:: 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;