ((1,2) --> (N3,I))) /\ the carrier of (Topen_unit_circle c[-10]);
reconsider W = (product ((1,2) --> (N3,I))) /\ the carrier of (Topen_unit_circle c[-10]) as Subset of (Topen_unit_circle c[-10]) by XBOOLE_1:17;
take W ; ::_thesis: ( W is open & p in W & Circle2IntervalL .: W c= V )
reconsider KK = product ((1,2) --> (N3,I)) as Subset of (Trectangle (p0,p1,p0,p1)) by TOPREALA:38;
reconsider I3 = I as open Subset of (Closed-Interval-TSpace ((- 1),1)) by TOPREALA:25;
A46: ((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N3 c= ((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N by RELAT_1:123, XBOOLE_1:17;
R^1 N = N ;
then reconsider M3 = N3 as open Subset of (Closed-Interval-TSpace ((- 1),1)) by A2, TOPS_2:24;
KK = product ((1,2) --> (M3,I3)) ;
then KK is open by TOPREALA:39;
hence W is open by A1, Lm17, TOPS_2:24; ::_thesis: ( p in W & Circle2IntervalL .: W c= V )
x in N by RCOMP_1:16;
then A47: x in N3 by A10, XBOOLE_0:def_4;
y <= 1 by A7, Th26;
then y in I by A34, XXREAL_1:2;
then p in product ((1,2) --> (N3,I)) by A42, A47, HILBERT3:11;
hence p in W by XBOOLE_0:def_4; ::_thesis: Circle2IntervalL .: W c= V
let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in Circle2IntervalL .: W or m in V )
assume m in Circle2IntervalL .: W ; ::_thesis: m in V
then consider c being Element of (Topen_unit_circle c[-10]) such that
A48: c in W and
A49: m = Circle2IntervalL . c by FUNCT_2:65;
A50: c in product ((1,2) --> (N3,I)) by A48, XBOOLE_0:def_4;
then A51: c . 1 in N3 by TOPREALA:3;
consider c1, c2 being real number such that
A52: c = |[c1,c2]| and
A53: ( c2 >= 0 implies Circle2IntervalL . c = 1 + ((arccos c1) / (2 * PI)) ) and
( c2 <= 0 implies Circle2IntervalL . c = 1 - ((arccos c1) / (2 * PI)) ) by Def14;
c . 2 in I by A50, TOPREALA:3;
then c2 in I by A52, TOPREALA:29;
then A54: 1 + ((1 * (arccos c1)) * (1 / (2 * PI))) = m by A49, A53, XCMPLX_1:74, XXREAL_1:2;
((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) . (c . 1) = (h1 + ((1 / (2 * PI)) (#) arccos)) . (c . 1) by A45, A51, FUNCT_1:49
.= (h1 . (c . 1)) + (((1 / (2 * PI)) (#) arccos) . (c . 1)) by A39, A45, A51, VALUED_1:def_1
.= 1 + (((1 / (2 * PI)) (#) arccos) . (c . 1)) by A51, FUNCOP_1:7
.= 1 + ((arccos . (c . 1)) * (1 / (2 * PI))) by A11, A45, A51, SIN_COS6:86, VALUED_1:def_5
.= 1 + ((arccos . c1) * (1 / (2 * PI))) by A52, TOPREALA:29
.= 1 + ((arccos c1) * (1 / (2 * PI))) by SIN_COS6:def_4 ;
then m in ((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N3 by A45, A51, A40, A54, FUNCT_1:def_6;
then m in ((h1 + ((1 / (2 * PI)) (#) arccos)) | [.(- 1),1.]) .: N by A46;
then m in N1 by A44;
hence m in V by A37; ::_thesis: verum
end;
hence Circle2IntervalL is_continuous_at p by TMAP_1:43; ::_thesis: verum
end;
end;
end;
hence Circle2IntervalL is continuous by TMAP_1:44; ::_thesis: verum
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
CircleMap . (R^1 0) = c[10] by Th32;
hence CircleMap (R^1 0) is open by Th42, TOPREALA:14; ::_thesis: verum
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
set F = AffineMap (1,(- i));
set f = (AffineMap (1,(- i))) | ].(0 + i),((0 + i) + p1).[;
A1: the carrier of (R^1 | (R^1 ].0,1.[)) = R^1 ].0,1.[ by PRE_TOPC:8;
dom (AffineMap (1,(- i))) = REAL by FUNCT_2:def_1;
then A2: dom ((AffineMap (1,(- i))) | ].(0 + i),((0 + i) + p1).[) = ].i,(i + 1).[ by RELAT_1:62;
A3: rng ((AffineMap (1,(- i))) | ].(0 + i),((0 + i) + p1).[) = ].0,(0 + 1).[ by Lm24;
the carrier of (R^1 | (R^1 ].i,(i + 1).[)) = R^1 ].i,(i + 1).[ by PRE_TOPC:8;
then reconsider f = (AffineMap (1,(- i))) | ].(0 + i),((0 + i) + p1).[ as Function of (R^1 | (R^1 ].i,(i + p1).[)),(R^1 | (R^1 ].0,(0 + p1).[)) by A1, A2, A3, FUNCT_2:2;
A4: CircleMap (R^1 (0 + i)) = (CircleMap (R^1 0)) * f by Th41;
A5: R^1 | (R^1 (rng (AffineMap (1,(- i))))) = R^1 by Lm12;
A6: CircleMap . (R^1 i) = c[10] by Th32
.= CircleMap . (R^1 0) by Th32 ;
A7: R^1 | (R^1 (dom (AffineMap (1,(- i))))) = R^1 by Lm12;
A8: R^1 (AffineMap (1,(- i))) = AffineMap (1,(- i)) ;
f is onto by A1, A3, FUNCT_2:def_3;
then f is open by A7, A5, A8, TOPREALA:10;
hence CircleMap (R^1 i) is open by A4, A6, Lm63, TOPREALA:11; ::_thesis: verum
end;
cluster CircleMap (R^1 ((1 / 2) + i)) -> open ;
coherence
CircleMap (R^1 ((1 / 2) + i)) is open
proof
(1 / 2) - 1 < 0 ;
then [\(1 / 2)/] = 0 by INT_1:def_6;
then A9: (1 / 2) - [\(1 / 2)/] = 1 / 2 ;
frac ((1 / 2) + i) = frac (1 / 2) by INT_1:66
.= 1 / 2 by A9, INT_1:def_8 ;
then A10: CircleMap . (R^1 ((1 / 2) + i)) = CircleMap . (R^1 ((1 / 2) + 0)) by Lm19, Th34;
set F = AffineMap (1,(- i));
set f = (AffineMap (1,(- i))) | ].((1 / 2) + i),(((1 / 2) + i) + p1).[;
A11: the carrier of (R^1 | (R^1 ].(1 / 2),(3 / 2).[)) = R^1 ].(1 / 2),(3 / 2).[ by PRE_TOPC:8;
dom (AffineMap (1,(- i))) = REAL by FUNCT_2:def_1;
then A12: dom ((AffineMap (1,(- i))) | ].((1 / 2) + i),(((1 / 2) + i) + p1).[) = ].((1 / 2) + i),(((1 / 2) + i) + 1).[ by RELAT_1:62;
A13: rng ((AffineMap (1,(- i))) | ].((1 / 2) + i),(((1 / 2) + i) + p1).[) = ].(1 / 2),((1 / 2) + 1).[ by Lm24;
the carrier of (R^1 | (R^1 ].((1 / 2) + i),(((1 / 2) + i) + 1).[)) = R^1 ].((1 / 2) + i),(((1 / 2) + i) + 1).[ by PRE_TOPC:8;
then reconsider f = (AffineMap (1,(- i))) | ].((1 / 2) + i),(((1 / 2) + i) + p1).[ as Function of (R^1 | (R^1 ].((1 / 2) + i),(((1 / 2) + i) + p1).[)),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by A11, A12, A13, FUNCT_2:2;
A14: CircleMap (R^1 ((1 / 2) + i)) = (CircleMap (R^1 (1 / 2))) * f by Th41;
A15: R^1 | (R^1 (rng (AffineMap (1,(- i))))) = R^1 by Lm12;
A16: R^1 | (R^1 (dom (AffineMap (1,(- i))))) = R^1 by Lm12;
A17: R^1 (AffineMap (1,(- i))) = AffineMap (1,(- i)) ;
f is onto by A11, A13, FUNCT_2:def_3;
then f is open by A16, A15, A17, TOPREALA:10;
hence CircleMap (R^1 ((1 / 2) + i)) is open by A14, A10, Lm64, TOPREALA:11; ::_thesis: verum
end;
end;
registration
cluster Circle2IntervalR -> open ;
coherence
Circle2IntervalR is open
proof
CircleMap . (R^1 0) = c[10] by Th32;
hence Circle2IntervalR is open by Th42, TOPREALA:13; ::_thesis: verum
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
reconsider r = 0 as Integer ;
CircleMap (R^1 ((1 / 2) + r)) is open ;
hence CircleMap (R^1 (1 / 2)) is being_homeomorphism by TOPREALA:16; ::_thesis: verum
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
set D2 = IntIntervals ((1 / 2),(3 / 2));
set D1 = IntIntervals (0,1);
set F1 = CircleMap .: (union (IntIntervals (0,1)));
set F2 = CircleMap .: (union (IntIntervals ((1 / 2),(3 / 2))));
set F = {(CircleMap .: (union (IntIntervals (0,1)))),(CircleMap .: (union (IntIntervals ((1 / 2),(3 / 2)))))};
reconsider F = {(CircleMap .: (union (IntIntervals (0,1)))),(CircleMap .: (union (IntIntervals ((1 / 2),(3 / 2)))))} as Subset-Family of (Tunit_circle 2) ;
take F ; ::_thesis: ( 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 ) ) ) ) ) )
].((1 / 2) + 0),((3 / 2) + 0).[ in IntIntervals ((1 / 2),(3 / 2)) by Lm1;
then A1: CircleMap .: (union (IntIntervals ((1 / 2),(3 / 2)))) = CircleMap .: ].(1 / 2),(3 / 2).[ by Th40;
A2: ].(0 + 0),(1 + 0).[ in IntIntervals (0,1) by Lm1;
hence F = {(CircleMap .: ].0,1.[),(CircleMap .: ].(1 / 2),(3 / 2).[)} by A1, Th40; ::_thesis: ( 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 ) ) ) ) ) )
thus F is Cover of (Tunit_circle 2) ::_thesis: ( 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
reconsider A = [.0,(0 + 1).[ as Subset of R^1 by TOPMETR:17;
reconsider f = CircleMap | A as Function of (R^1 | A),(Tunit_circle 2) by Lm21;
let a be set ; :: according to TARSKI:def_3,SETFAM_1:def_11 ::_thesis: ( not a in the carrier of (Tunit_circle 2) or a in union F )
A3: CircleMap .: (union (IntIntervals ((1 / 2),(3 / 2)))) in F by TARSKI:def_2;
f is onto by Th38;
then A4: rng f = the carrier of (Tunit_circle 2) by FUNCT_2:def_3;
assume a in the carrier of (Tunit_circle 2) ; ::_thesis: a in union F
then consider x being set such that
A5: x in dom f and
A6: f . x = a by A4, FUNCT_1:def_3;
A7: dom f = A by Lm18, RELAT_1:62;
then reconsider x = x as Real by A5;
A8: CircleMap . x = f . x by A5, FUNCT_1:47;
percases ( x = 0 or ( 0 < x & x < 1 ) or ( x >= 1 & x < 1 ) ) by A5, A7, XXREAL_1:3;
supposeA9: x = 0 ; ::_thesis: a in union F
0 in A by XXREAL_1:3;
then A10: f . 0 = CircleMap . 0 by FUNCT_1:49
.= c[10] by Th32
.= CircleMap . 1 by Th32 ;
1 in ].(1 / 2),(3 / 2).[ by XXREAL_1:4;
then a in CircleMap .: ].(1 / 2),(3 / 2).[ by A6, A9, A10, Lm18, FUNCT_1:def_6;
hence a in union F by A1, A3, TARSKI:def_4; ::_thesis: verum
end;
supposeA11: ( 0 < x & x < 1 ) ; ::_thesis: a in union F
A12: ].(0 + 0),(1 + 0).[ in IntIntervals (0,1) by Lm1;
x in ].0,1.[ by A11, XXREAL_1:4;
then x in union (IntIntervals (0,1)) by A12, TARSKI:def_4;
then A13: a in CircleMap .: (union (IntIntervals (0,1))) by A6, A8, Lm18, FUNCT_1:def_6;
CircleMap .: (union (IntIntervals (0,1))) in F by TARSKI:def_2;
hence a in union F by A13, TARSKI:def_4; ::_thesis: verum
end;
suppose ( x >= 1 & x < 1 ) ; ::_thesis: a in union F
hence a in union F ; ::_thesis: verum
end;
end;
end;
A14: CircleMap .: (union (IntIntervals (0,1))) = CircleMap .: ].0,1.[ by A2, Th40;
thus F is open ::_thesis: 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
reconsider r = 0 as Integer ;
A15: now__::_thesis:_for_A_being_Subset_of_REAL_holds_A_is_Subset_of_(R^1_|_(R^1_A))
let A be Subset of REAL; ::_thesis: A is Subset of (R^1 | (R^1 A))
A c= A ;
hence A is Subset of (R^1 | (R^1 A)) by PRE_TOPC:8; ::_thesis: verum
end;
then reconsider M = ].0,1.[ as Subset of (R^1 | (R^1 ].r,(r + 1).[)) ;
reconsider N = ].(1 / 2),(3 / 2).[ as Subset of (R^1 | (R^1 ].((1 / 2) + r),(((1 / 2) + r) + 1).[)) by A15;
let P be Subset of (Tunit_circle 2); :: according to TOPS_2:def_1 ::_thesis: ( not P in F or P is open )
A16: now__::_thesis:_for_A_being_open_Subset_of_REAL_holds_A_is_open_Subset_of_(R^1_|_(R^1_A))
let A be open Subset of REAL; ::_thesis: A is open Subset of (R^1 | (R^1 A))
reconsider B = A as Subset of (R^1 | (R^1 A)) by A15;
the carrier of (R^1 | (R^1 A)) = A by PRE_TOPC:8;
then B = ([#] (R^1 | (R^1 A))) /\ (R^1 A) ;
hence A is open Subset of (R^1 | (R^1 A)) by TOPS_2:24; ::_thesis: verum
end;
then M is open ;
then A17: (CircleMap (R^1 r)) .: M is open by T_0TOPSP:def_2;
N is open by A16;
then A18: (CircleMap (R^1 ((1 / 2) + r))) .: N is open by T_0TOPSP:def_2;
CircleMap .: ].(1 / 2),(3 / 2).[ = (CircleMap (R^1 (1 / 2))) .: ].(1 / 2),(3 / 2).[ by RELAT_1:129;
then A19: CircleMap .: (union (IntIntervals ((1 / 2),(3 / 2)))) is open by A1, A18, TSEP_1:17;
CircleMap .: ].0,1.[ = (CircleMap (R^1 0)) .: ].0,1.[ by RELAT_1:129;
then CircleMap .: (union (IntIntervals (0,1))) is open by A14, A17, TSEP_1:17;
hence ( not P in F or P is open ) by A19, TARSKI:def_2; ::_thesis: verum
end;
let U be Subset of (Tunit_circle 2); ::_thesis: ( ( 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 ) ) ) )
A20: c[10] in {c[10]} by TARSKI:def_1;
thus ( 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 ) ) ) ::_thesis: ( 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
assume A21: U = CircleMap .: ].0,1.[ ; ::_thesis: ( 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 ) )
then reconsider U1 = U as non empty Subset of (Tunit_circle 2) by Lm18;
A22: [#] ((Tunit_circle 2) | U) = U by PRE_TOPC:def_5;
].(0 + 0),(1 + 0).[ in IntIntervals (0,1) by Lm1;
then A23: CircleMap .: ].0,1.[ = CircleMap .: (union (IntIntervals (0,1))) by Th40;
now__::_thesis:_for_x1,_x2_being_Element_of_R^1_st_x1_in_union_(IntIntervals_(0,1))_&_CircleMap_._x1_=_CircleMap_._x2_holds_
x2_in_union_(IntIntervals_(0,1))
let x1, x2 be Element of R^1; ::_thesis: ( x1 in union (IntIntervals (0,1)) & CircleMap . x1 = CircleMap . x2 implies x2 in union (IntIntervals (0,1)) )
set k = [\x2/];
set K = ].(0 + [\x2/]),(1 + [\x2/]).[;
assume x1 in union (IntIntervals (0,1)) ; ::_thesis: ( CircleMap . x1 = CircleMap . x2 implies x2 in union (IntIntervals (0,1)) )
then consider Z being set such that
A24: x1 in Z and
A25: Z in IntIntervals (0,1) by TARSKI:def_4;
consider n being Element of INT such that
A26: Z = ].(0 + n),(1 + n).[ by A25;
x1 < 1 + n by A24, A26, XXREAL_1:4;
then A27: x1 - 1 < (1 + n) - 1 by XREAL_1:9;
then (x1 - 1) - n < n - n by XREAL_1:9;
then ((x1 - n) - 1) + 1 < 0 + 1 by XREAL_1:8;
then A28: (2 * PI) * (x1 - n) < (2 * PI) * 1 by XREAL_1:68;
A29: CircleMap . (x2 - [\x2/]) = |[(cos ((2 * PI) * (x2 - [\x2/]))),(sin ((2 * PI) * (x2 - [\x2/])))]| by Def11;
x2 - 1 < [\x2/] by INT_1:def_6;
then (x2 - 1) - [\x2/] < [\x2/] - [\x2/] by XREAL_1:9;
then ((x2 - 1) - [\x2/]) + 1 < 0 + 1 by XREAL_1:6;
then A30: (2 * PI) * (x2 - [\x2/]) < (2 * PI) * 1 by XREAL_1:68;
assume A31: CircleMap . x1 = CircleMap . x2 ; ::_thesis: x2 in union (IntIntervals (0,1))
A32: n < x1 by A24, A26, XXREAL_1:4;
then A33: 0 < x1 - n by XREAL_1:50;
[\x2/] in INT by INT_1:def_2;
then A34: ].(0 + [\x2/]),(1 + [\x2/]).[ in IntIntervals (0,1) ;
A35: CircleMap . x2 = CircleMap . (x2 + (- [\x2/])) by Th31;
[\x1/] = n by A32, A27, INT_1:def_6;
then A36: not x1 in INT by A32, INT_1:26;
A37: now__::_thesis:_not_[\x2/]_=_x2
assume [\x2/] = x2 ; ::_thesis: contradiction
then CircleMap . x1 = c[10] by A31, Th32;
hence contradiction by A20, A36, Lm18, Th33, FUNCT_1:def_7, TOPMETR:17; ::_thesis: verum
end;
A38: CircleMap . (x1 - n) = |[(cos ((2 * PI) * (x1 - n))),(sin ((2 * PI) * (x1 - n)))]| by Def11;
A39: CircleMap . x1 = CircleMap . (x1 + (- n)) by Th31;
then A40: cos ((2 * PI) * (x1 - n)) = cos ((2 * PI) * (x2 - [\x2/])) by A31, A35, A38, A29, SPPOL_2:1;
A41: sin ((2 * PI) * (x1 - n)) = sin ((2 * PI) * (x2 - [\x2/])) by A31, A39, A35, A38, A29, SPPOL_2:1;
[\x2/] <= x2 by INT_1:def_6;
then A42: [\x2/] < x2 by A37, XXREAL_0:1;
then 0 <= x2 - [\x2/] by XREAL_1:50;
then (2 * PI) * (x1 - n) = (2 * PI) * (x2 - [\x2/]) by A33, A28, A30, A40, A41, COMPLEX2:11;
then x1 - n = x2 - [\x2/] by XCMPLX_1:5;
then A43: x2 = (x1 - n) + [\x2/] ;
x1 < 1 + n by A24, A26, XXREAL_1:4;
then x1 - n < 1 by XREAL_1:19;
then x2 < 1 + [\x2/] by A43, XREAL_1:6;
then x2 in ].(0 + [\x2/]),(1 + [\x2/]).[ by A42, XXREAL_1:4;
hence x2 in union (IntIntervals (0,1)) by A34, TARSKI:def_4; ::_thesis: verum
end;
hence union (IntIntervals (0,1)) = CircleMap " U by A21, A23, T_0TOPSP:1; ::_thesis: 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
let d be Subset of R^1; ::_thesis: ( d in IntIntervals (0,1) implies for f being Function of (R^1 | d),((Tunit_circle 2) | U) st f = CircleMap | d holds
f is being_homeomorphism )
assume A44: d in IntIntervals (0,1) ; ::_thesis: for f being Function of (R^1 | d),((Tunit_circle 2) | U) st f = CircleMap | d holds
f is being_homeomorphism
then consider n being Element of INT such that
A45: d = ].(0 + n),(1 + n).[ ;
reconsider d1 = d as non empty Subset of R^1 by A45;
reconsider J = ].n,(n + p1).[ as non empty Subset of R^1 by TOPMETR:17;
A46: CircleMap | d = (CircleMap | J) | d1 by A45, RELAT_1:74;
let f be Function of (R^1 | d),((Tunit_circle 2) | U); ::_thesis: ( f = CircleMap | d implies f is being_homeomorphism )
reconsider f1 = f as Function of (R^1 | d1),((Tunit_circle 2) | U1) ;
assume A47: f = CircleMap | d ; ::_thesis: f is being_homeomorphism
then A48: f is continuous by TOPREALA:8;
d c= J by A45;
then reconsider d2 = d as Subset of (R^1 | J) by PRE_TOPC:8;
A49: (R^1 | J) | d2 = R^1 | d by A45, PRE_TOPC:7;
reconsider F = CircleMap | J as Function of (R^1 | J),(Tunit_circle 2) by Lm21;
CircleMap (R^1 n) is open ;
then A50: F is open by TOPREALA:12;
A51: CircleMap .: (union (IntIntervals (0,1))) = CircleMap .: d by A44, Th40;
A52: f1 is onto
proof
thus rng f1 c= the carrier of ((Tunit_circle 2) | U1) ; :: according to XBOOLE_0:def_10,FUNCT_2:def_3 ::_thesis: the carrier of ((Tunit_circle 2) | U1) c= rng f1
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in the carrier of ((Tunit_circle 2) | U1) or b in rng f1 )
A53: dom (CircleMap | d) = d by Lm18, RELAT_1:62, TOPMETR:17;
assume b in the carrier of ((Tunit_circle 2) | U1) ; ::_thesis: b in rng f1
then consider a being Element of R^1 such that
A54: a in d and
A55: b = CircleMap . a by A21, A23, A22, A51, FUNCT_2:65;
(CircleMap | d) . a = CircleMap . a by A54, FUNCT_1:49;
hence b in rng f1 by A47, A54, A55, A53, FUNCT_1:def_3; ::_thesis: verum
end;
f is one-to-one by A44, A47, Lm3, Th39;
hence f is being_homeomorphism by A47, A48, A49, A46, A52, A50, TOPREALA:10, TOPREALA:16; ::_thesis: verum
end;
assume A56: U = CircleMap .: ].(1 / 2),(3 / 2).[ ; ::_thesis: ( 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 ) )
then reconsider U1 = U as non empty Subset of (Tunit_circle 2) by Lm18;
now__::_thesis:_for_x1,_x2_being_Element_of_R^1_st_x1_in_union_(IntIntervals_((1_/_2),(3_/_2)))_&_CircleMap_._x1_=_CircleMap_._x2_holds_
x2_in_union_(IntIntervals_((1_/_2),(3_/_2)))
let x1, x2 be Element of R^1; ::_thesis: ( x1 in union (IntIntervals ((1 / 2),(3 / 2))) & CircleMap . x1 = CircleMap . x2 implies b2 in union (IntIntervals ((1 / 2),(3 / 2))) )
set k = [\x2/];
A57: [\x2/] <= x2 by INT_1:def_6;
assume x1 in union (IntIntervals ((1 / 2),(3 / 2))) ; ::_thesis: ( CircleMap . x1 = CircleMap . x2 implies b2 in union (IntIntervals ((1 / 2),(3 / 2))) )
then consider Z being set such that
A58: x1 in Z and
A59: Z in IntIntervals ((1 / 2),(3 / 2)) by TARSKI:def_4;
consider n being Element of INT such that
A60: Z = ].((1 / 2) + n),((3 / 2) + n).[ by A59;
A61: (1 / 2) + n < x1 by A58, A60, XXREAL_1:4;
0 + n < (1 / 2) + n by XREAL_1:8;
then A62: n < x1 by A61, XXREAL_0:2;
assume A63: CircleMap . x1 = CircleMap . x2 ; ::_thesis: b2 in union (IntIntervals ((1 / 2),(3 / 2)))
A64: x1 < (3 / 2) + n by A58, A60, XXREAL_1:4;
then A65: x1 - n < 3 / 2 by XREAL_1:19;
percases ( x1 = 1 + n or x1 < 1 + n or x1 > 1 + n ) by XXREAL_0:1;
suppose x1 = 1 + n ; ::_thesis: b2 in union (IntIntervals ((1 / 2),(3 / 2)))
then CircleMap . x2 = c[10] by A63, Th32;
then reconsider w = x2 as Element of INT by A20, Lm18, Th33, FUNCT_1:def_7, TOPMETR:17;
A66: 0 + w < (1 / 2) + w by XREAL_1:8;
w - 1 in INT by INT_1:def_2;
then A67: ].((1 / 2) + (w - 1)),((3 / 2) + (w - 1)).[ in IntIntervals ((1 / 2),(3 / 2)) ;
(- (1 / 2)) + w < 0 + w by XREAL_1:8;
then x2 in ].((1 / 2) + (w - 1)),((3 / 2) + (w - 1)).[ by A66, XXREAL_1:4;
hence x2 in union (IntIntervals ((1 / 2),(3 / 2))) by A67, TARSKI:def_4; ::_thesis: verum
end;
suppose x1 < 1 + n ; ::_thesis: b2 in union (IntIntervals ((1 / 2),(3 / 2)))
then x1 - 1 < (n + 1) - 1 by XREAL_1:9;
then (x1 - 1) - n < n - n by XREAL_1:9;
then ((x1 - n) - 1) + 1 < 0 + 1 by XREAL_1:8;
then A68: (2 * PI) * (x1 - n) < (2 * PI) * 1 by XREAL_1:68;
set K = ].((1 / 2) + [\x2/]),((3 / 2) + [\x2/]).[;
[\x2/] in INT by INT_1:def_2;
then A69: ].((1 / 2) + [\x2/]),((3 / 2) + [\x2/]).[ in IntIntervals ((1 / 2),(3 / 2)) ;
A70: x2 - x2 <= x2 - [\x2/] by A57, XREAL_1:13;
A71: 0 < x1 - n by A62, XREAL_1:50;
A72: CircleMap . (x2 - [\x2/]) = |[(cos ((2 * PI) * (x2 - [\x2/]))),(sin ((2 * PI) * (x2 - [\x2/])))]| by Def11;
x2 - 1 < [\x2/] by INT_1:def_6;
then (x2 - 1) - [\x2/] < [\x2/] - [\x2/] by XREAL_1:9;
then ((x2 - 1) - [\x2/]) + 1 < 0 + 1 by XREAL_1:6;
then A73: (2 * PI) * (x2 - [\x2/]) < (2 * PI) * 1 by XREAL_1:68;
A74: CircleMap . x2 = CircleMap . (x2 + (- [\x2/])) by Th31;
((1 / 2) + n) - n < x1 - n by A61, XREAL_1:9;
then A75: (1 / 2) + [\x2/] < (x1 - n) + [\x2/] by XREAL_1:8;
A76: CircleMap . (x1 - n) = |[(cos ((2 * PI) * (x1 - n))),(sin ((2 * PI) * (x1 - n)))]| by Def11;
A77: CircleMap . x1 = CircleMap . (x1 + (- n)) by Th31;
then A78: sin ((2 * PI) * (x1 - n)) = sin ((2 * PI) * (x2 - [\x2/])) by A63, A74, A76, A72, SPPOL_2:1;
cos ((2 * PI) * (x1 - n)) = cos ((2 * PI) * (x2 - [\x2/])) by A63, A77, A74, A76, A72, SPPOL_2:1;
then (2 * PI) * (x1 - n) = (2 * PI) * (x2 - [\x2/]) by A78, A71, A68, A70, A73, COMPLEX2:11;
then A79: x1 - n = x2 - [\x2/] by XCMPLX_1:5;
then x2 = (x1 - n) + [\x2/] ;
then x2 < (3 / 2) + [\x2/] by A65, XREAL_1:6;
then x2 in ].((1 / 2) + [\x2/]),((3 / 2) + [\x2/]).[ by A79, A75, XXREAL_1:4;
hence x2 in union (IntIntervals ((1 / 2),(3 / 2))) by A69, TARSKI:def_4; ::_thesis: verum
end;
suppose x1 > 1 + n ; ::_thesis: b2 in union (IntIntervals ((1 / 2),(3 / 2)))
then A80: (n + 1) - 1 < x1 - 1 by XREAL_1:9;
then A81: n - n < (x1 - 1) - n by XREAL_1:9;
set K = ].((1 / 2) + ([\x2/] - 1)),((3 / 2) + ([\x2/] - 1)).[;
A82: - (1 / 2) < 0 ;
n - n < (x1 - 1) - n by A80, XREAL_1:9;
then A83: (- (1 / 2)) + [\x2/] < ((x1 - 1) - n) + [\x2/] by A82, XREAL_1:8;
[\x2/] - 1 in INT by INT_1:def_2;
then A84: ].((1 / 2) + ([\x2/] - 1)),((3 / 2) + ([\x2/] - 1)).[ in IntIntervals ((1 / 2),(3 / 2)) ;
A85: (x1 - n) - 1 < (3 / 2) - 1 by A65, XREAL_1:9;
A86: x2 - x2 <= x2 - [\x2/] by A57, XREAL_1:13;
A87: CircleMap . (x2 - [\x2/]) = |[(cos ((2 * PI) * (x2 - [\x2/]))),(sin ((2 * PI) * (x2 - [\x2/])))]| by Def11;
x1 - 1 < ((3 / 2) + n) - 1 by A64, XREAL_1:9;
then (x1 - 1) - n < ((1 / 2) + n) - n by XREAL_1:9;
then (x1 - 1) - n < 1 by XXREAL_0:2;
then A88: (2 * PI) * ((x1 - 1) - n) < (2 * PI) * 1 by XREAL_1:68;
A89: CircleMap . x2 = CircleMap . (x2 + (- [\x2/])) by Th31;
x2 - 1 < [\x2/] by INT_1:def_6;
then (x2 - 1) - [\x2/] < [\x2/] - [\x2/] by XREAL_1:9;
then ((x2 - 1) - [\x2/]) + 1 < 0 + 1 by XREAL_1:6;
then A90: (2 * PI) * (x2 - [\x2/]) < (2 * PI) * 1 by XREAL_1:68;
A91: CircleMap . ((x1 - 1) - n) = |[(cos ((2 * PI) * ((x1 - 1) - n))),(sin ((2 * PI) * ((x1 - 1) - n)))]| by Def11;
A92: CircleMap . x1 = CircleMap . (x1 + ((- 1) - n)) by Th31;
then A93: sin ((2 * PI) * ((x1 - 1) - n)) = sin ((2 * PI) * (x2 - [\x2/])) by A63, A89, A91, A87, SPPOL_2:1;
cos ((2 * PI) * ((x1 - 1) - n)) = cos ((2 * PI) * (x2 - [\x2/])) by A63, A92, A89, A91, A87, SPPOL_2:1;
then (2 * PI) * ((x1 - 1) - n) = (2 * PI) * (x2 - [\x2/]) by A93, A81, A88, A86, A90, COMPLEX2:11;
then A94: (x1 - 1) - n = x2 - [\x2/] by XCMPLX_1:5;
then x2 = ((x1 - 1) - n) + [\x2/] ;
then x2 < (1 / 2) + [\x2/] by A85, XREAL_1:6;
then x2 in ].((1 / 2) + ([\x2/] - 1)),((3 / 2) + ([\x2/] - 1)).[ by A94, A83, XXREAL_1:4;
hence x2 in union (IntIntervals ((1 / 2),(3 / 2))) by A84, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
hence union (IntIntervals ((1 / 2),(3 / 2))) = CircleMap " U by A1, A56, T_0TOPSP:1; ::_thesis: 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
let d be Subset of R^1; ::_thesis: ( d in IntIntervals ((1 / 2),(3 / 2)) implies for f being Function of (R^1 | d),((Tunit_circle 2) | U) st f = CircleMap | d holds
f is being_homeomorphism )
assume A95: d in IntIntervals ((1 / 2),(3 / 2)) ; ::_thesis: for f being Function of (R^1 | d),((Tunit_circle 2) | U) st f = CircleMap | d holds
f is being_homeomorphism
then consider n being Element of INT such that
A96: d = ].((1 / 2) + n),((3 / 2) + n).[ ;
A97: 1 + n < (3 / 2) + n by XREAL_1:6;
(1 / 2) + n < 1 + n by XREAL_1:6;
then reconsider d1 = d as non empty Subset of R^1 by A96, A97, XXREAL_1:4;
A98: [#] ((Tunit_circle 2) | U) = U by PRE_TOPC:def_5;
let f be Function of (R^1 | d),((Tunit_circle 2) | U); ::_thesis: ( f = CircleMap | d implies f is being_homeomorphism )
reconsider f1 = f as Function of (R^1 | d1),((Tunit_circle 2) | U1) ;
assume A99: f = CircleMap | d ; ::_thesis: f is being_homeomorphism
then A100: f is continuous by TOPREALA:8;
reconsider J = ].((1 / 2) + n),(((1 / 2) + n) + p1).[ as non empty Subset of R^1 by TOPMETR:17;
A101: CircleMap | d = (CircleMap | J) | d1 by A96, RELAT_1:74;
d c= J by A96;
then reconsider d2 = d as Subset of (R^1 | J) by PRE_TOPC:8;
A102: (R^1 | J) | d2 = R^1 | d by A96, PRE_TOPC:7;
A103: CircleMap .: (union (IntIntervals ((1 / 2),(3 / 2)))) = CircleMap .: d by A95, Th40;
A104: f1 is onto
proof
thus rng f1 c= the carrier of ((Tunit_circle 2) | U1) ; :: according to XBOOLE_0:def_10,FUNCT_2:def_3 ::_thesis: the carrier of ((Tunit_circle 2) | U1) c= rng f1
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in the carrier of ((Tunit_circle 2) | U1) or b in rng f1 )
A105: dom (CircleMap | d) = d by Lm18, RELAT_1:62, TOPMETR:17;
assume b in the carrier of ((Tunit_circle 2) | U1) ; ::_thesis: b in rng f1
then consider a being Element of R^1 such that
A106: a in d and
A107: b = CircleMap . a by A1, A56, A98, A103, FUNCT_2:65;
(CircleMap | d) . a = CircleMap . a by A106, FUNCT_1:49;
hence b in rng f1 by A99, A106, A107, A105, FUNCT_1:def_3; ::_thesis: verum
end;
reconsider F = CircleMap | J as Function of (R^1 | J),(Tunit_circle 2) by Lm21;
CircleMap (R^1 ((1 / 2) + n)) is open ;
then A108: F is open by TOPREALA:12;
f is one-to-one by A95, A99, Lm4, Th39;
hence f is being_homeomorphism by A99, A100, A102, A101, A104, A108, TOPREALA:10, TOPREALA:16; ::_thesis: verum
end;