:: JGRAPH_2 semantic presentation begin theorem Th1: :: JGRAPH_2:1 for T1, S, T2, T being non empty TopSpace for f being Function of T1,S for g being Function of T2,S for F1, F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds f . p = g . p ) holds ex h being Function of T,S st ( h = f +* g & h is continuous ) proof let T1, S, T2, T be non empty TopSpace; ::_thesis: for f being Function of T1,S for g being Function of T2,S for F1, F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds f . p = g . p ) holds ex h being Function of T,S st ( h = f +* g & h is continuous ) let f be Function of T1,S; ::_thesis: for g being Function of T2,S for F1, F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds f . p = g . p ) holds ex h being Function of T,S st ( h = f +* g & h is continuous ) let g be Function of T2,S; ::_thesis: for F1, F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds f . p = g . p ) holds ex h being Function of T,S st ( h = f +* g & h is continuous ) let F1, F2 be Subset of T; ::_thesis: ( T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds f . p = g . p ) implies ex h being Function of T,S st ( h = f +* g & h is continuous ) ) assume that A1: T1 is SubSpace of T and A2: T2 is SubSpace of T and A3: F1 = [#] T1 and A4: F2 = [#] T2 and A5: ([#] T1) \/ ([#] T2) = [#] T and A6: F1 is closed and A7: F2 is closed and A8: f is continuous and A9: g is continuous and A10: for p being set st p in ([#] T1) /\ ([#] T2) holds f . p = g . p ; ::_thesis: ex h being Function of T,S st ( h = f +* g & h is continuous ) set h = f +* g; A11: dom g = the carrier of T2 by FUNCT_2:def_1 .= [#] T2 ; A12: dom f = the carrier of T1 by FUNCT_2:def_1 .= [#] T1 ; then A13: dom (f +* g) = [#] T by A5, A11, FUNCT_4:def_1 .= the carrier of T ; rng (f +* g) c= (rng f) \/ (rng g) by FUNCT_4:17; then reconsider h = f +* g as Function of T,S by A13, FUNCT_2:2, XBOOLE_1:1; take h ; ::_thesis: ( h = f +* g & h is continuous ) thus h = f +* g ; ::_thesis: h is continuous for P being Subset of S st P is closed holds h " P is closed proof let P be Subset of S; ::_thesis: ( P is closed implies h " P is closed ) set P3 = f " P; set P4 = g " P; [#] T1 c= [#] T by A5, XBOOLE_1:7; then reconsider P1 = f " P as Subset of T by XBOOLE_1:1; [#] T2 c= [#] T by A5, XBOOLE_1:7; then reconsider P2 = g " P as Subset of T by XBOOLE_1:1; A14: dom h = (dom f) \/ (dom g) by FUNCT_4:def_1; A15: now__::_thesis:_for_x_being_set_holds_ (_(_x_in_(h_"_P)_/\_([#]_T2)_implies_x_in_g_"_P_)_&_(_x_in_g_"_P_implies_x_in_(h_"_P)_/\_([#]_T2)_)_) let x be set ; ::_thesis: ( ( x in (h " P) /\ ([#] T2) implies x in g " P ) & ( x in g " P implies x in (h " P) /\ ([#] T2) ) ) thus ( x in (h " P) /\ ([#] T2) implies x in g " P ) ::_thesis: ( x in g " P implies x in (h " P) /\ ([#] T2) ) proof assume A16: x in (h " P) /\ ([#] T2) ; ::_thesis: x in g " P then x in h " P by XBOOLE_0:def_4; then A17: h . x in P by FUNCT_1:def_7; g . x = h . x by A11, A16, FUNCT_4:13; hence x in g " P by A11, A16, A17, FUNCT_1:def_7; ::_thesis: verum end; assume A18: x in g " P ; ::_thesis: x in (h " P) /\ ([#] T2) then A19: x in dom g by FUNCT_1:def_7; g . x in P by A18, FUNCT_1:def_7; then A20: h . x in P by A19, FUNCT_4:13; x in dom h by A14, A19, XBOOLE_0:def_3; then x in h " P by A20, FUNCT_1:def_7; hence x in (h " P) /\ ([#] T2) by A18, XBOOLE_0:def_4; ::_thesis: verum end; A21: for x being set st x in [#] T1 holds h . x = f . x proof let x be set ; ::_thesis: ( x in [#] T1 implies h . x = f . x ) assume A22: x in [#] T1 ; ::_thesis: h . x = f . x now__::_thesis:_h_._x_=_f_._x percases ( x in [#] T2 or not x in [#] T2 ) ; supposeA23: x in [#] T2 ; ::_thesis: h . x = f . x then x in ([#] T1) /\ ([#] T2) by A22, XBOOLE_0:def_4; then f . x = g . x by A10; hence h . x = f . x by A11, A23, FUNCT_4:13; ::_thesis: verum end; suppose not x in [#] T2 ; ::_thesis: h . x = f . x hence h . x = f . x by A11, FUNCT_4:11; ::_thesis: verum end; end; end; hence h . x = f . x ; ::_thesis: verum end; now__::_thesis:_for_x_being_set_holds_ (_(_x_in_(h_"_P)_/\_([#]_T1)_implies_x_in_f_"_P_)_&_(_x_in_f_"_P_implies_x_in_(h_"_P)_/\_([#]_T1)_)_) let x be set ; ::_thesis: ( ( x in (h " P) /\ ([#] T1) implies x in f " P ) & ( x in f " P implies x in (h " P) /\ ([#] T1) ) ) thus ( x in (h " P) /\ ([#] T1) implies x in f " P ) ::_thesis: ( x in f " P implies x in (h " P) /\ ([#] T1) ) proof assume A24: x in (h " P) /\ ([#] T1) ; ::_thesis: x in f " P then x in h " P by XBOOLE_0:def_4; then A25: h . x in P by FUNCT_1:def_7; f . x = h . x by A21, A24; hence x in f " P by A12, A24, A25, FUNCT_1:def_7; ::_thesis: verum end; assume A26: x in f " P ; ::_thesis: x in (h " P) /\ ([#] T1) then x in dom f by FUNCT_1:def_7; then A27: x in dom h by A14, XBOOLE_0:def_3; f . x in P by A26, FUNCT_1:def_7; then h . x in P by A21, A26; then x in h " P by A27, FUNCT_1:def_7; hence x in (h " P) /\ ([#] T1) by A26, XBOOLE_0:def_4; ::_thesis: verum end; then A28: (h " P) /\ ([#] T1) = f " P by TARSKI:1; assume A29: P is closed ; ::_thesis: h " P is closed then f " P is closed by A8, PRE_TOPC:def_6; then ex F01 being Subset of T st ( F01 is closed & f " P = F01 /\ ([#] T1) ) by A1, PRE_TOPC:13; then A30: P1 is closed by A3, A6; g " P is closed by A9, A29, PRE_TOPC:def_6; then ex F02 being Subset of T st ( F02 is closed & g " P = F02 /\ ([#] T2) ) by A2, PRE_TOPC:13; then A31: P2 is closed by A4, A7; h " P = (h " P) /\ (([#] T1) \/ ([#] T2)) by A12, A11, A14, RELAT_1:132, XBOOLE_1:28 .= ((h " P) /\ ([#] T1)) \/ ((h " P) /\ ([#] T2)) by XBOOLE_1:23 ; then h " P = (f " P) \/ (g " P) by A28, A15, TARSKI:1; hence h " P is closed by A30, A31; ::_thesis: verum end; hence h is continuous by PRE_TOPC:def_6; ::_thesis: verum end; theorem Th2: :: JGRAPH_2:2 for n being Element of NAT for q2 being Point of (Euclid n) for q being Point of (TOP-REAL n) for r being real number st q = q2 holds Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } proof let n be Element of NAT ; ::_thesis: for q2 being Point of (Euclid n) for q being Point of (TOP-REAL n) for r being real number st q = q2 holds Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } let q2 be Point of (Euclid n); ::_thesis: for q being Point of (TOP-REAL n) for r being real number st q = q2 holds Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } let q be Point of (TOP-REAL n); ::_thesis: for r being real number st q = q2 holds Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } let r be real number ; ::_thesis: ( q = q2 implies Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } ) assume A1: q = q2 ; ::_thesis: Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } A2: { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } c= { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } or x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } ) assume x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } ; ::_thesis: x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } then consider q4 being Element of (Euclid n) such that A3: ( q4 = x & dist (q2,q4) < r ) ; reconsider q44 = q4 as Point of (TOP-REAL n) by TOPREAL3:8; dist (q2,q4) = |.(q - q44).| by A1, JGRAPH_1:28; hence x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } by A3; ::_thesis: verum end; A4: { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } c= { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } or x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } ) assume x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } ; ::_thesis: x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } then consider q3 being Point of (TOP-REAL n) such that A5: ( x = q3 & |.(q - q3).| < r ) ; reconsider q34 = q3 as Point of (Euclid n) by TOPREAL3:8; dist (q2,q34) = |.(q - q3).| by A1, JGRAPH_1:28; hence x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } by A5; ::_thesis: verum end; Ball (q2,r) = { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } by METRIC_1:17; hence Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } by A2, A4, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th3: :: JGRAPH_2:3 ( (0. (TOP-REAL 2)) `1 = 0 & (0. (TOP-REAL 2)) `2 = 0 ) by EUCLID:52, EUCLID:54; theorem Th4: :: JGRAPH_2:4 1.REAL 2 = <*1,1*> by FINSEQ_2:61; theorem Th5: :: JGRAPH_2:5 ( (1.REAL 2) `1 = 1 & (1.REAL 2) `2 = 1 ) by Th4, EUCLID:52; theorem Th6: :: JGRAPH_2:6 ( dom proj1 = the carrier of (TOP-REAL 2) & dom proj1 = REAL 2 ) proof thus dom proj1 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; ::_thesis: dom proj1 = REAL 2 hence dom proj1 = REAL 2 by EUCLID:22; ::_thesis: verum end; theorem Th7: :: JGRAPH_2:7 ( dom proj2 = the carrier of (TOP-REAL 2) & dom proj2 = REAL 2 ) proof thus dom proj2 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; ::_thesis: dom proj2 = REAL 2 hence dom proj2 = REAL 2 by EUCLID:22; ::_thesis: verum end; theorem Th8: :: JGRAPH_2:8 for p being Point of (TOP-REAL 2) holds p = |[(proj1 . p),(proj2 . p)]| proof let p be Point of (TOP-REAL 2); ::_thesis: p = |[(proj1 . p),(proj2 . p)]| ( p = |[(p `1),(p `2)]| & p `1 = proj1 . p ) by EUCLID:53, PSCOMP_1:def_5; hence p = |[(proj1 . p),(proj2 . p)]| by PSCOMP_1:def_6; ::_thesis: verum end; theorem Th9: :: JGRAPH_2:9 for B being Subset of (TOP-REAL 2) st B = {(0. (TOP-REAL 2))} holds ( B ` <> {} & the carrier of (TOP-REAL 2) \ B <> {} ) proof let B be Subset of (TOP-REAL 2); ::_thesis: ( B = {(0. (TOP-REAL 2))} implies ( B ` <> {} & the carrier of (TOP-REAL 2) \ B <> {} ) ) assume A1: B = {(0. (TOP-REAL 2))} ; ::_thesis: ( B ` <> {} & the carrier of (TOP-REAL 2) \ B <> {} ) now__::_thesis:_not_|[0,1]|_in_B assume |[0,1]| in B ; ::_thesis: contradiction then |[0,1]| `2 = 0 by A1, Th3, TARSKI:def_1; hence contradiction by EUCLID:52; ::_thesis: verum end; then |[0,1]| in the carrier of (TOP-REAL 2) \ B by XBOOLE_0:def_5; hence ( B ` <> {} & the carrier of (TOP-REAL 2) \ B <> {} ) by SUBSET_1:def_4; ::_thesis: verum end; theorem Th10: :: JGRAPH_2:10 for X, Y being non empty TopSpace for f being Function of X,Y holds ( f is continuous iff for p being Point of X for V being Subset of Y st f . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & f .: W c= V ) ) proof let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y holds ( f is continuous iff for p being Point of X for V being Subset of Y st f . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & f .: W c= V ) ) let f be Function of X,Y; ::_thesis: ( f is continuous iff for p being Point of X for V being Subset of Y st f . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & f .: W c= V ) ) A1: [#] Y <> {} ; A2: dom f = the carrier of X by FUNCT_2:def_1; hereby ::_thesis: ( ( for p being Point of X for V being Subset of Y st f . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & f .: W c= V ) ) implies f is continuous ) assume A3: f is continuous ; ::_thesis: for p being Point of X for V being Subset of Y st f . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & f .: W c= V ) thus for p being Point of X for V being Subset of Y st f . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & f .: W c= V ) ::_thesis: verum proof let p be Point of X; ::_thesis: for V being Subset of Y st f . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & f .: W c= V ) let V be Subset of Y; ::_thesis: ( f . p in V & V is open implies ex W being Subset of X st ( p in W & W is open & f .: W c= V ) ) assume ( f . p in V & V is open ) ; ::_thesis: ex W being Subset of X st ( p in W & W is open & f .: W c= V ) then A4: ( f " V is open & p in f " V ) by A2, A1, A3, FUNCT_1:def_7, TOPS_2:43; f .: (f " V) c= V by FUNCT_1:75; hence ex W being Subset of X st ( p in W & W is open & f .: W c= V ) by A4; ::_thesis: verum end; end; assume A5: for p being Point of X for V being Subset of Y st f . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & f .: W c= V ) ; ::_thesis: f is continuous for G being Subset of Y st G is open holds f " G is open proof let G be Subset of Y; ::_thesis: ( G is open implies f " G is open ) assume A6: G is open ; ::_thesis: f " G is open for z being set holds ( z in f " G iff ex Q being Subset of X st ( Q is open & Q c= f " G & z in Q ) ) proof let z be set ; ::_thesis: ( z in f " G iff ex Q being Subset of X st ( Q is open & Q c= f " G & z in Q ) ) now__::_thesis:_(_z_in_f_"_G_implies_ex_Q_being_Subset_of_X_st_ (_Q_is_open_&_Q_c=_f_"_G_&_z_in_Q_)_) assume A7: z in f " G ; ::_thesis: ex Q being Subset of X st ( Q is open & Q c= f " G & z in Q ) then reconsider p = z as Point of X ; f . z in G by A7, FUNCT_1:def_7; then consider W being Subset of X such that A8: ( p in W & W is open ) and A9: f .: W c= G by A5, A6; A10: W c= f " (f .: W) by A2, FUNCT_1:76; f " (f .: W) c= f " G by A9, RELAT_1:143; hence ex Q being Subset of X st ( Q is open & Q c= f " G & z in Q ) by A8, A10, XBOOLE_1:1; ::_thesis: verum end; hence ( z in f " G iff ex Q being Subset of X st ( Q is open & Q c= f " G & z in Q ) ) ; ::_thesis: verum end; hence f " G is open by TOPS_1:25; ::_thesis: verum end; hence f is continuous by A1, TOPS_2:43; ::_thesis: verum end; theorem Th11: :: JGRAPH_2:11 for p being Point of (TOP-REAL 2) for G being Subset of (TOP-REAL 2) st G is open & p in G holds ex r being real number st ( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G ) proof let p be Point of (TOP-REAL 2); ::_thesis: for G being Subset of (TOP-REAL 2) st G is open & p in G holds ex r being real number st ( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G ) let G be Subset of (TOP-REAL 2); ::_thesis: ( G is open & p in G implies ex r being real number st ( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G ) ) assume that A1: G is open and A2: p in G ; ::_thesis: ex r being real number st ( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G ) reconsider GG = G as Subset of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) ; reconsider q2 = p as Point of (Euclid 2) by TOPREAL3:8; ( TopSpaceMetr (Euclid 2) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) & GG is open ) by A1, EUCLID:def_8, PRE_TOPC:30; then consider r being real number such that A3: r > 0 and A4: Ball (q2,r) c= GG by A2, TOPMETR:15; set s = r / (sqrt 2); A5: Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL 2) : |.(p - q3).| < r } by Th2; A6: { q where q is Point of (TOP-REAL 2) : ( (p `1) - (r / (sqrt 2)) < q `1 & q `1 < (p `1) + (r / (sqrt 2)) & (p `2) - (r / (sqrt 2)) < q `2 & q `2 < (p `2) + (r / (sqrt 2)) ) } c= Ball (q2,r) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL 2) : ( (p `1) - (r / (sqrt 2)) < q `1 & q `1 < (p `1) + (r / (sqrt 2)) & (p `2) - (r / (sqrt 2)) < q `2 & q `2 < (p `2) + (r / (sqrt 2)) ) } or x in Ball (q2,r) ) assume x in { q where q is Point of (TOP-REAL 2) : ( (p `1) - (r / (sqrt 2)) < q `1 & q `1 < (p `1) + (r / (sqrt 2)) & (p `2) - (r / (sqrt 2)) < q `2 & q `2 < (p `2) + (r / (sqrt 2)) ) } ; ::_thesis: x in Ball (q2,r) then consider q being Point of (TOP-REAL 2) such that A7: q = x and A8: (p `1) - (r / (sqrt 2)) < q `1 and A9: q `1 < (p `1) + (r / (sqrt 2)) and A10: (p `2) - (r / (sqrt 2)) < q `2 and A11: q `2 < (p `2) + (r / (sqrt 2)) ; ((p `1) + (r / (sqrt 2))) - (r / (sqrt 2)) > (q `1) - (r / (sqrt 2)) by A9, XREAL_1:14; then A12: (p `1) - (q `1) > ((q `1) + (- (r / (sqrt 2)))) - (q `1) by XREAL_1:14; ((p `2) + (r / (sqrt 2))) - (r / (sqrt 2)) > (q `2) - (r / (sqrt 2)) by A11, XREAL_1:14; then A13: (p `2) - (q `2) > ((q `2) + (- (r / (sqrt 2)))) - (q `2) by XREAL_1:14; ((p `2) - (r / (sqrt 2))) + (r / (sqrt 2)) < (q `2) + (r / (sqrt 2)) by A10, XREAL_1:8; then (p `2) - (q `2) < ((q `2) + (r / (sqrt 2))) - (q `2) by XREAL_1:14; then A14: ((p `2) - (q `2)) ^2 < (r / (sqrt 2)) ^2 by A13, SQUARE_1:50; (r / (sqrt 2)) ^2 = (r ^2) / ((sqrt 2) ^2) by XCMPLX_1:76 .= (r ^2) / 2 by SQUARE_1:def_2 ; then A15: ((r / (sqrt 2)) ^2) + ((r / (sqrt 2)) ^2) = r ^2 ; ((p `1) - (r / (sqrt 2))) + (r / (sqrt 2)) < (q `1) + (r / (sqrt 2)) by A8, XREAL_1:8; then (p `1) - (q `1) < ((q `1) + (r / (sqrt 2))) - (q `1) by XREAL_1:14; then A16: ( (p - q) `2 = (p `2) - (q `2) & ((p `1) - (q `1)) ^2 < (r / (sqrt 2)) ^2 ) by A12, SQUARE_1:50, TOPREAL3:3; ( |.(p - q).| ^2 = (((p - q) `1) ^2) + (((p - q) `2) ^2) & (p - q) `1 = (p `1) - (q `1) ) by JGRAPH_1:29, TOPREAL3:3; then |.(p - q).| ^2 < r ^2 by A16, A14, A15, XREAL_1:8; then |.(p - q).| < r by A3, SQUARE_1:48; hence x in Ball (q2,r) by A5, A7; ::_thesis: verum end; sqrt 2 > 0 by SQUARE_1:25; then r / (sqrt 2) > 0 by A3, XREAL_1:139; hence ex r being real number st ( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G ) by A4, A6, XBOOLE_1:1; ::_thesis: verum end; theorem Th12: :: JGRAPH_2:12 for X, Y, Z being non empty TopSpace for B being Subset of Y for C being Subset of Z for f being Function of X,Y for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds ex g being Function of X,Z st ( g is continuous & g = h * f ) proof let X, Y, Z be non empty TopSpace; ::_thesis: for B being Subset of Y for C being Subset of Z for f being Function of X,Y for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds ex g being Function of X,Z st ( g is continuous & g = h * f ) let B be Subset of Y; ::_thesis: for C being Subset of Z for f being Function of X,Y for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds ex g being Function of X,Z st ( g is continuous & g = h * f ) let C be Subset of Z; ::_thesis: for f being Function of X,Y for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds ex g being Function of X,Z st ( g is continuous & g = h * f ) let f be Function of X,Y; ::_thesis: for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds ex g being Function of X,Z st ( g is continuous & g = h * f ) let h be Function of (Y | B),(Z | C); ::_thesis: ( f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} implies ex g being Function of X,Z st ( g is continuous & g = h * f ) ) assume that A1: f is continuous and A2: h is continuous and A3: rng f c= B and A4: B <> {} and A5: C <> {} ; ::_thesis: ex g being Function of X,Z st ( g is continuous & g = h * f ) A6: the carrier of X = dom f by FUNCT_2:def_1; the carrier of (Y | B) = [#] (Y | B) .= B by PRE_TOPC:def_5 ; then reconsider u = f as Function of X,(Y | B) by A3, A6, FUNCT_2:2; reconsider V = B as non empty Subset of Y by A4; not Y | V is empty ; then reconsider H = Y | B as non empty TopSpace ; reconsider F = C as non empty Subset of Z by A5; reconsider k = u as Function of X,H ; not Z | F is empty ; then reconsider G = Z | C as non empty TopSpace ; reconsider j = h as Function of H,G ; A7: the carrier of (Z | C) = [#] (Z | C) .= C by PRE_TOPC:def_5 ; j * k is Function of X,G ; then reconsider v = h * u as Function of X,Z by A7, FUNCT_2:7; u is continuous by A1, TOPMETR:6; then v is continuous by A2, A4, A5, PRE_TOPC:26; hence ex g being Function of X,Z st ( g is continuous & g = h * f ) ; ::_thesis: verum end; definition func Out_In_Sq -> Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) means :Def1: :: JGRAPH_2:def 1 for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies it . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or it . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ); existence ex b1 being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) st for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) proof reconsider BP = NonZero (TOP-REAL 2) as non empty set by Th9; defpred S1[ set , set ] means for p being Point of (TOP-REAL 2) st p = $1 holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies $2 = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or $2 = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ); A1: for x being Element of BP ex y being Element of BP st S1[x,y] proof let x be Element of BP; ::_thesis: ex y being Element of BP st S1[x,y] reconsider q = x as Point of (TOP-REAL 2) by TARSKI:def_3; now__::_thesis:_(_(_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_)_&_ex_y_being_Element_of_BP_st_S1[x,y]_)_or_(_not_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_&_not_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_&_ex_y_being_Element_of_BP_st_S1[x,y]_)_) percases ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) or ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; caseA2: ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: ex y being Element of BP st S1[x,y] now__::_thesis:_not_|[(1_/_(q_`1)),(((q_`2)_/_(q_`1))_/_(q_`1))]|_in_{(0._(TOP-REAL_2))} assume |[(1 / (q `1)),(((q `2) / (q `1)) / (q `1))]| in {(0. (TOP-REAL 2))} ; ::_thesis: contradiction then 0. (TOP-REAL 2) = |[(1 / (q `1)),(((q `2) / (q `1)) / (q `1))]| by TARSKI:def_1; then 0 = 1 / (q `1) by Th3, EUCLID:52; then A3: 0 = (1 / (q `1)) * (q `1) ; now__::_thesis:_(_(_q_`1_=_0_&_contradiction_)_or_(_q_`1_<>_0_&_contradiction_)_) percases ( q `1 = 0 or q `1 <> 0 ) ; caseA4: q `1 = 0 ; ::_thesis: contradiction then q `2 = 0 by A2; then q = 0. (TOP-REAL 2) by A4, EUCLID:53, EUCLID:54; then q in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence contradiction by XBOOLE_0:def_5; ::_thesis: verum end; case q `1 <> 0 ; ::_thesis: contradiction hence contradiction by A3, XCMPLX_1:87; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; then reconsider r = |[(1 / (q `1)),(((q `2) / (q `1)) / (q `1))]| as Element of BP by XBOOLE_0:def_5; for p being Point of (TOP-REAL 2) st p = x holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies r = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or r = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) by A2; hence ex y being Element of BP st S1[x,y] ; ::_thesis: verum end; caseA5: ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: ex y being Element of BP st S1[x,y] now__::_thesis:_not_|[(((q_`1)_/_(q_`2))_/_(q_`2)),(1_/_(q_`2))]|_in_{(0._(TOP-REAL_2))} assume |[(((q `1) / (q `2)) / (q `2)),(1 / (q `2))]| in {(0. (TOP-REAL 2))} ; ::_thesis: contradiction then 0. (TOP-REAL 2) = |[(((q `1) / (q `2)) / (q `2)),(1 / (q `2))]| by TARSKI:def_1; then (0. (TOP-REAL 2)) `2 = 1 / (q `2) by EUCLID:52; then A6: 0 = (1 / (q `2)) * (q `2) by Th3; q `2 <> 0 by A5; hence contradiction by A6, XCMPLX_1:87; ::_thesis: verum end; then reconsider r = |[(((q `1) / (q `2)) / (q `2)),(1 / (q `2))]| as Element of BP by XBOOLE_0:def_5; for p being Point of (TOP-REAL 2) st p = x holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies r = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or r = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) by A5; hence ex y being Element of BP st S1[x,y] ; ::_thesis: verum end; end; end; hence ex y being Element of BP st S1[x,y] ; ::_thesis: verum end; ex h being Function of BP,BP st for x being Element of BP holds S1[x,h . x] from FUNCT_2:sch_3(A1); then consider h being Function of BP,BP such that A7: for x being Element of BP for p being Point of (TOP-REAL 2) st p = x holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . x = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . x = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ; for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p <> 0. (TOP-REAL 2) implies ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) assume p <> 0. (TOP-REAL 2) ; ::_thesis: ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) then not p in {(0. (TOP-REAL 2))} by TARSKI:def_1; then p in NonZero (TOP-REAL 2) by XBOOLE_0:def_5; hence ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) by A7; ::_thesis: verum end; hence ex b1 being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) st for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) st ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) & ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b2 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b2 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) holds b1 = b2 proof let h1, h2 be Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)); ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) & ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h2 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h2 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) implies h1 = h2 ) assume that A8: for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) and A9: for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h2 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h2 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ; ::_thesis: h1 = h2 for x being set st x in NonZero (TOP-REAL 2) holds h1 . x = h2 . x proof let x be set ; ::_thesis: ( x in NonZero (TOP-REAL 2) implies h1 . x = h2 . x ) assume A10: x in NonZero (TOP-REAL 2) ; ::_thesis: h1 . x = h2 . x then reconsider q = x as Point of (TOP-REAL 2) ; not q in {(0. (TOP-REAL 2))} by A10, XBOOLE_0:def_5; then A11: q <> 0. (TOP-REAL 2) by TARSKI:def_1; now__::_thesis:_(_(_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_)_&_h1_._x_=_h2_._x_)_or_(_not_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_&_not_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_&_h1_._x_=_h2_._x_)_) percases ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) or ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; caseA12: ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: h1 . x = h2 . x then h1 . q = |[(1 / (q `1)),(((q `2) / (q `1)) / (q `1))]| by A8, A11; hence h1 . x = h2 . x by A9, A11, A12; ::_thesis: verum end; caseA13: ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: h1 . x = h2 . x then h1 . q = |[(((q `1) / (q `2)) / (q `2)),(1 / (q `2))]| by A8, A11; hence h1 . x = h2 . x by A9, A11, A13; ::_thesis: verum end; end; end; hence h1 . x = h2 . x ; ::_thesis: verum end; hence h1 = h2 by FUNCT_2:12; ::_thesis: verum end; end; :: deftheorem Def1 defines Out_In_Sq JGRAPH_2:def_1_:_ for b1 being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) holds ( b1 = Out_In_Sq iff for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ); theorem Th13: :: JGRAPH_2:13 for p being Point of (TOP-REAL 2) holds ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) proof let p be Point of (TOP-REAL 2); ::_thesis: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) A1: ( - (p `1) < p `2 implies - (- (p `1)) > - (p `2) ) by XREAL_1:24; A2: ( - (p `1) > p `2 implies - (- (p `1)) < - (p `2) ) by XREAL_1:24; assume ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) hence ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by A1, A2; ::_thesis: verum end; theorem Th14: :: JGRAPH_2:14 for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) implies Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) ) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p <> 0. (TOP-REAL 2) implies ( ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) implies Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) ) ) assume A1: p <> 0. (TOP-REAL 2) ; ::_thesis: ( ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) implies Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) ) hereby ::_thesis: ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) assume A2: ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| now__::_thesis:_(_(_p_`1_<=_p_`2_&_-_(p_`2)_<=_p_`1_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_or_(_p_`1_>=_p_`2_&_p_`1_<=_-_(p_`2)_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_) percases ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by A2; caseA3: ( p `1 <= p `2 & - (p `2) <= p `1 ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| now__::_thesis:_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_implies_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_) assume A4: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| A5: now__::_thesis:_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_) percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A4; case ( p `2 <= p `1 & - (p `1) <= p `2 ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) ) hence ( p `1 = p `2 or p `1 = - (p `2) ) by A3, XXREAL_0:1; ::_thesis: verum end; case ( p `2 >= p `1 & p `2 <= - (p `1) ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) ) then - (p `2) >= - (- (p `1)) by XREAL_1:24; hence ( p `1 = p `2 or p `1 = - (p `2) ) by A3, XXREAL_0:1; ::_thesis: verum end; end; end; now__::_thesis:_(_(_p_`1_=_p_`2_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_or_(_p_`1_=_-_(p_`2)_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_) percases ( p `1 = p `2 or p `1 = - (p `2) ) by A5; caseA6: p `1 = p `2 ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| then p `1 <> 0 by A1, EUCLID:53, EUCLID:54; then ((p `1) / (p `2)) / (p `2) = 1 / (p `1) by A6, XCMPLX_1:60; hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, A4, A6, Def1; ::_thesis: verum end; caseA7: p `1 = - (p `2) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| then A8: p `2 <> 0 by A1, EUCLID:53, EUCLID:54; A9: ((p `1) / (p `2)) / (p `2) = (- ((p `2) / (p `2))) / (p `2) by A7 .= (- 1) / (p `2) by A8, XCMPLX_1:60 .= 1 / (p `1) by A7, XCMPLX_1:192 ; - (p `1) = p `2 by A7; then 1 / (p `2) = - (1 / (p `1)) by XCMPLX_1:188 .= - (((p `2) / (p `1)) / (- (p `1))) by A7, A9, XCMPLX_1:192 .= - (- (((p `2) / (p `1)) / (p `1))) by XCMPLX_1:188 .= ((p `2) / (p `1)) / (p `1) ; hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, A4, A9, Def1; ::_thesis: verum end; end; end; hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ; ::_thesis: verum end; hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, Def1; ::_thesis: verum end; caseA10: ( p `1 >= p `2 & p `1 <= - (p `2) ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| now__::_thesis:_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_implies_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_) assume A11: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| A12: now__::_thesis:_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_) percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A11; case ( p `2 <= p `1 & - (p `1) <= p `2 ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) ) then - (- (p `1)) >= - (p `2) by XREAL_1:24; hence ( p `1 = p `2 or p `1 = - (p `2) ) by A10, XXREAL_0:1; ::_thesis: verum end; case ( p `2 >= p `1 & p `2 <= - (p `1) ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) ) hence ( p `1 = p `2 or p `1 = - (p `2) ) by A10, XXREAL_0:1; ::_thesis: verum end; end; end; now__::_thesis:_(_(_p_`1_=_p_`2_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_or_(_p_`1_=_-_(p_`2)_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_) percases ( p `1 = p `2 or p `1 = - (p `2) ) by A12; caseA13: p `1 = p `2 ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| then p `1 <> 0 by A1, EUCLID:53, EUCLID:54; then ((p `1) / (p `2)) / (p `2) = 1 / (p `1) by A13, XCMPLX_1:60; hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, A11, A13, Def1; ::_thesis: verum end; caseA14: p `1 = - (p `2) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| then A15: p `2 <> 0 by A1, EUCLID:53, EUCLID:54; A16: ((p `1) / (p `2)) / (p `2) = (- ((p `2) / (p `2))) / (p `2) by A14 .= (- 1) / (p `2) by A15, XCMPLX_1:60 .= 1 / (p `1) by A14, XCMPLX_1:192 ; - (p `1) = p `2 by A14; then 1 / (p `2) = - (((p `1) / (p `2)) / (p `2)) by A16, XCMPLX_1:188 .= - (((p `2) / (p `1)) / (- (p `1))) by A14, XCMPLX_1:191 .= - (- (((p `2) / (p `1)) / (p `1))) by XCMPLX_1:188 .= ((p `2) / (p `1)) / (p `1) ; hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, A11, A16, Def1; ::_thesis: verum end; end; end; hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ; ::_thesis: verum end; hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, Def1; ::_thesis: verum end; end; end; hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ; ::_thesis: verum end; hereby ::_thesis: verum A17: ( - (p `2) > p `1 implies - (- (p `2)) < - (p `1) ) by XREAL_1:24; A18: ( - (p `2) < p `1 implies - (- (p `2)) > - (p `1) ) by XREAL_1:24; assume ( not ( p `1 <= p `2 & - (p `2) <= p `1 ) & not ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ; ::_thesis: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| hence Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by A1, A18, A17, Def1; ::_thesis: verum end; end; theorem Th15: :: JGRAPH_2:15 for D being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | D) st K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) proof let D be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | D) st K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) let K0 be Subset of ((TOP-REAL 2) | D); ::_thesis: ( K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } implies rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) ) A1: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D) .= D by PRE_TOPC:def_5 ; then reconsider K00 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A2: K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K00) holds q `1 <> 0 proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K00) implies q `1 <> 0 ) A4: the carrier of ((TOP-REAL 2) | K00) = [#] ((TOP-REAL 2) | K00) .= K0 by PRE_TOPC:def_5 ; assume q in the carrier of ((TOP-REAL 2) | K00) ; ::_thesis: q `1 <> 0 then A5: ex p3 being Point of (TOP-REAL 2) st ( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A4; now__::_thesis:_not_q_`1_=_0 assume A6: q `1 = 0 ; ::_thesis: contradiction then q `2 = 0 by A5; hence contradiction by A5, A6, EUCLID:53, EUCLID:54; ::_thesis: verum end; hence q `1 <> 0 ; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Out_In_Sq | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) ) assume y in rng (Out_In_Sq | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0) then consider x being set such that A7: x in dom (Out_In_Sq | K0) and A8: y = (Out_In_Sq | K0) . x by FUNCT_1:def_3; A9: x in (dom Out_In_Sq) /\ K0 by A7, RELAT_1:61; then A10: x in K0 by XBOOLE_0:def_4; K0 c= the carrier of (TOP-REAL 2) by A1, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A10; A11: Out_In_Sq . p = y by A8, A10, FUNCT_1:49; A12: ex px being Point of (TOP-REAL 2) st ( x = px & ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) by A2, A10; then A13: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by Def1; set p9 = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]|; K00 = [#] ((TOP-REAL 2) | K00) by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | K00) ; then A14: p in the carrier of ((TOP-REAL 2) | K00) by A9, XBOOLE_0:def_4; A15: |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) by EUCLID:52; A16: now__::_thesis:_not_|[(1_/_(p_`1)),(((p_`2)_/_(p_`1))_/_(p_`1))]|_=_0._(TOP-REAL_2) assume |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then 0 * (p `1) = (1 / (p `1)) * (p `1) by A15, EUCLID:52, EUCLID:54; hence contradiction by A14, A3, XCMPLX_1:87; ::_thesis: verum end; A17: p `1 <> 0 by A14, A3; now__::_thesis:_(_(_p_`1_>=_0_&_y_in_K0_)_or_(_p_`1_<_0_&_y_in_K0_)_) percases ( p `1 >= 0 or p `1 < 0 ) ; caseA18: p `1 >= 0 ; ::_thesis: y in K0 then ( ( (p `2) / (p `1) <= (p `1) / (p `1) & (- (1 * (p `1))) / (p `1) <= (p `2) / (p `1) ) or ( p `2 >= p `1 & p `2 <= - (1 * (p `1)) ) ) by A12, XREAL_1:72; then A19: ( ( (p `2) / (p `1) <= 1 & ((- 1) * (p `1)) / (p `1) <= (p `2) / (p `1) ) or ( p `2 >= p `1 & p `2 <= - (1 * (p `1)) ) ) by A14, A3, XCMPLX_1:60; then ( ( (p `2) / (p `1) <= 1 & - 1 <= (p `2) / (p `1) ) or ( (p `2) / (p `1) >= 1 & (p `2) / (p `1) <= ((- 1) * (p `1)) / (p `1) ) ) by A17, A18, XCMPLX_1:89; then (- 1) / (p `1) <= ((p `2) / (p `1)) / (p `1) by A18, XREAL_1:72; then A20: ( ( ((p `2) / (p `1)) / (p `1) <= 1 / (p `1) & - (1 / (p `1)) <= ((p `2) / (p `1)) / (p `1) ) or ( ((p `2) / (p `1)) / (p `1) >= 1 / (p `1) & ((p `2) / (p `1)) / (p `1) <= - (1 / (p `1)) ) ) by A17, A18, A19, XREAL_1:72; ( |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) & |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 = ((p `2) / (p `1)) / (p `1) ) by EUCLID:52; hence y in K0 by A2, A11, A16, A13, A20; ::_thesis: verum end; caseA21: p `1 < 0 ; ::_thesis: y in K0 A22: ( not (p `2) / (p `1) >= 1 or not (p `2) / (p `1) <= - 1 ) ; ( ( p `2 <= p `1 & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= (p `1) / (p `1) & (p `2) / (p `1) >= (- (1 * (p `1))) / (p `1) ) ) by A12, A21, XREAL_1:73; then A23: ( ( p `2 <= p `1 & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= 1 & (p `2) / (p `1) >= ((- 1) * (p `1)) / (p `1) ) ) by A21, XCMPLX_1:60; then ( ( (p `2) / (p `1) >= (p `1) / (p `1) & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= 1 & (p `2) / (p `1) >= - 1 ) ) by A21, XCMPLX_1:89; then (- 1) / (p `1) >= ((p `2) / (p `1)) / (p `1) by A21, A22, XCMPLX_1:60, XREAL_1:73; then A24: ( ( ((p `2) / (p `1)) / (p `1) <= 1 / (p `1) & - (1 / (p `1)) <= ((p `2) / (p `1)) / (p `1) ) or ( ((p `2) / (p `1)) / (p `1) >= 1 / (p `1) & ((p `2) / (p `1)) / (p `1) <= - (1 / (p `1)) ) ) by A21, A23, XREAL_1:73; ( |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) & |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 = ((p `2) / (p `1)) / (p `1) ) by EUCLID:52; hence y in K0 by A2, A11, A16, A13, A24; ::_thesis: verum end; end; end; then y in [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5; hence y in the carrier of (((TOP-REAL 2) | D) | K0) ; ::_thesis: verum end; theorem Th16: :: JGRAPH_2:16 for D being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | D) st K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) proof let D be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | D) st K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) let K0 be Subset of ((TOP-REAL 2) | D); ::_thesis: ( K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } implies rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) ) A1: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D) .= D by PRE_TOPC:def_5 ; then reconsider K00 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A2: K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K00) holds q `2 <> 0 proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K00) implies q `2 <> 0 ) A4: the carrier of ((TOP-REAL 2) | K00) = [#] ((TOP-REAL 2) | K00) .= K0 by PRE_TOPC:def_5 ; assume q in the carrier of ((TOP-REAL 2) | K00) ; ::_thesis: q `2 <> 0 then A5: ex p3 being Point of (TOP-REAL 2) st ( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A4; now__::_thesis:_not_q_`2_=_0 assume A6: q `2 = 0 ; ::_thesis: contradiction then q `1 = 0 by A5; hence contradiction by A5, A6, EUCLID:53, EUCLID:54; ::_thesis: verum end; hence q `2 <> 0 ; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Out_In_Sq | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) ) assume y in rng (Out_In_Sq | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0) then consider x being set such that A7: x in dom (Out_In_Sq | K0) and A8: y = (Out_In_Sq | K0) . x by FUNCT_1:def_3; x in (dom Out_In_Sq) /\ K0 by A7, RELAT_1:61; then A9: x in K0 by XBOOLE_0:def_4; K0 c= the carrier of (TOP-REAL 2) by A1, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A9; A10: Out_In_Sq . p = y by A8, A9, FUNCT_1:49; A11: ex px being Point of (TOP-REAL 2) st ( x = px & ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) by A2, A9; then A12: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by Th14; A13: K00 = [#] ((TOP-REAL 2) | K00) by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | K00) ; set p9 = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|; A14: |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) by EUCLID:52; A15: now__::_thesis:_not_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_=_0._(TOP-REAL_2) assume |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then 0 * (p `2) = (1 / (p `2)) * (p `2) by A14, EUCLID:52, EUCLID:54; hence contradiction by A9, A13, A3, XCMPLX_1:87; ::_thesis: verum end; A16: p `2 <> 0 by A9, A13, A3; now__::_thesis:_(_(_p_`2_>=_0_&_y_in_K0_)_or_(_p_`2_<_0_&_y_in_K0_)_) percases ( p `2 >= 0 or p `2 < 0 ) ; caseA17: p `2 >= 0 ; ::_thesis: y in K0 then ( ( (p `1) / (p `2) <= (p `2) / (p `2) & (- (1 * (p `2))) / (p `2) <= (p `1) / (p `2) ) or ( p `1 >= p `2 & p `1 <= - (1 * (p `2)) ) ) by A11, XREAL_1:72; then A18: ( ( (p `1) / (p `2) <= 1 & ((- 1) * (p `2)) / (p `2) <= (p `1) / (p `2) ) or ( p `1 >= p `2 & p `1 <= - (1 * (p `2)) ) ) by A9, A13, A3, XCMPLX_1:60; then ( ( (p `1) / (p `2) <= 1 & - 1 <= (p `1) / (p `2) ) or ( (p `1) / (p `2) >= 1 & (p `1) / (p `2) <= ((- 1) * (p `2)) / (p `2) ) ) by A16, A17, XCMPLX_1:89; then (- 1) / (p `2) <= ((p `1) / (p `2)) / (p `2) by A17, XREAL_1:72; then A19: ( ( ((p `1) / (p `2)) / (p `2) <= 1 / (p `2) & - (1 / (p `2)) <= ((p `1) / (p `2)) / (p `2) ) or ( ((p `1) / (p `2)) / (p `2) >= 1 / (p `2) & ((p `1) / (p `2)) / (p `2) <= - (1 / (p `2)) ) ) by A16, A17, A18, XREAL_1:72; ( |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) & |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 = ((p `1) / (p `2)) / (p `2) ) by EUCLID:52; hence y in K0 by A2, A10, A15, A12, A19; ::_thesis: verum end; caseA20: p `2 < 0 ; ::_thesis: y in K0 then ( ( p `1 <= p `2 & - (1 * (p `2)) <= p `1 ) or ( (p `1) / (p `2) <= (p `2) / (p `2) & (p `1) / (p `2) >= (- (1 * (p `2))) / (p `2) ) ) by A11, XREAL_1:73; then A21: ( ( p `1 <= p `2 & - (1 * (p `2)) <= p `1 ) or ( (p `1) / (p `2) <= 1 & (p `1) / (p `2) >= ((- 1) * (p `2)) / (p `2) ) ) by A20, XCMPLX_1:60; then ( ( (p `1) / (p `2) >= 1 & ((- 1) * (p `2)) / (p `2) >= (p `1) / (p `2) ) or ( (p `1) / (p `2) <= 1 & (p `1) / (p `2) >= - 1 ) ) by A20, XCMPLX_1:89; then (- 1) / (p `2) >= ((p `1) / (p `2)) / (p `2) by A20, XREAL_1:73; then A22: ( ( ((p `1) / (p `2)) / (p `2) <= 1 / (p `2) & - (1 / (p `2)) <= ((p `1) / (p `2)) / (p `2) ) or ( ((p `1) / (p `2)) / (p `2) >= 1 / (p `2) & ((p `1) / (p `2)) / (p `2) <= - (1 / (p `2)) ) ) by A20, A21, XREAL_1:73; ( |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) & |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 = ((p `1) / (p `2)) / (p `2) ) by EUCLID:52; hence y in K0 by A2, A10, A15, A12, A22; ::_thesis: verum end; end; end; then y in [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5; hence y in the carrier of (((TOP-REAL 2) | D) | K0) ; ::_thesis: verum end; Lm1: 0. (TOP-REAL 2) = 0.REAL 2 by EUCLID:66; theorem Th17: :: JGRAPH_2:17 for K0a being set for D being non empty Subset of (TOP-REAL 2) st K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) proof A1: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19; let K0a be set ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} implies ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) ) assume that A2: K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } and A3: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) ( ( (1.REAL 2) `2 <= (1.REAL 2) `1 & - ((1.REAL 2) `1) <= (1.REAL 2) `2 ) or ( (1.REAL 2) `2 >= (1.REAL 2) `1 & (1.REAL 2) `2 <= - ((1.REAL 2) `1) ) ) by Th5; then A4: 1.REAL 2 in K0a by A2, A1; A5: K0a c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0a or x in D ) A6: D = (D `) ` .= NonZero (TOP-REAL 2) by A3, SUBSET_1:def_4 ; assume x in K0a ; ::_thesis: x in D then A7: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in D by A7, A6, XBOOLE_0:def_5; ::_thesis: verum end; the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D) .= D by PRE_TOPC:def_5 ; hence K0a is non empty Subset of ((TOP-REAL 2) | D) by A4, A5; ::_thesis: K0a is non empty Subset of (TOP-REAL 2) thus K0a is non empty Subset of (TOP-REAL 2) by A4, A5, XBOOLE_1:1; ::_thesis: verum end; theorem Th18: :: JGRAPH_2:18 for K0a being set for D being non empty Subset of (TOP-REAL 2) st K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) proof A1: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19; let K0a be set ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} implies ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) ) assume that A2: K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } and A3: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) ( ( (1.REAL 2) `1 <= (1.REAL 2) `2 & - ((1.REAL 2) `2) <= (1.REAL 2) `1 ) or ( (1.REAL 2) `1 >= (1.REAL 2) `2 & (1.REAL 2) `1 <= - ((1.REAL 2) `2) ) ) by Th5; then A4: 1.REAL 2 in K0a by A2, A1; A5: K0a c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0a or x in D ) A6: D = (D `) ` .= NonZero (TOP-REAL 2) by A3, SUBSET_1:def_4 ; assume x in K0a ; ::_thesis: x in D then A7: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in D by A7, A6, XBOOLE_0:def_5; ::_thesis: verum end; the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D) .= D by PRE_TOPC:def_5 ; hence K0a is non empty Subset of ((TOP-REAL 2) | D) by A4, A5; ::_thesis: K0a is non empty Subset of (TOP-REAL 2) thus K0a is non empty Subset of (TOP-REAL 2) by A4, A5, XBOOLE_1:1; ::_thesis: verum end; theorem Th19: :: JGRAPH_2:19 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 + r2 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 + r2 ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 + r2 ) & g is continuous ) ) assume that A1: f1 is continuous and A2: f2 is continuous ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 + r2 ) & g is continuous ) defpred S1[ set , set ] means for r1, r2 being real number st f1 . $1 = r1 & f2 . $1 = r2 holds $2 = r1 + r2; A3: for x being Element of X ex y being Element of REAL st S1[x,y] proof let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y] reconsider r1 = f1 . x as Real by TOPMETR:17; reconsider r2 = f2 . x as Real by TOPMETR:17; set r3 = r1 + r2; for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds r1 + r2 = r1 + r2 ; hence ex y being Element of REAL st for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds y = r1 + r2 ; ::_thesis: verum end; ex f being Function of the carrier of X,REAL st for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A3); then consider f being Function of the carrier of X,REAL such that A4: for x being Element of X for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds f . x = r1 + r2 ; reconsider g0 = f as Function of X,R^1 by TOPMETR:17; for p being Point of X for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) proof let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ) reconsider r = g0 . p as Real by TOPMETR:17; reconsider r1 = f1 . p as Real by TOPMETR:17; reconsider r2 = f2 . p as Real by TOPMETR:17; assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) then consider r0 being Real such that A5: r0 > 0 and A6: ].(r - r0),(r + r0).[ c= V by FRECHET:8; reconsider G1 = ].(r1 - (r0 / 2)),(r1 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17; A7: r1 < r1 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215; then r1 - (r0 / 2) < r1 by XREAL_1:19; then A8: f1 . p in G1 by A7, XXREAL_1:4; reconsider G2 = ].(r2 - (r0 / 2)),(r2 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17; A9: r2 < r2 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215; then r2 - (r0 / 2) < r2 by XREAL_1:19; then A10: f2 . p in G2 by A9, XXREAL_1:4; G2 is open by JORDAN6:35; then consider W2 being Subset of X such that A11: ( p in W2 & W2 is open ) and A12: f2 .: W2 c= G2 by A2, A10, Th10; G1 is open by JORDAN6:35; then consider W1 being Subset of X such that A13: ( p in W1 & W1 is open ) and A14: f1 .: W1 c= G1 by A1, A8, Th10; set W = W1 /\ W2; A15: g0 .: (W1 /\ W2) c= ].(r - r0),(r + r0).[ proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: (W1 /\ W2) or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: (W1 /\ W2) ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A16: z in dom g0 and A17: z in W1 /\ W2 and A18: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A16; reconsider aa2 = f2 . pz as Real by TOPMETR:17; reconsider aa1 = f1 . pz as Real by TOPMETR:17; A19: pz in the carrier of X ; then A20: pz in dom f2 by FUNCT_2:def_1; z in W2 by A17, XBOOLE_0:def_4; then A21: f2 . pz in f2 .: W2 by A20, FUNCT_1:def_6; then A22: r2 - (r0 / 2) < aa2 by A12, XXREAL_1:4; A23: pz in dom f1 by A19, FUNCT_2:def_1; z in W1 by A17, XBOOLE_0:def_4; then A24: f1 . pz in f1 .: W1 by A23, FUNCT_1:def_6; then r1 - (r0 / 2) < aa1 by A14, XXREAL_1:4; then (r1 - (r0 / 2)) + (r2 - (r0 / 2)) < aa1 + aa2 by A22, XREAL_1:8; then (r1 + r2) - ((r0 / 2) + (r0 / 2)) < aa1 + aa2 ; then A25: r - r0 < aa1 + aa2 by A4; A26: aa2 < r2 + (r0 / 2) by A12, A21, XXREAL_1:4; A27: x = aa1 + aa2 by A4, A18; then reconsider rx = x as Real ; aa1 < r1 + (r0 / 2) by A14, A24, XXREAL_1:4; then aa1 + aa2 < (r1 + (r0 / 2)) + (r2 + (r0 / 2)) by A26, XREAL_1:8; then aa1 + aa2 < (r1 + r2) + ((r0 / 2) + (r0 / 2)) ; then rx < r + r0 by A4, A27; hence x in ].(r - r0),(r + r0).[ by A27, A25, XXREAL_1:4; ::_thesis: verum end; ( W1 /\ W2 is open & p in W1 /\ W2 ) by A13, A11, XBOOLE_0:def_4; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A6, A15, XBOOLE_1:1; ::_thesis: verum end; then A28: g0 is continuous by Th10; for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g0 . p = r1 + r2 by A4; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 + r2 ) & g is continuous ) by A28; ::_thesis: verum end; theorem :: JGRAPH_2:20 for X being non empty TopSpace for a being real number ex g being Function of X,R^1 st ( ( for p being Point of X holds g . p = a ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for a being real number ex g being Function of X,R^1 st ( ( for p being Point of X holds g . p = a ) & g is continuous ) let a be real number ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X holds g . p = a ) & g is continuous ) reconsider a1 = a as Element of R^1 by TOPMETR:17, XREAL_0:def_1; set g1 = the carrier of X --> a1; reconsider g0 = the carrier of X --> a1 as Function of X,R^1 ; for p being Point of X for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) proof set f1 = g0; let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ) assume that A1: g0 . p in V and V is open ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) set G1 = V; g0 .: ([#] X) c= V proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in g0 .: ([#] X) or y in V ) assume y in g0 .: ([#] X) ; ::_thesis: y in V then ex x being set st ( x in dom g0 & x in [#] X & y = g0 . x ) by FUNCT_1:def_6; then y = a by FUNCOP_1:7; hence y in V by A1, FUNCOP_1:7; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum end; then ( ( for p being Point of X holds ( the carrier of X --> a1) . p = a ) & g0 is continuous ) by Th10, FUNCOP_1:7; hence ex g being Function of X,R^1 st ( ( for p being Point of X holds g . p = a ) & g is continuous ) ; ::_thesis: verum end; theorem Th21: :: JGRAPH_2:21 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 - r2 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 - r2 ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 - r2 ) & g is continuous ) ) assume that A1: f1 is continuous and A2: f2 is continuous ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 - r2 ) & g is continuous ) defpred S1[ set , set ] means for r1, r2 being real number st f1 . $1 = r1 & f2 . $1 = r2 holds $2 = r1 - r2; A3: for x being Element of X ex y being Element of REAL st S1[x,y] proof let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y] reconsider r1 = f1 . x as Real by TOPMETR:17; reconsider r2 = f2 . x as Real by TOPMETR:17; set r3 = r1 - r2; for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds r1 - r2 = r1 - r2 ; hence ex y being Element of REAL st for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds y = r1 - r2 ; ::_thesis: verum end; ex f being Function of the carrier of X,REAL st for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A3); then consider f being Function of the carrier of X,REAL such that A4: for x being Element of X for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds f . x = r1 - r2 ; reconsider g0 = f as Function of X,R^1 by TOPMETR:17; for p being Point of X for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) proof let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ) reconsider r = g0 . p as Real by TOPMETR:17; reconsider r1 = f1 . p as Real by TOPMETR:17; reconsider r2 = f2 . p as Real by TOPMETR:17; assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) then consider r0 being Real such that A5: r0 > 0 and A6: ].(r - r0),(r + r0).[ c= V by FRECHET:8; reconsider G1 = ].(r1 - (r0 / 2)),(r1 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17; A7: r1 < r1 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215; then r1 - (r0 / 2) < r1 by XREAL_1:19; then A8: f1 . p in G1 by A7, XXREAL_1:4; reconsider G2 = ].(r2 - (r0 / 2)),(r2 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17; A9: r2 < r2 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215; then r2 - (r0 / 2) < r2 by XREAL_1:19; then A10: f2 . p in G2 by A9, XXREAL_1:4; G2 is open by JORDAN6:35; then consider W2 being Subset of X such that A11: ( p in W2 & W2 is open ) and A12: f2 .: W2 c= G2 by A2, A10, Th10; G1 is open by JORDAN6:35; then consider W1 being Subset of X such that A13: ( p in W1 & W1 is open ) and A14: f1 .: W1 c= G1 by A1, A8, Th10; set W = W1 /\ W2; A15: g0 .: (W1 /\ W2) c= ].(r - r0),(r + r0).[ proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: (W1 /\ W2) or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: (W1 /\ W2) ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A16: z in dom g0 and A17: z in W1 /\ W2 and A18: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A16; reconsider aa2 = f2 . pz as Real by TOPMETR:17; reconsider aa1 = f1 . pz as Real by TOPMETR:17; A19: pz in the carrier of X ; then A20: pz in dom f1 by FUNCT_2:def_1; A21: pz in dom f2 by A19, FUNCT_2:def_1; z in W2 by A17, XBOOLE_0:def_4; then A22: f2 . pz in f2 .: W2 by A21, FUNCT_1:def_6; then A23: r2 - (r0 / 2) < aa2 by A12, XXREAL_1:4; A24: aa2 < r2 + (r0 / 2) by A12, A22, XXREAL_1:4; z in W1 by A17, XBOOLE_0:def_4; then A25: f1 . pz in f1 .: W1 by A20, FUNCT_1:def_6; then r1 - (r0 / 2) < aa1 by A14, XXREAL_1:4; then (r1 - (r0 / 2)) - (r2 + (r0 / 2)) < aa1 - aa2 by A24, XREAL_1:14; then (r1 - r2) - ((r0 / 2) + (r0 / 2)) < aa1 - aa2 ; then A26: r - r0 < aa1 - aa2 by A4; A27: x = aa1 - aa2 by A4, A18; then reconsider rx = x as Real ; aa1 < r1 + (r0 / 2) by A14, A25, XXREAL_1:4; then aa1 - aa2 < (r1 + (r0 / 2)) - (r2 - (r0 / 2)) by A23, XREAL_1:14; then aa1 - aa2 < (r1 - r2) + ((r0 / 2) + (r0 / 2)) ; then rx < r + r0 by A4, A27; hence x in ].(r - r0),(r + r0).[ by A27, A26, XXREAL_1:4; ::_thesis: verum end; ( W1 /\ W2 is open & p in W1 /\ W2 ) by A13, A11, XBOOLE_0:def_4; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A6, A15, XBOOLE_1:1; ::_thesis: verum end; then A28: g0 is continuous by Th10; for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g0 . p = r1 - r2 by A4; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 - r2 ) & g is continuous ) by A28; ::_thesis: verum end; theorem Th22: :: JGRAPH_2:22 for X being non empty TopSpace for f1 being Function of X,R^1 st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 * r1 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 * r1 ) & g is continuous ) let f1 be Function of X,R^1; ::_thesis: ( f1 is continuous implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 * r1 ) & g is continuous ) ) defpred S1[ set , set ] means for r1 being real number st f1 . $1 = r1 holds $2 = r1 * r1; A1: for x being Element of X ex y being Element of REAL st S1[x,y] proof let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y] reconsider r1 = f1 . x as Real by TOPMETR:17; set r3 = r1 * r1; for r1 being real number st f1 . x = r1 holds r1 * r1 = r1 * r1 ; hence ex y being Element of REAL st for r1 being real number st f1 . x = r1 holds y = r1 * r1 ; ::_thesis: verum end; ex f being Function of the carrier of X,REAL st for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A1); then consider f being Function of the carrier of X,REAL such that A2: for x being Element of X for r1 being real number st f1 . x = r1 holds f . x = r1 * r1 ; reconsider g0 = f as Function of X,R^1 by TOPMETR:17; assume A3: f1 is continuous ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 * r1 ) & g is continuous ) for p being Point of X for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) proof let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ) reconsider r = g0 . p as Real by TOPMETR:17; reconsider r1 = f1 . p as Real by TOPMETR:17; assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) then consider r0 being Real such that A4: r0 > 0 and A5: ].(r - r0),(r + r0).[ c= V by FRECHET:8; A6: r = r1 ^2 by A2; A7: r = r1 * r1 by A2; then A8: 0 <= r by XREAL_1:63; then A9: (sqrt (r + r0)) ^2 = r + r0 by A4, SQUARE_1:def_2; now__::_thesis:_(_(_r1_>=_0_&_ex_W_being_Subset_of_X_st_ (_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_or_(_r1_<_0_&_ex_W_being_Subset_of_X_st_ (_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_) percases ( r1 >= 0 or r1 < 0 ) ; caseA10: r1 >= 0 ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) set r4 = (sqrt (r + r0)) - (sqrt r); reconsider G1 = ].(r1 - ((sqrt (r + r0)) - (sqrt r))),(r1 + ((sqrt (r + r0)) - (sqrt r))).[ as Subset of R^1 by TOPMETR:17; A11: G1 is open by JORDAN6:35; r + r0 > r by A4, XREAL_1:29; then sqrt (r + r0) > sqrt r by A7, SQUARE_1:27, XREAL_1:63; then A12: (sqrt (r + r0)) - (sqrt r) > 0 by XREAL_1:50; then A13: r1 < r1 + ((sqrt (r + r0)) - (sqrt r)) by XREAL_1:29; then r1 - ((sqrt (r + r0)) - (sqrt r)) < r1 by XREAL_1:19; then f1 . p in G1 by A13, XXREAL_1:4; then consider W1 being Subset of X such that A14: ( p in W1 & W1 is open ) and A15: f1 .: W1 c= G1 by A3, A11, Th10; A16: r1 = sqrt r by A6, A10, SQUARE_1:def_2; set W = W1; A17: ((sqrt (r + r0)) - (sqrt r)) ^2 = (((sqrt (r + r0)) ^2) - ((2 * (sqrt (r + r0))) * (sqrt r))) + ((sqrt r) ^2) .= ((r + r0) - ((2 * (sqrt (r + r0))) * (sqrt r))) + ((sqrt r) ^2) by A4, A8, SQUARE_1:def_2 .= (r + (r0 - ((2 * (sqrt (r + r0))) * (sqrt r)))) + r by A8, SQUARE_1:def_2 .= ((2 * r) + r0) - ((2 * (sqrt (r + r0))) * (sqrt r)) ; g0 .: W1 c= ].(r - r0),(r + r0).[ proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A18: z in dom g0 and A19: z in W1 and A20: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A18; reconsider aa1 = f1 . pz as Real by TOPMETR:17; pz in the carrier of X ; then pz in dom f1 by FUNCT_2:def_1; then A21: f1 . pz in f1 .: W1 by A19, FUNCT_1:def_6; then A22: r1 - ((sqrt (r + r0)) - (sqrt r)) < aa1 by A15, XXREAL_1:4; A23: now__::_thesis:_(_(_0_<=_r1_-_((sqrt_(r_+_r0))_-_(sqrt_r))_&_r_-_r0_<_aa1_*_aa1_)_or_(_0_>_r1_-_((sqrt_(r_+_r0))_-_(sqrt_r))_&_r_-_r0_<_aa1_*_aa1_)_) percases ( 0 <= r1 - ((sqrt (r + r0)) - (sqrt r)) or 0 > r1 - ((sqrt (r + r0)) - (sqrt r)) ) ; caseA24: 0 <= r1 - ((sqrt (r + r0)) - (sqrt r)) ; ::_thesis: r - r0 < aa1 * aa1 (- 2) * (((sqrt (r + r0)) - (sqrt r)) ^2) <= 0 by XREAL_1:63, XREAL_1:131; then A25: (((r1 - ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2)) + (- (2 * (((sqrt (r + r0)) - (sqrt r)) ^2))) <= (((r1 - ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2)) + 0 by XREAL_1:7; (r1 - ((sqrt (r + r0)) - (sqrt r))) ^2 < aa1 ^2 by A22, A24, SQUARE_1:16; then (((r1 - ((sqrt (r + r0)) - (sqrt r))) ^2) - (2 * (((sqrt (r + r0)) - (sqrt r)) ^2))) - (aa1 ^2) < 0 by A25, XREAL_1:49; hence r - r0 < aa1 * aa1 by A7, A16, A17, XREAL_1:48; ::_thesis: verum end; case 0 > r1 - ((sqrt (r + r0)) - (sqrt r)) ; ::_thesis: r - r0 < aa1 * aa1 then r1 < (sqrt (r + r0)) - (sqrt r) by XREAL_1:48; then r1 ^2 < ((sqrt (r + r0)) - (sqrt r)) ^2 by A10, SQUARE_1:16; then (r1 ^2) - (((sqrt (r + r0)) - (sqrt r)) ^2) < 0 by XREAL_1:49; then ((r1 ^2) - (((sqrt (r + r0)) - (sqrt r)) ^2)) - ((2 * r1) * ((sqrt (r + r0)) - (sqrt r))) < 0 - 0 by A10, A12; hence r - r0 < aa1 * aa1 by A7, A16, A17, XREAL_1:63; ::_thesis: verum end; end; end; (- r1) - ((sqrt (r + r0)) - (sqrt r)) <= r1 - ((sqrt (r + r0)) - (sqrt r)) by A10, XREAL_1:9; then - (r1 + ((sqrt (r + r0)) - (sqrt r))) < aa1 by A22, XXREAL_0:2; then A26: aa1 - (- (r1 + ((sqrt (r + r0)) - (sqrt r)))) > 0 by XREAL_1:50; aa1 < r1 + ((sqrt (r + r0)) - (sqrt r)) by A15, A21, XXREAL_1:4; then (r1 + ((sqrt (r + r0)) - (sqrt r))) - aa1 > 0 by XREAL_1:50; then ((r1 + ((sqrt (r + r0)) - (sqrt r))) - aa1) * ((r1 + ((sqrt (r + r0)) - (sqrt r))) + aa1) > 0 by A26, XREAL_1:129; then ((r1 + ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2) > 0 ; then A27: aa1 ^2 < (r1 + ((sqrt (r + r0)) - (sqrt r))) ^2 by XREAL_1:47; x = aa1 * aa1 by A2, A20; hence x in ].(r - r0),(r + r0).[ by A7, A16, A17, A27, A23, XXREAL_1:4; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A5, A14, XBOOLE_1:1; ::_thesis: verum end; caseA28: r1 < 0 ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) set r4 = (sqrt (r + r0)) - (sqrt r); reconsider G1 = ].(r1 - ((sqrt (r + r0)) - (sqrt r))),(r1 + ((sqrt (r + r0)) - (sqrt r))).[ as Subset of R^1 by TOPMETR:17; A29: G1 is open by JORDAN6:35; r + r0 > r by A4, XREAL_1:29; then sqrt (r + r0) > sqrt r by A7, SQUARE_1:27, XREAL_1:63; then A30: (sqrt (r + r0)) - (sqrt r) > 0 by XREAL_1:50; then A31: r1 < r1 + ((sqrt (r + r0)) - (sqrt r)) by XREAL_1:29; then r1 - ((sqrt (r + r0)) - (sqrt r)) < r1 by XREAL_1:19; then f1 . p in G1 by A31, XXREAL_1:4; then consider W1 being Subset of X such that A32: ( p in W1 & W1 is open ) and A33: f1 .: W1 c= G1 by A3, A29, Th10; A34: (- r1) ^2 = r1 ^2 ; then A35: - r1 = sqrt r by A7, A28, SQUARE_1:22; set W = W1; A36: ((sqrt (r + r0)) - (sqrt r)) ^2 = ((r + r0) - ((2 * (sqrt (r + r0))) * (sqrt r))) + ((sqrt r) ^2) by A9 .= (r + (r0 - ((2 * (sqrt (r + r0))) * (sqrt r)))) + r by A7, A28, A34, SQUARE_1:22 .= ((2 * r) + r0) - ((2 * (sqrt (r + r0))) * (sqrt r)) ; then A37: (- ((2 * r1) * ((sqrt (r + r0)) - (sqrt r)))) + (((sqrt (r + r0)) - (sqrt r)) ^2) = r0 by A7, A35; g0 .: W1 c= ].(r - r0),(r + r0).[ proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A38: z in dom g0 and A39: z in W1 and A40: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A38; reconsider aa1 = f1 . pz as Real by TOPMETR:17; pz in the carrier of X ; then pz in dom f1 by FUNCT_2:def_1; then A41: f1 . pz in f1 .: W1 by A39, FUNCT_1:def_6; then A42: aa1 < r1 + ((sqrt (r + r0)) - (sqrt r)) by A33, XXREAL_1:4; A43: now__::_thesis:_(_(_0_>=_r1_+_((sqrt_(r_+_r0))_-_(sqrt_r))_&_r_-_r0_<_aa1_*_aa1_)_or_(_0_<_r1_+_((sqrt_(r_+_r0))_-_(sqrt_r))_&_r_-_r0_<_aa1_*_aa1_)_) percases ( 0 >= r1 + ((sqrt (r + r0)) - (sqrt r)) or 0 < r1 + ((sqrt (r + r0)) - (sqrt r)) ) ; caseA44: 0 >= r1 + ((sqrt (r + r0)) - (sqrt r)) ; ::_thesis: r - r0 < aa1 * aa1 (- 2) * (((sqrt (r + r0)) - (sqrt r)) ^2) <= 0 by XREAL_1:63, XREAL_1:131; then A45: (((r1 + ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2)) + (- (2 * (((sqrt (r + r0)) - (sqrt r)) ^2))) <= (((r1 + ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2)) + 0 by XREAL_1:7; - aa1 > - (r1 + ((sqrt (r + r0)) - (sqrt r))) by A42, XREAL_1:24; then (- (r1 + ((sqrt (r + r0)) - (sqrt r)))) ^2 < (- aa1) ^2 by A44, SQUARE_1:16; then (((r1 + ((sqrt (r + r0)) - (sqrt r))) ^2) - (2 * (((sqrt (r + r0)) - (sqrt r)) ^2))) - (aa1 ^2) < 0 by A45, XREAL_1:49; hence r - r0 < aa1 * aa1 by A7, A37, XREAL_1:48; ::_thesis: verum end; case 0 < r1 + ((sqrt (r + r0)) - (sqrt r)) ; ::_thesis: r - r0 < aa1 * aa1 then 0 + (- r1) < (r1 + ((sqrt (r + r0)) - (sqrt r))) + (- r1) by XREAL_1:8; then (- r1) ^2 < ((sqrt (r + r0)) - (sqrt r)) ^2 by A28, SQUARE_1:16; then (r1 ^2) - (r1 ^2) > (r1 ^2) - (((sqrt (r + r0)) - (sqrt r)) ^2) by XREAL_1:15; then ((r1 ^2) - (((sqrt (r + r0)) - (sqrt r)) ^2)) + ((2 * r1) * ((sqrt (r + r0)) - (sqrt r))) < 0 + 0 by A28, A30; hence r - r0 < aa1 * aa1 by A7, A35, A36, XREAL_1:63; ::_thesis: verum end; end; end; r1 - ((sqrt (r + r0)) - (sqrt r)) < aa1 by A33, A41, XXREAL_1:4; then aa1 - (r1 - ((sqrt (r + r0)) - (sqrt r))) > 0 by XREAL_1:50; then - ((- aa1) + (r1 - ((sqrt (r + r0)) - (sqrt r)))) > 0 ; then A46: (r1 - ((sqrt (r + r0)) - (sqrt r))) + (- aa1) < 0 ; (- r1) - ((sqrt (r + r0)) - (sqrt r)) >= r1 - ((sqrt (r + r0)) - (sqrt r)) by A28, XREAL_1:9; then - ((- r1) - ((sqrt (r + r0)) - (sqrt r))) <= - (r1 - ((sqrt (r + r0)) - (sqrt r))) by XREAL_1:24; then - (r1 - ((sqrt (r + r0)) - (sqrt r))) > aa1 by A42, XXREAL_0:2; then (- (r1 - ((sqrt (r + r0)) - (sqrt r)))) + (r1 - ((sqrt (r + r0)) - (sqrt r))) > aa1 + (r1 - ((sqrt (r + r0)) - (sqrt r))) by XREAL_1:8; then ((r1 - ((sqrt (r + r0)) - (sqrt r))) - aa1) * ((r1 - ((sqrt (r + r0)) - (sqrt r))) + aa1) > 0 by A46, XREAL_1:130; then ((r1 - ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2) > 0 ; then A47: aa1 ^2 < (r1 - ((sqrt (r + r0)) - (sqrt r))) ^2 by XREAL_1:47; x = aa1 * aa1 by A2, A40; hence x in ].(r - r0),(r + r0).[ by A7, A37, A47, A43, XXREAL_1:4; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A5, A32, XBOOLE_1:1; ::_thesis: verum end; end; end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum end; then A48: g0 is continuous by Th10; for p being Point of X for r1 being real number st f1 . p = r1 holds g0 . p = r1 * r1 by A2; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 * r1 ) & g is continuous ) by A48; ::_thesis: verum end; theorem Th23: :: JGRAPH_2:23 for X being non empty TopSpace for f1 being Function of X,R^1 for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a * r1 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a * r1 ) & g is continuous ) let f1 be Function of X,R^1; ::_thesis: for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a * r1 ) & g is continuous ) let a be real number ; ::_thesis: ( f1 is continuous implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a * r1 ) & g is continuous ) ) defpred S1[ set , set ] means for r1 being Real st f1 . $1 = r1 holds $2 = a * r1; A1: for x being Element of X ex y being Element of REAL st S1[x,y] proof let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y] reconsider r1 = f1 . x as Real by TOPMETR:17; reconsider r3 = a * r1 as Element of REAL ; for r1 being Real st f1 . x = r1 holds r3 = a * r1 ; hence ex y being Element of REAL st for r1 being Real st f1 . x = r1 holds y = a * r1 ; ::_thesis: verum end; ex f being Function of the carrier of X,REAL st for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A1); then consider f being Function of the carrier of X,REAL such that A2: for x being Element of X for r1 being Real st f1 . x = r1 holds f . x = a * r1 ; reconsider g0 = f as Function of X,R^1 by TOPMETR:17; A3: for p being Point of X for r1 being real number st f1 . p = r1 holds g0 . p = a * r1 proof let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds g0 . p = a * r1 let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g0 . p = a * r1 ) assume A4: f1 . p = r1 ; ::_thesis: g0 . p = a * r1 reconsider r1 = r1 as Element of REAL by XREAL_0:def_1; g0 . p = a * r1 by A2, A4; hence g0 . p = a * r1 ; ::_thesis: verum end; assume A5: f1 is continuous ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a * r1 ) & g is continuous ) for p being Point of X for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) proof let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ) reconsider r = g0 . p as Real by TOPMETR:17; reconsider r1 = f1 . p as Real by TOPMETR:17; assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) then consider r0 being Real such that A6: r0 > 0 and A7: ].(r - r0),(r + r0).[ c= V by FRECHET:8; A8: r = a * r1 by A2; A9: r = a * r1 by A2; now__::_thesis:_(_(_a_>=_0_&_ex_W_being_Subset_of_X_st_ (_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_or_(_a_<_0_&_ex_W_being_Subset_of_X_st_ (_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_) percases ( a >= 0 or a < 0 ) ; caseA10: a >= 0 ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) now__::_thesis:_(_(_a_>_0_&_ex_W_being_Subset_of_X_st_ (_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_or_(_a_=_0_&_ex_W_being_Subset_of_X_st_ (_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_) percases ( a > 0 or a = 0 ) by A10; caseA11: a > 0 ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) set r4 = r0 / a; reconsider G1 = ].(r1 - (r0 / a)),(r1 + (r0 / a)).[ as Subset of R^1 by TOPMETR:17; A12: r1 < r1 + (r0 / a) by A6, A11, XREAL_1:29, XREAL_1:139; then r1 - (r0 / a) < r1 by XREAL_1:19; then A13: f1 . p in G1 by A12, XXREAL_1:4; G1 is open by JORDAN6:35; then consider W1 being Subset of X such that A14: ( p in W1 & W1 is open ) and A15: f1 .: W1 c= G1 by A5, A13, Th10; set W = W1; g0 .: W1 c= ].(r - r0),(r + r0).[ proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A16: z in dom g0 and A17: z in W1 and A18: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A16; reconsider aa1 = f1 . pz as Real by TOPMETR:17; A19: x = a * aa1 by A2, A18; pz in the carrier of X ; then pz in dom f1 by FUNCT_2:def_1; then A20: f1 . pz in f1 .: W1 by A17, FUNCT_1:def_6; then r1 - (r0 / a) < aa1 by A15, XXREAL_1:4; then A21: a * (r1 - (r0 / a)) < a * aa1 by A11, XREAL_1:68; reconsider rx = x as Real by A18, XREAL_0:def_1; A22: a * (r1 + (r0 / a)) = (a * r1) + (a * (r0 / a)) .= r + r0 by A8, A11, XCMPLX_1:87 ; A23: a * (r1 - (r0 / a)) = (a * r1) - (a * (r0 / a)) .= r - r0 by A8, A11, XCMPLX_1:87 ; aa1 < r1 + (r0 / a) by A15, A20, XXREAL_1:4; then rx < r + r0 by A11, A19, A22, XREAL_1:68; hence x in ].(r - r0),(r + r0).[ by A19, A21, A23, XXREAL_1:4; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A7, A14, XBOOLE_1:1; ::_thesis: verum end; caseA24: a = 0 ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) set r4 = r0; reconsider G1 = ].(r1 - r0),(r1 + r0).[ as Subset of R^1 by TOPMETR:17; A25: r1 < r1 + r0 by A6, XREAL_1:29; then r1 - r0 < r1 by XREAL_1:19; then A26: f1 . p in G1 by A25, XXREAL_1:4; G1 is open by JORDAN6:35; then consider W1 being Subset of X such that A27: ( p in W1 & W1 is open ) and f1 .: W1 c= G1 by A5, A26, Th10; set W = W1; g0 .: W1 c= ].(r - r0),(r + r0).[ proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A28: z in dom g0 and z in W1 and A29: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A28; reconsider aa1 = f1 . pz as Real by TOPMETR:17; x = a * aa1 by A2, A29 .= 0 by A24 ; hence x in ].(r - r0),(r + r0).[ by A6, A9, A24, XXREAL_1:4; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A7, A27, XBOOLE_1:1; ::_thesis: verum end; end; end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum end; caseA30: a < 0 ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) set r4 = r0 / (- a); reconsider G1 = ].(r1 - (r0 / (- a))),(r1 + (r0 / (- a))).[ as Subset of R^1 by TOPMETR:17; - a > 0 by A30, XREAL_1:58; then A31: r1 < r1 + (r0 / (- a)) by A6, XREAL_1:29, XREAL_1:139; then r1 - (r0 / (- a)) < r1 by XREAL_1:19; then A32: f1 . p in G1 by A31, XXREAL_1:4; G1 is open by JORDAN6:35; then consider W1 being Subset of X such that A33: ( p in W1 & W1 is open ) and A34: f1 .: W1 c= G1 by A5, A32, Th10; set W = W1; - a <> 0 by A30; then A35: (- a) * (r0 / (- a)) = r0 by XCMPLX_1:87; g0 .: W1 c= ].(r - r0),(r + r0).[ proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A36: z in dom g0 and A37: z in W1 and A38: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A36; reconsider aa1 = f1 . pz as Real by TOPMETR:17; pz in the carrier of X ; then pz in dom f1 by FUNCT_2:def_1; then A39: f1 . pz in f1 .: W1 by A37, FUNCT_1:def_6; then r1 - (r0 / (- a)) < aa1 by A34, XXREAL_1:4; then A40: a * aa1 < a * (r1 - (r0 / (- a))) by A30, XREAL_1:69; A41: a * (r1 + (r0 / (- a))) = (a * r1) - (- (a * (r0 / (- a)))) .= r - r0 by A3, A35 ; A42: a * (r1 - (r0 / (- a))) = (a * r1) + (- (a * (r0 / (- a)))) .= r + r0 by A3, A35 ; aa1 < r1 + (r0 / (- a)) by A34, A39, XXREAL_1:4; then A43: r - r0 < a * aa1 by A30, A41, XREAL_1:69; x = a * aa1 by A2, A38; hence x in ].(r - r0),(r + r0).[ by A40, A42, A43, XXREAL_1:4; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A7, A33, XBOOLE_1:1; ::_thesis: verum end; end; end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum end; then g0 is continuous by Th10; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a * r1 ) & g is continuous ) by A3; ::_thesis: verum end; theorem Th24: :: JGRAPH_2:24 for X being non empty TopSpace for f1 being Function of X,R^1 for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 + a ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 + a ) & g is continuous ) let f1 be Function of X,R^1; ::_thesis: for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 + a ) & g is continuous ) let a be real number ; ::_thesis: ( f1 is continuous implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 + a ) & g is continuous ) ) defpred S1[ set , set ] means for r1 being Real st f1 . $1 = r1 holds $2 = r1 + a; A1: for x being Element of X ex y being Element of REAL st S1[x,y] proof reconsider r2 = a as Element of REAL by XREAL_0:def_1; let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y] reconsider r1 = f1 . x as Real by TOPMETR:17; set r3 = r1 + r2; for r1 being Real st f1 . x = r1 holds r1 + r2 = r1 + r2 ; hence ex y being Element of REAL st for r1 being Real st f1 . x = r1 holds y = r1 + a ; ::_thesis: verum end; ex f being Function of the carrier of X,REAL st for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A1); then consider f being Function of the carrier of X,REAL such that A2: for x being Element of X for r1 being Real st f1 . x = r1 holds f . x = r1 + a ; reconsider g0 = f as Function of X,R^1 by TOPMETR:17; assume A3: f1 is continuous ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 + a ) & g is continuous ) for p being Point of X for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) proof let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ) reconsider r = g0 . p as Real by TOPMETR:17; reconsider r1 = f1 . p as Real by TOPMETR:17; assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) then consider r0 being Real such that A4: r0 > 0 and A5: ].(r - r0),(r + r0).[ c= V by FRECHET:8; set r4 = r0; reconsider G1 = ].(r1 - r0),(r1 + r0).[ as Subset of R^1 by TOPMETR:17; A6: r1 < r1 + r0 by A4, XREAL_1:29; then r1 - r0 < r1 by XREAL_1:19; then A7: f1 . p in G1 by A6, XXREAL_1:4; G1 is open by JORDAN6:35; then consider W1 being Subset of X such that A8: ( p in W1 & W1 is open ) and A9: f1 .: W1 c= G1 by A3, A7, Th10; set W = W1; g0 .: W1 c= ].(r - r0),(r + r0).[ proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A10: z in dom g0 and A11: z in W1 and A12: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A10; reconsider aa1 = f1 . pz as Real by TOPMETR:17; pz in the carrier of X ; then pz in dom f1 by FUNCT_2:def_1; then A13: f1 . pz in f1 .: W1 by A11, FUNCT_1:def_6; then r1 - r0 < aa1 by A9, XXREAL_1:4; then A14: (r1 - r0) + a < aa1 + a by XREAL_1:8; A15: (r1 - r0) + a = (r1 + a) - r0 .= r - r0 by A2 ; aa1 < r1 + r0 by A9, A13, XXREAL_1:4; then A16: (r1 + r0) + a > aa1 + a by XREAL_1:8; x = aa1 + a by A2, A12; hence x in ].(r - r0),(r + r0).[ by A16, A14, A15, XXREAL_1:4; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A5, A8, XBOOLE_1:1; ::_thesis: verum end; then A17: g0 is continuous by Th10; for p being Point of X for r1 being real number st f1 . p = r1 holds g0 . p = r1 + a proof let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds g0 . p = r1 + a let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g0 . p = r1 + a ) assume A18: f1 . p = r1 ; ::_thesis: g0 . p = r1 + a reconsider r1 = r1 as Element of REAL by XREAL_0:def_1; g0 . p = r1 + a by A2, A18; hence g0 . p = r1 + a ; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 + a ) & g is continuous ) by A17; ::_thesis: verum end; theorem Th25: :: JGRAPH_2:25 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 * r2 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 * r2 ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 * r2 ) & g is continuous ) ) assume A1: ( f1 is continuous & f2 is continuous ) ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 * r2 ) & g is continuous ) then consider g1 being Function of X,R^1 such that A2: for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g1 . p = r1 + r2 and A3: g1 is continuous by Th19; consider g3 being Function of X,R^1 such that A4: for p being Point of X for r1 being real number st g1 . p = r1 holds g3 . p = r1 * r1 and A5: g3 is continuous by A3, Th22; consider g2 being Function of X,R^1 such that A6: for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g2 . p = r1 - r2 and A7: g2 is continuous by A1, Th21; consider g4 being Function of X,R^1 such that A8: for p being Point of X for r1 being real number st g2 . p = r1 holds g4 . p = r1 * r1 and A9: g4 is continuous by A7, Th22; consider g5 being Function of X,R^1 such that A10: for p being Point of X for r1, r2 being real number st g3 . p = r1 & g4 . p = r2 holds g5 . p = r1 - r2 and A11: g5 is continuous by A5, A9, Th21; consider g6 being Function of X,R^1 such that A12: for p being Point of X for r1 being real number st g5 . p = r1 holds g6 . p = (1 / 4) * r1 and A13: g6 is continuous by A11, Th23; for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g6 . p = r1 * r2 proof let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g6 . p = r1 * r2 let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g6 . p = r1 * r2 ) assume A14: ( f1 . p = r1 & f2 . p = r2 ) ; ::_thesis: g6 . p = r1 * r2 then g2 . p = r1 - r2 by A6; then A15: g4 . p = (r1 - r2) ^2 by A8; g1 . p = r1 + r2 by A2, A14; then g3 . p = (r1 + r2) ^2 by A4; then g5 . p = ((r1 + r2) ^2) - ((r1 - r2) ^2) by A10, A15; then g6 . p = (1 / 4) * ((((r1 ^2) + ((2 * r1) * r2)) + (r2 ^2)) - ((r1 - r2) ^2)) by A12 .= r1 * r2 ; hence g6 . p = r1 * r2 ; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 * r2 ) & g is continuous ) by A13; ::_thesis: verum end; theorem Th26: :: JGRAPH_2:26 for X being non empty TopSpace for f1 being Function of X,R^1 st f1 is continuous & ( for q being Point of X holds f1 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = 1 / r1 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 st f1 is continuous & ( for q being Point of X holds f1 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = 1 / r1 ) & g is continuous ) let f1 be Function of X,R^1; ::_thesis: ( f1 is continuous & ( for q being Point of X holds f1 . q <> 0 ) implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = 1 / r1 ) & g is continuous ) ) assume that A1: f1 is continuous and A2: for q being Point of X holds f1 . q <> 0 ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = 1 / r1 ) & g is continuous ) defpred S1[ set , set ] means for r1 being Real st f1 . $1 = r1 holds $2 = 1 / r1; A3: for x being Element of X ex y being Element of REAL st S1[x,y] proof let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y] reconsider r1 = f1 . x as Real by TOPMETR:17; set r3 = 1 / r1; for r1 being Real st f1 . x = r1 holds 1 / r1 = 1 / r1 ; hence ex y being Element of REAL st for r1 being Real st f1 . x = r1 holds y = 1 / r1 ; ::_thesis: verum end; ex f being Function of the carrier of X,REAL st for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A3); then consider f being Function of the carrier of X,REAL such that A4: for x being Element of X for r1 being Real st f1 . x = r1 holds f . x = 1 / r1 ; reconsider g0 = f as Function of X,R^1 by TOPMETR:17; for p being Point of X for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) proof let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ) reconsider r = g0 . p as Real by TOPMETR:17; reconsider r1 = f1 . p as Real by TOPMETR:17; assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) then consider r0 being Real such that A5: r0 > 0 and A6: ].(r - r0),(r + r0).[ c= V by FRECHET:8; A7: r = 1 / r1 by A4; A8: r1 <> 0 by A2; now__::_thesis:_(_(_r1_>=_0_&_ex_W_being_Subset_of_X_st_ (_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_or_(_r1_<_0_&_ex_W_being_Subset_of_X_st_ (_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_) percases ( r1 >= 0 or r1 < 0 ) ; caseA9: r1 >= 0 ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) set r4 = (r0 / r) / (r + r0); reconsider G1 = ].(r1 - ((r0 / r) / (r + r0))),(r1 + ((r0 / r) / (r + r0))).[ as Subset of R^1 by TOPMETR:17; r0 / r > 0 by A5, A8, A7, A9, XREAL_1:139; then A10: r1 < r1 + ((r0 / r) / (r + r0)) by A5, A7, A9, XREAL_1:29, XREAL_1:139; then r1 - ((r0 / r) / (r + r0)) < r1 by XREAL_1:19; then A11: f1 . p in G1 by A10, XXREAL_1:4; A12: r / (r + r0) > 0 by A5, A8, A7, A9, XREAL_1:139; G1 is open by JORDAN6:35; then consider W1 being Subset of X such that A13: ( p in W1 & W1 is open ) and A14: f1 .: W1 c= G1 by A1, A11, Th10; set W = W1; r1 - ((r0 / r) / (r + r0)) = (1 / r) - ((r0 / (r + r0)) / r) by A7 .= (1 - (r0 / (r + r0))) / r .= (((r + r0) / (r + r0)) - (r0 / (r + r0))) / r by A5, A7, A9, XCMPLX_1:60 .= (((r + r0) - r0) / (r + r0)) / r .= (r / (r + r0)) / r ; then A15: r1 - ((r0 / r) / (r + r0)) > 0 by A8, A7, A9, A12, XREAL_1:139; g0 .: W1 c= ].(r - r0),(r + r0).[ proof 0 < r0 ^2 by A5, SQUARE_1:12; then r0 * r < (r0 * r) + ((r0 * r0) + (r0 * r0)) by XREAL_1:29; then (r0 * r) - ((r0 * r0) + (r0 * r0)) < r0 * r by XREAL_1:19; then ((r0 * r) - ((r0 * r0) + (r0 * r0))) + (r * r) < (r * r) + (r0 * r) by XREAL_1:8; then ((r - r0) * ((r + r0) + r0)) / ((r + r0) + r0) < (r * (r + r0)) / ((r + r0) + r0) by A5, A7, A9, XREAL_1:74; then r - r0 < (r * (r + r0)) / ((r + r0) + r0) by A5, A7, A9, XCMPLX_1:89; then r - r0 < r / (((r + r0) + r0) / (r + r0)) by XCMPLX_1:77; then r - r0 < r / (((r + r0) / (r + r0)) + (r0 / (r + r0))) ; then r - r0 < (r * 1) / (1 + (r0 / (r + r0))) by A5, A7, A9, XCMPLX_1:60; then A16: r - r0 < 1 / ((1 + (r0 / (r + r0))) / r) by XCMPLX_1:77; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A17: z in dom g0 and A18: z in W1 and A19: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A17; reconsider aa1 = f1 . pz as Real by TOPMETR:17; A20: x = 1 / aa1 by A4, A19; pz in the carrier of X ; then pz in dom f1 by FUNCT_2:def_1; then A21: f1 . pz in f1 .: W1 by A18, FUNCT_1:def_6; then A22: r1 - ((r0 / r) / (r + r0)) < aa1 by A14, XXREAL_1:4; then A23: 1 / aa1 < 1 / (r1 - ((r0 / r) / (r + r0))) by A15, XREAL_1:88; aa1 < r1 + ((r0 / r) / (r + r0)) by A14, A21, XXREAL_1:4; then 1 / ((1 / r) + ((r0 / r) / (r + r0))) < 1 / aa1 by A7, A15, A22, XREAL_1:76; then A24: r - r0 < 1 / aa1 by A16, XXREAL_0:2; 1 / (r1 - ((r0 / r) / (r + r0))) = 1 / (r1 - ((r0 * (r ")) / (r + r0))) .= 1 / (r1 - ((r0 * (1 / r)) / (r + r0))) .= 1 / (r1 - (r0 / ((r + r0) / r1))) by A7, XCMPLX_1:77 .= 1 / ((r1 * 1) - (r1 * (r0 / (r + r0)))) by XCMPLX_1:81 .= 1 / ((1 - (r0 / (r + r0))) * r1) .= 1 / ((((r + r0) / (r + r0)) - (r0 / (r + r0))) * r1) by A5, A7, A9, XCMPLX_1:60 .= 1 / ((((r + r0) - r0) / (r + r0)) * r1) .= 1 / (r / ((r + r0) / r1)) by XCMPLX_1:81 .= 1 / ((r * r1) / (r + r0)) by XCMPLX_1:77 .= ((r + r0) / (r * r1)) * 1 by XCMPLX_1:80 .= (r + r0) / 1 by A8, A7, XCMPLX_0:def_7 .= r + r0 ; hence x in ].(r - r0),(r + r0).[ by A20, A24, A23, XXREAL_1:4; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A6, A13, XBOOLE_1:1; ::_thesis: verum end; caseA25: r1 < 0 ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) set r4 = (r0 / (- r)) / ((- r) + r0); reconsider G1 = ].(r1 - ((r0 / (- r)) / ((- r) + r0))),(r1 + ((r0 / (- r)) / ((- r) + r0))).[ as Subset of R^1 by TOPMETR:17; A26: G1 is open by JORDAN6:35; A27: 0 < - r by A7, A25, XREAL_1:58; then (- r) / ((- r) + r0) > 0 by A5, XREAL_1:139; then - (r / ((- r) + r0)) > 0 ; then A28: r / ((- r) + r0) < 0 ; r0 / (- r) > 0 by A5, A27, XREAL_1:139; then A29: r1 < r1 + ((r0 / (- r)) / ((- r) + r0)) by A5, A7, A25, XREAL_1:29, XREAL_1:139; then r1 - ((r0 / (- r)) / ((- r) + r0)) < r1 by XREAL_1:19; then f1 . p in G1 by A29, XXREAL_1:4; then consider W1 being Subset of X such that A30: ( p in W1 & W1 is open ) and A31: f1 .: W1 c= G1 by A1, A26, Th10; set W = W1; r1 * ((- r) * (1 / (- r))) = r1 * 1 by A27, XCMPLX_1:87; then (- (r * r1)) * (1 / (- r)) = r1 ; then A32: (- 1) * (1 / (- r)) = r1 by A2, A7, XCMPLX_1:87; then r1 + ((r0 / (- r)) / ((- r) + r0)) = (- (1 / (- r))) + ((r0 / ((- r) + r0)) / (- r)) .= ((- 1) / (- r)) + ((r0 / ((- r) + r0)) / (- r)) .= ((- 1) + (r0 / ((- r) + r0))) / (- r) .= ((- (((- r) + r0) / ((- r) + r0))) + (r0 / ((- r) + r0))) / (- r) by A5, A7, A25, XCMPLX_1:60 .= (((- ((- r) + r0)) / ((- r) + r0)) + (r0 / ((- r) + r0))) / (- r) .= (((r - r0) + r0) / ((- r) + r0)) / (- r) .= (r / ((- r) + r0)) / (- r) ; then A33: r1 + ((r0 / (- r)) / ((- r) + r0)) < 0 by A27, A28, XREAL_1:141; g0 .: W1 c= ].(r - r0),(r + r0).[ proof 0 < r0 ^2 by A5, SQUARE_1:12; then r0 * (- r) < (r0 * (- r)) + ((r0 * r0) + (r0 * r0)) by XREAL_1:29; then (r0 * (- r)) - ((r0 * r0) + (r0 * r0)) < r0 * (- r) by XREAL_1:19; then ((r0 * (- r)) - ((r0 * r0) + (r0 * r0))) + ((- r) * (- r)) < (r0 * (- r)) + ((- r) * (- r)) by XREAL_1:8; then (((- r) - r0) * (((- r) + r0) + r0)) / (((- r) + r0) + r0) < ((- r) * ((- r) + r0)) / (((- r) + r0) + r0) by A5, A7, A25, XREAL_1:74; then (- r) - r0 < ((- r) * ((- r) + r0)) / (((- r) + r0) + r0) by A5, A7, A25, XCMPLX_1:89; then (- r) - r0 < (- r) / ((((- r) + r0) + r0) / ((- r) + r0)) by XCMPLX_1:77; then (- r) - r0 < (- r) / ((((- r) + r0) / ((- r) + r0)) + (r0 / ((- r) + r0))) ; then (- r) - r0 < ((- r) * 1) / (1 + (r0 / ((- r) + r0))) by A5, A7, A25, XCMPLX_1:60; then (- r) - r0 < 1 / ((1 + (r0 / ((- r) + r0))) / (- r)) by XCMPLX_1:77; then - (r + r0) < 1 / ((1 / (- r)) + ((r0 / (- r)) / ((- r) + r0))) ; then r + r0 > - (1 / ((1 / (- r)) + ((r0 / (- r)) / ((- r) + r0)))) by XREAL_1:25; then A34: r + r0 > 1 / (- ((1 / (- r)) + ((r0 / (- r)) / ((- r) + r0)))) by XCMPLX_1:188; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ ) assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[ then consider z being set such that A35: z in dom g0 and A36: z in W1 and A37: g0 . z = x by FUNCT_1:def_6; reconsider pz = z as Point of X by A35; reconsider aa1 = f1 . pz as Real by TOPMETR:17; A38: x = 1 / aa1 by A4, A37; pz in the carrier of X ; then pz in dom f1 by FUNCT_2:def_1; then A39: f1 . pz in f1 .: W1 by A36, FUNCT_1:def_6; then A40: aa1 < r1 + ((r0 / (- r)) / ((- r) + r0)) by A31, XXREAL_1:4; then A41: 1 / aa1 > 1 / (r1 + ((r0 / (- r)) / ((- r) + r0))) by A33, XREAL_1:87; r1 - ((r0 / (- r)) / ((- r) + r0)) < aa1 by A31, A39, XXREAL_1:4; then 1 / ((- (1 / (- r))) - ((r0 / (- r)) / ((- r) + r0))) > 1 / aa1 by A32, A33, A40, XREAL_1:99; then A42: r + r0 > 1 / aa1 by A34, XXREAL_0:2; 1 / (r1 + ((r0 / (- r)) / ((- r) + r0))) = 1 / (r1 + ((r0 * ((- r) ")) / ((- r) + r0))) .= 1 / (r1 + ((r0 * (1 / (- r))) / ((- r) + r0))) .= 1 / (r1 + ((- (r1 * r0)) / ((- r) + r0))) by A32 .= 1 / (r1 + (- ((r1 * r0) / ((- r) + r0)))) .= 1 / (r1 - ((r1 * r0) / ((- r) + r0))) .= 1 / (r1 - (r0 / (((- r) + r0) / r1))) by XCMPLX_1:77 .= 1 / ((r1 * 1) - (r1 * (r0 / ((- r) + r0)))) by XCMPLX_1:81 .= 1 / (r1 * (1 - (r0 / ((- r) + r0)))) .= 1 / (((((- r) + r0) / ((- r) + r0)) - (r0 / ((- r) + r0))) * r1) by A5, A7, A25, XCMPLX_1:60 .= 1 / (((((- r) + r0) - r0) / (- (r - r0))) * r1) .= 1 / ((- ((((- r) + r0) - r0) / (r - r0))) * r1) by XCMPLX_1:188 .= 1 / (((((- r) + r0) - r0) / (r - r0)) * (- r1)) .= 1 / ((- r) / ((r - r0) / (- r1))) by XCMPLX_1:81 .= 1 / (((- r) * (- r1)) / (r - r0)) by XCMPLX_1:77 .= ((r - r0) / ((- r) * (- r1))) * 1 by XCMPLX_1:80 .= (r - r0) / ((- r) * ((- r) ")) by A32 .= (r - r0) / 1 by A27, XCMPLX_0:def_7 .= r - r0 ; hence x in ].(r - r0),(r + r0).[ by A38, A42, A41, XXREAL_1:4; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A6, A30, XBOOLE_1:1; ::_thesis: verum end; end; end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum end; then A43: g0 is continuous by Th10; for p being Point of X for r1 being real number st f1 . p = r1 holds g0 . p = 1 / r1 proof let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds g0 . p = 1 / r1 let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g0 . p = 1 / r1 ) assume A44: f1 . p = r1 ; ::_thesis: g0 . p = 1 / r1 reconsider r1 = r1 as Element of REAL by XREAL_0:def_1; g0 . p = 1 / r1 by A4, A44; hence g0 . p = 1 / r1 ; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = 1 / r1 ) & g is continuous ) by A43; ::_thesis: verum end; theorem Th27: :: JGRAPH_2:27 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 / r2 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 / r2 ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 / r2 ) & g is continuous ) ) assume that A1: f1 is continuous and A2: ( f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) ) ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 / r2 ) & g is continuous ) consider g1 being Function of X,R^1 such that A3: for p being Point of X for r2 being real number st f2 . p = r2 holds g1 . p = 1 / r2 and A4: g1 is continuous by A2, Th26; consider g2 being Function of X,R^1 such that A5: for p being Point of X for r1, r2 being real number st f1 . p = r1 & g1 . p = r2 holds g2 . p = r1 * r2 and A6: g2 is continuous by A1, A4, Th25; for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g2 . p = r1 / r2 proof let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g2 . p = r1 / r2 let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g2 . p = r1 / r2 ) assume that A7: f1 . p = r1 and A8: f2 . p = r2 ; ::_thesis: g2 . p = r1 / r2 g1 . p = 1 / r2 by A3, A8; then g2 . p = r1 * (1 / r2) by A5, A7 .= r1 / r2 ; hence g2 . p = r1 / r2 ; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 / r2 ) & g is continuous ) by A6; ::_thesis: verum end; theorem Th28: :: JGRAPH_2:28 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = (r1 / r2) / r2 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = (r1 / r2) / r2 ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = (r1 / r2) / r2 ) & g is continuous ) ) assume that A1: f1 is continuous and A2: ( f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) ) ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = (r1 / r2) / r2 ) & g is continuous ) consider g2 being Function of X,R^1 such that A3: for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g2 . p = r1 / r2 and A4: g2 is continuous by A1, A2, Th27; consider g3 being Function of X,R^1 such that A5: for p being Point of X for r1, r2 being real number st g2 . p = r1 & f2 . p = r2 holds g3 . p = r1 / r2 and A6: g3 is continuous by A2, A4, Th27; for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g3 . p = (r1 / r2) / r2 proof let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g3 . p = (r1 / r2) / r2 let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = (r1 / r2) / r2 ) assume that A7: f1 . p = r1 and A8: f2 . p = r2 ; ::_thesis: g3 . p = (r1 / r2) / r2 g2 . p = r1 / r2 by A3, A7, A8; hence g3 . p = (r1 / r2) / r2 by A5, A8; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = (r1 / r2) / r2 ) & g is continuous ) by A6; ::_thesis: verum end; theorem Th29: :: JGRAPH_2:29 for K0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj1 . p ) holds f is continuous proof reconsider g = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; let K0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj1 . p ) holds f is continuous let f be Function of ((TOP-REAL 2) | K0),R^1; ::_thesis: ( ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj1 . p ) implies f is continuous ) A1: ( dom f = the carrier of ((TOP-REAL 2) | K0) & the carrier of (TOP-REAL 2) /\ K0 = K0 ) by FUNCT_2:def_1, XBOOLE_1:28; A2: g is continuous by JORDAN5A:27; assume for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj1 . p ; ::_thesis: f is continuous then A3: for x being set st x in dom f holds f . x = proj1 . x ; the carrier of ((TOP-REAL 2) | K0) = [#] ((TOP-REAL 2) | K0) .= K0 by PRE_TOPC:def_5 ; then f = g | K0 by A1, A3, Th6, FUNCT_1:46; hence f is continuous by A2, TOPMETR:7; ::_thesis: verum end; theorem Th30: :: JGRAPH_2:30 for K0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj2 . p ) holds f is continuous proof let K0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj2 . p ) holds f is continuous let f be Function of ((TOP-REAL 2) | K0),R^1; ::_thesis: ( ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj2 . p ) implies f is continuous ) A1: ( dom f = the carrier of ((TOP-REAL 2) | K0) & the carrier of (TOP-REAL 2) /\ K0 = K0 ) by FUNCT_2:def_1, XBOOLE_1:28; assume for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj2 . p ; ::_thesis: f is continuous then A2: for x being set st x in dom f holds f . x = proj2 . x ; the carrier of ((TOP-REAL 2) | K0) = [#] ((TOP-REAL 2) | K0) .= K0 by PRE_TOPC:def_5 ; then f = proj2 | K0 by A1, A2, Th7, FUNCT_1:46; hence f is continuous by JORDAN5A:27; ::_thesis: verum end; theorem Th31: :: JGRAPH_2:31 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = 1 / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) holds f is continuous proof let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = 1 / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = 1 / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) implies f is continuous ) assume that A1: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = 1 / (p `1) and A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ; ::_thesis: f is continuous reconsider g1 = proj1 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17; A3: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = proj1 . q proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q = proj1 . q ( q in the carrier of ((TOP-REAL 2) | K1) & dom proj1 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1; then q in (dom proj1) /\ K1 by A3, XBOOLE_0:def_4; hence g1 . q = proj1 . q by FUNCT_1:48; ::_thesis: verum end; A5: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0 q in the carrier of ((TOP-REAL 2) | K1) ; then reconsider q2 = q as Point of (TOP-REAL 2) by A3; g1 . q = proj1 . q by A4 .= q2 `1 by PSCOMP_1:def_5 ; hence g1 . q <> 0 by A2; ::_thesis: verum end; g1 is continuous by A4, Th29; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A6: for q being Point of ((TOP-REAL 2) | K1) for r2 being real number st g1 . q = r2 holds g3 . q = 1 / r2 and A7: g3 is continuous by A5, Th26; A8: for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in [#] ((TOP-REAL 2) | K1) by A9; then x in K1 by PRE_TOPC:def_5; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( g1 . s = proj1 . s & proj1 . r = r `1 ) by A4, PSCOMP_1:def_5; f . r = 1 / (r `1) by A1, A9; hence f . x = g3 . x by A6, A10; ::_thesis: verum end; dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then dom f = dom g3 by FUNCT_2:def_1; hence f is continuous by A7, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th32: :: JGRAPH_2:32 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = 1 / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) holds f is continuous proof let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = 1 / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = 1 / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) implies f is continuous ) assume that A1: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = 1 / (p `2) and A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ; ::_thesis: f is continuous reconsider g1 = proj2 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17; A3: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = proj2 . q proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q = proj2 . q ( q in the carrier of ((TOP-REAL 2) | K1) & dom proj2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1; then q in (dom proj2) /\ K1 by A3, XBOOLE_0:def_4; hence g1 . q = proj2 . q by FUNCT_1:48; ::_thesis: verum end; A5: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0 q in the carrier of ((TOP-REAL 2) | K1) ; then reconsider q2 = q as Point of (TOP-REAL 2) by A3; g1 . q = proj2 . q by A4 .= q2 `2 by PSCOMP_1:def_6 ; hence g1 . q <> 0 by A2; ::_thesis: verum end; g1 is continuous by A4, Th30; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A6: for q being Point of ((TOP-REAL 2) | K1) for r2 being real number st g1 . q = r2 holds g3 . q = 1 / r2 and A7: g3 is continuous by A5, Th26; A8: for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in [#] ((TOP-REAL 2) | K1) by A9; then x in K1 by PRE_TOPC:def_5; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( g1 . s = proj2 . s & proj2 . r = r `2 ) by A4, PSCOMP_1:def_6; f . r = 1 / (r `2) by A1, A9; hence f . x = g3 . x by A6, A10; ::_thesis: verum end; dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then dom f = dom g3 by FUNCT_2:def_1; hence f is continuous by A7, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th33: :: JGRAPH_2:33 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = ((p `2) / (p `1)) / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) holds f is continuous proof let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = ((p `2) / (p `1)) / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = ((p `2) / (p `1)) / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) implies f is continuous ) assume that A1: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = ((p `2) / (p `1)) / (p `1) and A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ; ::_thesis: f is continuous reconsider g2 = proj2 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17; reconsider g1 = proj1 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17; A3: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = proj1 . q proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q = proj1 . q ( q in the carrier of ((TOP-REAL 2) | K1) & dom proj1 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1; then q in (dom proj1) /\ K1 by A3, XBOOLE_0:def_4; hence g1 . q = proj1 . q by FUNCT_1:48; ::_thesis: verum end; then A5: g1 is continuous by Th29; A6: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0 q in the carrier of ((TOP-REAL 2) | K1) ; then reconsider q2 = q as Point of (TOP-REAL 2) by A3; g1 . q = proj1 . q by A4 .= q2 `1 by PSCOMP_1:def_5 ; hence g1 . q <> 0 by A2; ::_thesis: verum end; A7: for q being Point of ((TOP-REAL 2) | K1) holds g2 . q = proj2 . q proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g2 . q = proj2 . q ( q in the carrier of ((TOP-REAL 2) | K1) & dom proj2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1; then q in (dom proj2) /\ K1 by A3, XBOOLE_0:def_4; hence g2 . q = proj2 . q by FUNCT_1:48; ::_thesis: verum end; then g2 is continuous by Th30; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A8: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = (r1 / r2) / r2 and A9: g3 is continuous by A5, A6, Th28; A10: for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A11: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in [#] ((TOP-REAL 2) | K1) by A11; then x in K1 by PRE_TOPC:def_5; then reconsider r = x as Point of (TOP-REAL 2) ; A12: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6; A13: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by A7, A4; f . r = ((r `2) / (r `1)) / (r `1) by A1, A11; hence f . x = g3 . x by A8, A13, A12; ::_thesis: verum end; dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then dom f = dom g3 by FUNCT_2:def_1; hence f is continuous by A9, A10, FUNCT_1:2; ::_thesis: verum end; theorem Th34: :: JGRAPH_2:34 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = ((p `1) / (p `2)) / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) holds f is continuous proof let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = ((p `1) / (p `2)) / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = ((p `1) / (p `2)) / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) implies f is continuous ) assume that A1: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = ((p `1) / (p `2)) / (p `2) and A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ; ::_thesis: f is continuous reconsider g2 = proj1 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17; reconsider g1 = proj2 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17; A3: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = proj2 . q proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q = proj2 . q ( q in the carrier of ((TOP-REAL 2) | K1) & dom proj2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1; then q in (dom proj2) /\ K1 by A3, XBOOLE_0:def_4; hence g1 . q = proj2 . q by FUNCT_1:48; ::_thesis: verum end; then A5: g1 is continuous by Th30; A6: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0 q in the carrier of ((TOP-REAL 2) | K1) ; then reconsider q2 = q as Point of (TOP-REAL 2) by A3; g1 . q = proj2 . q by A4 .= q2 `2 by PSCOMP_1:def_6 ; hence g1 . q <> 0 by A2; ::_thesis: verum end; A7: for q being Point of ((TOP-REAL 2) | K1) holds g2 . q = proj1 . q proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g2 . q = proj1 . q ( q in the carrier of ((TOP-REAL 2) | K1) & dom proj1 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1; then q in (dom proj1) /\ K1 by A3, XBOOLE_0:def_4; hence g2 . q = proj1 . q by FUNCT_1:48; ::_thesis: verum end; then g2 is continuous by Th29; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A8: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = (r1 / r2) / r2 and A9: g3 is continuous by A5, A6, Th28; A10: for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A11: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in [#] ((TOP-REAL 2) | K1) by A11; then x in K1 by PRE_TOPC:def_5; then reconsider r = x as Point of (TOP-REAL 2) ; A12: ( proj1 . r = r `1 & proj2 . r = r `2 ) by PSCOMP_1:def_5, PSCOMP_1:def_6; A13: ( g2 . s = proj1 . s & g1 . s = proj2 . s ) by A7, A4; f . r = ((r `1) / (r `2)) / (r `2) by A1, A11; hence f . x = g3 . x by A8, A13, A12; ::_thesis: verum end; dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then dom f = dom g3 by FUNCT_2:def_1; hence f is continuous by A9, A10, FUNCT_1:2; ::_thesis: verum end; theorem Th35: :: JGRAPH_2:35 for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) for f1, f2 being Function of ((TOP-REAL 2) | K0),R^1 st f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & ( for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| ) holds f is continuous proof let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) for f1, f2 being Function of ((TOP-REAL 2) | K0),R^1 st f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & ( for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| ) holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: for f1, f2 being Function of ((TOP-REAL 2) | K0),R^1 st f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & ( for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| ) holds f is continuous let f1, f2 be Function of ((TOP-REAL 2) | K0),R^1; ::_thesis: ( f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & ( for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| ) implies f is continuous ) assume that A1: f1 is continuous and A2: f2 is continuous and A3: K0 <> {} and A4: B0 <> {} and A5: for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| ; ::_thesis: f is continuous reconsider B1 = B0 as non empty Subset of (TOP-REAL 2) by A4; reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) by A3; reconsider X = (TOP-REAL 2) | K1, Y = (TOP-REAL 2) | B1 as non empty TopSpace ; reconsider f0 = f as Function of X,Y ; for r being Point of X for V being Subset of Y st f0 . r in V & V is open holds ex W being Subset of X st ( r in W & W is open & f0 .: W c= V ) proof let r be Point of X; ::_thesis: for V being Subset of Y st f0 . r in V & V is open holds ex W being Subset of X st ( r in W & W is open & f0 .: W c= V ) let V be Subset of Y; ::_thesis: ( f0 . r in V & V is open implies ex W being Subset of X st ( r in W & W is open & f0 .: W c= V ) ) assume that A6: f0 . r in V and A7: V is open ; ::_thesis: ex W being Subset of X st ( r in W & W is open & f0 .: W c= V ) consider V2 being Subset of (TOP-REAL 2) such that A8: V2 is open and A9: V = V2 /\ ([#] Y) by A7, TOPS_2:24; A10: V2 /\ ([#] Y) c= V2 by XBOOLE_1:17; then f0 . r in V2 by A6, A9; then reconsider p = f0 . r as Point of (TOP-REAL 2) ; consider r2 being real number such that A11: r2 > 0 and A12: { q where q is Point of (TOP-REAL 2) : ( (p `1) - r2 < q `1 & q `1 < (p `1) + r2 & (p `2) - r2 < q `2 & q `2 < (p `2) + r2 ) } c= V2 by A6, A8, A9, A10, Th11; reconsider G1 = ].((p `1) - r2),((p `1) + r2).[, G2 = ].((p `2) - r2),((p `2) + r2).[ as Subset of R^1 by TOPMETR:17; A13: G1 is open by JORDAN6:35; A14: r in the carrier of X ; then r in dom f2 by FUNCT_2:def_1; then A15: f2 . r in rng f2 by FUNCT_1:3; r in dom f1 by A14, FUNCT_2:def_1; then f1 . r in rng f1 by FUNCT_1:3; then reconsider r3 = f1 . r, r4 = f2 . r as Real by A15, TOPMETR:17; A16: the carrier of X = [#] X .= K0 by PRE_TOPC:def_5 ; then r in K0 ; then reconsider pr = r as Point of (TOP-REAL 2) ; A17: r = |[(pr `1),(pr `2)]| by EUCLID:53; then A18: f0 . |[(pr `1),(pr `2)]| = |[r3,r4]| by A5, A16; A19: p `2 < (p `2) + r2 by A11, XREAL_1:29; then (p `2) - r2 < p `2 by XREAL_1:19; then p `2 in ].((p `2) - r2),((p `2) + r2).[ by A19, XXREAL_1:4; then ( G2 is open & f2 . r in G2 ) by A17, A18, EUCLID:52, JORDAN6:35; then consider W2 being Subset of X such that A20: r in W2 and A21: W2 is open and A22: f2 .: W2 c= G2 by A2, Th10; A23: p `1 < (p `1) + r2 by A11, XREAL_1:29; then (p `1) - r2 < p `1 by XREAL_1:19; then p `1 in ].((p `1) - r2),((p `1) + r2).[ by A23, XXREAL_1:4; then f1 . r in ].((p `1) - r2),((p `1) + r2).[ by A17, A18, EUCLID:52; then consider W1 being Subset of X such that A24: r in W1 and A25: W1 is open and A26: f1 .: W1 c= G1 by A1, A13, Th10; reconsider W5 = W1 /\ W2 as Subset of X ; f2 .: W5 c= f2 .: W2 by RELAT_1:123, XBOOLE_1:17; then A27: f2 .: W5 c= G2 by A22, XBOOLE_1:1; f1 .: W5 c= f1 .: W1 by RELAT_1:123, XBOOLE_1:17; then A28: f1 .: W5 c= G1 by A26, XBOOLE_1:1; A29: f0 .: W5 c= V proof let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in f0 .: W5 or v in V ) assume A30: v in f0 .: W5 ; ::_thesis: v in V then reconsider q2 = v as Point of Y ; consider k being set such that A31: k in dom f0 and A32: k in W5 and A33: q2 = f0 . k by A30, FUNCT_1:def_6; the carrier of X = [#] X .= K0 by PRE_TOPC:def_5 ; then k in K0 by A31; then reconsider r8 = k as Point of (TOP-REAL 2) ; A34: dom f0 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1 .= [#] ((TOP-REAL 2) | K1) .= K0 by PRE_TOPC:def_5 ; then A35: |[(r8 `1),(r8 `2)]| in K0 by A31, EUCLID:53; A36: dom f2 = the carrier of ((TOP-REAL 2) | K0) by FUNCT_2:def_1 .= [#] ((TOP-REAL 2) | K0) .= K0 by PRE_TOPC:def_5 ; then A37: f2 . |[(r8 `1),(r8 `2)]| in rng f2 by A35, FUNCT_1:def_3; A38: dom f1 = the carrier of ((TOP-REAL 2) | K0) by FUNCT_2:def_1 .= [#] ((TOP-REAL 2) | K0) .= K0 by PRE_TOPC:def_5 ; then f1 . |[(r8 `1),(r8 `2)]| in rng f1 by A35, FUNCT_1:def_3; then reconsider r7 = f1 . |[(r8 `1),(r8 `2)]|, s7 = f2 . |[(r8 `1),(r8 `2)]| as Real by A37, TOPMETR:17; A39: |[(r8 `1),(r8 `2)]| in W5 by A32, EUCLID:53; then f1 . |[(r8 `1),(r8 `2)]| in f1 .: W5 by A35, A38, FUNCT_1:def_6; then A40: ( (p `1) - r2 < r7 & r7 < (p `1) + r2 ) by A28, XXREAL_1:4; f2 . |[(r8 `1),(r8 `2)]| in f2 .: W5 by A35, A36, A39, FUNCT_1:def_6; then A41: ( (p `2) - r2 < s7 & s7 < (p `2) + r2 ) by A27, XXREAL_1:4; k = |[(r8 `1),(r8 `2)]| by EUCLID:53; then A42: v = |[r7,s7]| by A5, A31, A33, A34; ( |[r7,s7]| `1 = r7 & |[r7,s7]| `2 = s7 ) by EUCLID:52; then ( q2 in [#] Y & v in { q3 where q3 is Point of (TOP-REAL 2) : ( (p `1) - r2 < q3 `1 & q3 `1 < (p `1) + r2 & (p `2) - r2 < q3 `2 & q3 `2 < (p `2) + r2 ) } ) by A42, A40, A41; hence v in V by A9, A12, XBOOLE_0:def_4; ::_thesis: verum end; r in W5 by A24, A20, XBOOLE_0:def_4; hence ex W being Subset of X st ( r in W & W is open & f0 .: W c= V ) by A25, A21, A29; ::_thesis: verum end; hence f is continuous by Th10; ::_thesis: verum end; theorem Th36: :: JGRAPH_2:36 for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) A1: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19; assume A2: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous A3: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume A4: x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A2, A4, XBOOLE_0:def_5; ::_thesis: verum end; ( ( (1.REAL 2) `2 <= (1.REAL 2) `1 & - ((1.REAL 2) `1) <= (1.REAL 2) `2 ) or ( (1.REAL 2) `2 >= (1.REAL 2) `1 & (1.REAL 2) `2 <= - ((1.REAL 2) `1) ) ) by Th5; then A5: 1.REAL 2 in K0 by A2, A1; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A6: K1 c= NonZero (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) ) assume A7: z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2; then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence z in NonZero (TOP-REAL 2) by A7, XBOOLE_0:def_5; ::_thesis: verum end; A8: dom (Out_In_Sq | K1) c= dom (proj2 * (Out_In_Sq | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Out_In_Sq | K1) or x in dom (proj2 * (Out_In_Sq | K1)) ) assume A9: x in dom (Out_In_Sq | K1) ; ::_thesis: x in dom (proj2 * (Out_In_Sq | K1)) then x in (dom Out_In_Sq) /\ K1 by RELAT_1:61; then x in dom Out_In_Sq by XBOOLE_0:def_4; then Out_In_Sq . x in rng Out_In_Sq by FUNCT_1:3; then A10: ( dom proj2 = the carrier of (TOP-REAL 2) & Out_In_Sq . x in the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1, XBOOLE_0:def_5; (Out_In_Sq | K1) . x = Out_In_Sq . x by A9, FUNCT_1:47; hence x in dom (proj2 * (Out_In_Sq | K1)) by A9, A10, FUNCT_1:11; ::_thesis: verum end; A11: rng (proj2 * (Out_In_Sq | K1)) c= the carrier of R^1 by TOPMETR:17; A12: NonZero (TOP-REAL 2) <> {} by Th9; A13: dom (Out_In_Sq | K1) c= dom (proj1 * (Out_In_Sq | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Out_In_Sq | K1) or x in dom (proj1 * (Out_In_Sq | K1)) ) assume A14: x in dom (Out_In_Sq | K1) ; ::_thesis: x in dom (proj1 * (Out_In_Sq | K1)) then x in (dom Out_In_Sq) /\ K1 by RELAT_1:61; then x in dom Out_In_Sq by XBOOLE_0:def_4; then Out_In_Sq . x in rng Out_In_Sq by FUNCT_1:3; then A15: ( dom proj1 = the carrier of (TOP-REAL 2) & Out_In_Sq . x in the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1, XBOOLE_0:def_5; (Out_In_Sq | K1) . x = Out_In_Sq . x by A14, FUNCT_1:47; hence x in dom (proj1 * (Out_In_Sq | K1)) by A14, A15, FUNCT_1:11; ::_thesis: verum end; A16: rng (proj1 * (Out_In_Sq | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj1 * (Out_In_Sq | K1)) c= dom (Out_In_Sq | K1) by RELAT_1:25; then dom (proj1 * (Out_In_Sq | K1)) = dom (Out_In_Sq | K1) by A13, XBOOLE_0:def_10 .= (dom Out_In_Sq) /\ K1 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K1 by A12, FUNCT_2:def_1 .= K1 by A6, XBOOLE_1:28 .= [#] ((TOP-REAL 2) | K1) by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | K1) ; then reconsider g1 = proj1 * (Out_In_Sq | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A16, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = 1 / (p `1) proof A17: K1 c= NonZero (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) ) assume A18: z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2; then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence z in NonZero (TOP-REAL 2) by A18, XBOOLE_0:def_5; ::_thesis: verum end; A19: NonZero (TOP-REAL 2) <> {} by Th9; A20: dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K1 by A19, FUNCT_2:def_1 .= K1 by A17, XBOOLE_1:28 ; let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = 1 / (p `1) ) A21: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; assume A22: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = 1 / (p `1) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A21; then A23: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by Def1; (Out_In_Sq | K1) . p = Out_In_Sq . p by A22, A21, FUNCT_1:49; then g1 . p = proj1 . |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by A22, A20, A21, A23, FUNCT_1:13 .= |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 by PSCOMP_1:def_5 .= 1 / (p `1) by EUCLID:52 ; hence g1 . p = 1 / (p `1) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A24: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = 1 / (p `1) ; dom (proj2 * (Out_In_Sq | K1)) c= dom (Out_In_Sq | K1) by RELAT_1:25; then dom (proj2 * (Out_In_Sq | K1)) = dom (Out_In_Sq | K1) by A8, XBOOLE_0:def_10 .= (dom Out_In_Sq) /\ K1 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K1 by A12, FUNCT_2:def_1 .= K1 by A6, XBOOLE_1:28 .= [#] ((TOP-REAL 2) | K1) by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | K1) ; then reconsider g2 = proj2 * (Out_In_Sq | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A11, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = ((p `2) / (p `1)) / (p `1) proof A25: NonZero (TOP-REAL 2) <> {} by Th9; A26: dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K1 by A25, FUNCT_2:def_1 .= K1 by A6, XBOOLE_1:28 ; let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = ((p `2) / (p `1)) / (p `1) ) A27: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; assume A28: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = ((p `2) / (p `1)) / (p `1) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A27; then A29: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by Def1; (Out_In_Sq | K1) . p = Out_In_Sq . p by A28, A27, FUNCT_1:49; then g2 . p = proj2 . |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by A28, A26, A27, A29, FUNCT_1:13 .= |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 by PSCOMP_1:def_6 .= ((p `2) / (p `1)) / (p `1) by EUCLID:52 ; hence g2 . p = ((p `2) / (p `1)) / (p `1) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A30: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = ((p `2) / (p `1)) / (p `1) ; A31: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies q `1 <> 0 ) A32: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: q `1 <> 0 then A33: ex p3 being Point of (TOP-REAL 2) st ( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A32; now__::_thesis:_not_q_`1_=_0 assume A34: q `1 = 0 ; ::_thesis: contradiction then q `2 = 0 by A33; hence contradiction by A33, A34, EUCLID:53, EUCLID:54; ::_thesis: verum end; hence q `1 <> 0 ; ::_thesis: verum end; then A35: f1 is continuous by A24, Th31; A36: for x, y, r, s being real number st |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| proof let x, y, r, s be real number ; ::_thesis: ( |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| implies f . |[x,y]| = |[r,s]| ) assume that A37: |[x,y]| in K1 and A38: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[r,s]| set p99 = |[x,y]|; A39: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; then A40: f1 . |[x,y]| = 1 / (|[x,y]| `1) by A24, A37; A41: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A37; then ( ( ( |[x,y]| `2 <= |[x,y]| `1 & - (|[x,y]| `1) <= |[x,y]| `2 ) or ( |[x,y]| `2 >= |[x,y]| `1 & |[x,y]| `2 <= - (|[x,y]| `1) ) ) implies Out_In_Sq . |[x,y]| = |[(1 / (|[x,y]| `1)),(((|[x,y]| `2) / (|[x,y]| `1)) / (|[x,y]| `1))]| ) by Def1; then (Out_In_Sq | K0) . |[x,y]| = |[(1 / (|[x,y]| `1)),(((|[x,y]| `2) / (|[x,y]| `1)) / (|[x,y]| `1))]| by A37, A41, FUNCT_1:49 .= |[r,s]| by A30, A37, A38, A39, A40 ; hence f . |[x,y]| = |[r,s]| by A2; ::_thesis: verum end; f2 is continuous by A31, A30, Th33; hence f is continuous by A5, A3, A35, A36, Th35; ::_thesis: verum end; theorem Th37: :: JGRAPH_2:37 for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) A1: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19; assume A2: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous A3: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume A4: x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A2, A4, XBOOLE_0:def_5; ::_thesis: verum end; ( ( (1.REAL 2) `1 <= (1.REAL 2) `2 & - ((1.REAL 2) `2) <= (1.REAL 2) `1 ) or ( (1.REAL 2) `1 >= (1.REAL 2) `2 & (1.REAL 2) `1 <= - ((1.REAL 2) `2) ) ) by Th5; then A5: 1.REAL 2 in K0 by A2, A1; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A6: K1 c= NonZero (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) ) assume A7: z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2; then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence z in NonZero (TOP-REAL 2) by A7, XBOOLE_0:def_5; ::_thesis: verum end; A8: dom (Out_In_Sq | K1) c= dom (proj1 * (Out_In_Sq | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Out_In_Sq | K1) or x in dom (proj1 * (Out_In_Sq | K1)) ) assume A9: x in dom (Out_In_Sq | K1) ; ::_thesis: x in dom (proj1 * (Out_In_Sq | K1)) then x in (dom Out_In_Sq) /\ K1 by RELAT_1:61; then x in dom Out_In_Sq by XBOOLE_0:def_4; then Out_In_Sq . x in rng Out_In_Sq by FUNCT_1:3; then A10: ( dom proj1 = the carrier of (TOP-REAL 2) & Out_In_Sq . x in the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1, XBOOLE_0:def_5; (Out_In_Sq | K1) . x = Out_In_Sq . x by A9, FUNCT_1:47; hence x in dom (proj1 * (Out_In_Sq | K1)) by A9, A10, FUNCT_1:11; ::_thesis: verum end; A11: rng (proj1 * (Out_In_Sq | K1)) c= the carrier of R^1 by TOPMETR:17; A12: NonZero (TOP-REAL 2) <> {} by Th9; A13: dom (Out_In_Sq | K1) c= dom (proj2 * (Out_In_Sq | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Out_In_Sq | K1) or x in dom (proj2 * (Out_In_Sq | K1)) ) assume A14: x in dom (Out_In_Sq | K1) ; ::_thesis: x in dom (proj2 * (Out_In_Sq | K1)) then x in (dom Out_In_Sq) /\ K1 by RELAT_1:61; then x in dom Out_In_Sq by XBOOLE_0:def_4; then Out_In_Sq . x in rng Out_In_Sq by FUNCT_1:3; then A15: ( dom proj2 = the carrier of (TOP-REAL 2) & Out_In_Sq . x in the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1, XBOOLE_0:def_5; (Out_In_Sq | K1) . x = Out_In_Sq . x by A14, FUNCT_1:47; hence x in dom (proj2 * (Out_In_Sq | K1)) by A14, A15, FUNCT_1:11; ::_thesis: verum end; A16: rng (proj2 * (Out_In_Sq | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj2 * (Out_In_Sq | K1)) c= dom (Out_In_Sq | K1) by RELAT_1:25; then dom (proj2 * (Out_In_Sq | K1)) = dom (Out_In_Sq | K1) by A13, XBOOLE_0:def_10 .= (dom Out_In_Sq) /\ K1 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K1 by A12, FUNCT_2:def_1 .= K1 by A6, XBOOLE_1:28 .= [#] ((TOP-REAL 2) | K1) by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | K1) ; then reconsider g1 = proj2 * (Out_In_Sq | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A16, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = 1 / (p `2) proof A17: K1 c= NonZero (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) ) assume A18: z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2; then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence z in NonZero (TOP-REAL 2) by A18, XBOOLE_0:def_5; ::_thesis: verum end; A19: NonZero (TOP-REAL 2) <> {} by Th9; A20: dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K1 by A19, FUNCT_2:def_1 .= K1 by A17, XBOOLE_1:28 ; let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = 1 / (p `2) ) A21: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; assume A22: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = 1 / (p `2) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A21; then A23: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by Th14; (Out_In_Sq | K1) . p = Out_In_Sq . p by A22, A21, FUNCT_1:49; then g1 . p = proj2 . |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A22, A20, A21, A23, FUNCT_1:13 .= |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 by PSCOMP_1:def_6 .= 1 / (p `2) by EUCLID:52 ; hence g1 . p = 1 / (p `2) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A24: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = 1 / (p `2) ; dom (proj1 * (Out_In_Sq | K1)) c= dom (Out_In_Sq | K1) by RELAT_1:25; then dom (proj1 * (Out_In_Sq | K1)) = dom (Out_In_Sq | K1) by A8, XBOOLE_0:def_10 .= (dom Out_In_Sq) /\ K1 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K1 by A12, FUNCT_2:def_1 .= K1 by A6, XBOOLE_1:28 .= [#] ((TOP-REAL 2) | K1) by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | K1) ; then reconsider g2 = proj1 * (Out_In_Sq | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A11, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = ((p `1) / (p `2)) / (p `2) proof A25: NonZero (TOP-REAL 2) <> {} by Th9; A26: dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K1 by A25, FUNCT_2:def_1 .= K1 by A6, XBOOLE_1:28 ; let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = ((p `1) / (p `2)) / (p `2) ) A27: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; assume A28: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = ((p `1) / (p `2)) / (p `2) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A27; then A29: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by Th14; (Out_In_Sq | K1) . p = Out_In_Sq . p by A28, A27, FUNCT_1:49; then g2 . p = proj1 . |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A28, A26, A27, A29, FUNCT_1:13 .= |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 by PSCOMP_1:def_5 .= ((p `1) / (p `2)) / (p `2) by EUCLID:52 ; hence g2 . p = ((p `1) / (p `2)) / (p `2) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A30: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = ((p `1) / (p `2)) / (p `2) ; A31: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies q `2 <> 0 ) A32: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: q `2 <> 0 then A33: ex p3 being Point of (TOP-REAL 2) st ( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A32; now__::_thesis:_not_q_`2_=_0 assume A34: q `2 = 0 ; ::_thesis: contradiction then q `1 = 0 by A33; hence contradiction by A33, A34, EUCLID:53, EUCLID:54; ::_thesis: verum end; hence q `2 <> 0 ; ::_thesis: verum end; then A35: f1 is continuous by A24, Th32; A36: for x, y, s, r being real number st |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| holds f . |[x,y]| = |[s,r]| proof let x, y, s, r be real number ; ::_thesis: ( |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| implies f . |[x,y]| = |[s,r]| ) assume that A37: |[x,y]| in K1 and A38: ( s = f2 . |[x,y]| & r = f1 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[s,r]| set p99 = |[x,y]|; A39: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A37; A40: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1) .= K1 by PRE_TOPC:def_5 ; then A41: f1 . |[x,y]| = 1 / (|[x,y]| `2) by A24, A37; (Out_In_Sq | K0) . |[x,y]| = Out_In_Sq . |[x,y]| by A37, FUNCT_1:49 .= |[(((|[x,y]| `1) / (|[x,y]| `2)) / (|[x,y]| `2)),(1 / (|[x,y]| `2))]| by A39, Th14 .= |[s,r]| by A30, A37, A38, A40, A41 ; hence f . |[x,y]| = |[s,r]| by A2; ::_thesis: verum end; f2 is continuous by A31, A30, Th34; hence f is continuous by A5, A3, A35, A36, Th35; ::_thesis: verum end; scheme :: JGRAPH_2:sch 1 TopSubset{ P1[ set ] } : { p where p is Point of (TOP-REAL 2) : P1[p] } is Subset of (TOP-REAL 2) proof { p where p is Point of (TOP-REAL 2) : P1[p] } c= the carrier of (TOP-REAL 2) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : P1[p] } or x in the carrier of (TOP-REAL 2) ) assume x in { p where p is Point of (TOP-REAL 2) : P1[p] } ; ::_thesis: x in the carrier of (TOP-REAL 2) then ex p being Point of (TOP-REAL 2) st ( p = x & P1[p] ) ; hence x in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; hence { p where p is Point of (TOP-REAL 2) : P1[p] } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; scheme :: JGRAPH_2:sch 2 TopCompl{ P1[ set ], F1() -> Subset of (TOP-REAL 2) } : F1() ` = { p where p is Point of (TOP-REAL 2) : P1[p] } provided A1: F1() = { p where p is Point of (TOP-REAL 2) : P1[p] } proof thus F1() ` c= { p where p is Point of (TOP-REAL 2) : P1[p] } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of (TOP-REAL 2) : P1[p] } c= F1() ` proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F1() ` or x in { p where p is Point of (TOP-REAL 2) : P1[p] } ) assume A2: x in F1() ` ; ::_thesis: x in { p where p is Point of (TOP-REAL 2) : P1[p] } then reconsider qx = x as Point of (TOP-REAL 2) ; x in the carrier of (TOP-REAL 2) \ F1() by A2, SUBSET_1:def_4; then not x in F1() by XBOOLE_0:def_5; then P1[qx] by A1; hence x in { p where p is Point of (TOP-REAL 2) : P1[p] } ; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : P1[p] } or x in F1() ` ) assume x in { p7 where p7 is Point of (TOP-REAL 2) : P1[p7] } ; ::_thesis: x in F1() ` then A3: ex p7 being Point of (TOP-REAL 2) st ( p7 = x & P1[p7] ) ; then for q7 being Point of (TOP-REAL 2) holds ( not x = q7 or not P1[q7] ) ; then not x in F1() by A1; then x in the carrier of (TOP-REAL 2) \ F1() by A3, XBOOLE_0:def_5; hence x in F1() ` by SUBSET_1:def_4; ::_thesis: verum end; Lm2: now__::_thesis:_for_p01,_p02,_px1,_px2_being_real_number_st_(p01_-_px1)_-_(p02_-_px2)_<=_((p01_-_p02)_/_4)_-_(-_((p01_-_p02)_/_4))_holds_ (p01_-_p02)_/_2_<=_px1_-_px2 let p01, p02, px1, px2 be real number ; ::_thesis: ( (p01 - px1) - (p02 - px2) <= ((p01 - p02) / 4) - (- ((p01 - p02) / 4)) implies (p01 - p02) / 2 <= px1 - px2 ) set r0 = (p01 - p02) / 4; assume (p01 - px1) - (p02 - px2) <= ((p01 - p02) / 4) - (- ((p01 - p02) / 4)) ; ::_thesis: (p01 - p02) / 2 <= px1 - px2 then (p01 - p02) - (px1 - px2) <= ((p01 - p02) / 4) + ((p01 - p02) / 4) ; then p01 - p02 <= (px1 - px2) + (((p01 - p02) / 4) + ((p01 - p02) / 4)) by XREAL_1:20; then (p01 - p02) - ((p01 - p02) / 2) <= px1 - px2 by XREAL_1:20; hence (p01 - p02) / 2 <= px1 - px2 ; ::_thesis: verum end; scheme :: JGRAPH_2:sch 3 ClosedSubset{ F1( Point of (TOP-REAL 2)) -> real number , F2( Point of (TOP-REAL 2)) -> real number } : { p where p is Point of (TOP-REAL 2) : F1(p) <= F2(p) } is closed Subset of (TOP-REAL 2) provided A1: for p, q being Point of (TOP-REAL 2) holds ( F1((p - q)) = F1(p) - F1(q) & F2((p - q)) = F2(p) - F2(q) ) and A2: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (F1((p - q)) ^2) + (F2((p - q)) ^2) proof defpred S1[ Point of (TOP-REAL 2)] means F1($1) <= F2($1); reconsider K2 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A3: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; then reconsider K21 = K2 ` as Subset of (TopSpaceMetr (Euclid 2)) ; A4: K2 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } ; A5: K2 ` = { p7 where p7 is Point of (TOP-REAL 2) : not S1[p7] } from JGRAPH_2:sch_2(A4); for p being Point of (Euclid 2) st p in K21 holds ex r being real number st ( r > 0 & Ball (p,r) c= K21 ) proof let p be Point of (Euclid 2); ::_thesis: ( p in K21 implies ex r being real number st ( r > 0 & Ball (p,r) c= K21 ) ) assume A6: p in K21 ; ::_thesis: ex r being real number st ( r > 0 & Ball (p,r) c= K21 ) then reconsider p0 = p as Point of (TOP-REAL 2) ; set r0 = (F1(p0) - F2(p0)) / 4; ex p7 being Point of (TOP-REAL 2) st ( p0 = p7 & F1(p7) > F2(p7) ) by A5, A6; then A7: F1(p0) - F2(p0) > 0 by XREAL_1:50; then A8: (F1(p0) - F2(p0)) / 2 > 0 by XREAL_1:139; Ball (p,((F1(p0) - F2(p0)) / 4)) c= K2 ` proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (p,((F1(p0) - F2(p0)) / 4)) or x in K2 ` ) A9: Ball (p,((F1(p0) - F2(p0)) / 4)) = { q where q is Element of (Euclid 2) : dist (p,q) < (F1(p0) - F2(p0)) / 4 } by METRIC_1:17; assume A10: x in Ball (p,((F1(p0) - F2(p0)) / 4)) ; ::_thesis: x in K2 ` then reconsider px = x as Point of (TOP-REAL 2) by TOPREAL3:8; consider q being Element of (Euclid 2) such that A11: q = x and A12: dist (p,q) < (F1(p0) - F2(p0)) / 4 by A10, A9; dist (p,q) = |.(p0 - px).| by A11, JGRAPH_1:28; then A13: |.(p0 - px).| ^2 <= ((F1(p0) - F2(p0)) / 4) ^2 by A12, SQUARE_1:15; A14: F1((p0 - px)) = F1(p0) - F1(px) by A1; A15: |.(p0 - px).| ^2 = (F1((p0 - px)) ^2) + (F2((p0 - px)) ^2) by A2; F2((p0 - px)) ^2 >= 0 by XREAL_1:63; then 0 + (F1((p0 - px)) ^2) <= (F2((p0 - px)) ^2) + (F1((p0 - px)) ^2) by XREAL_1:7; then F1((p0 - px)) ^2 <= ((F1(p0) - F2(p0)) / 4) ^2 by A15, A13, XXREAL_0:2; then A16: F1(p0) - F1(px) <= (F1(p0) - F2(p0)) / 4 by A7, A14, SQUARE_1:47; A17: F2((p0 - px)) = F2(p0) - F2(px) by A1; F1((p0 - px)) ^2 >= 0 by XREAL_1:63; then (F2((p0 - px)) ^2) + 0 <= (F2((p0 - px)) ^2) + (F1((p0 - px)) ^2) by XREAL_1:7; then F2((p0 - px)) ^2 <= ((F1(p0) - F2(p0)) / 4) ^2 by A15, A13, XXREAL_0:2; then - ((F1(p0) - F2(p0)) / 4) <= F2(p0) - F2(px) by A7, A17, SQUARE_1:47; then (F1(p0) - F1(px)) - (F2(p0) - F2(px)) <= ((F1(p0) - F2(p0)) / 4) - (- ((F1(p0) - F2(p0)) / 4)) by A16, XREAL_1:13; then F1(px) - F2(px) > 0 by A8, Lm2; then F1(px) > F2(px) by XREAL_1:47; hence x in K2 ` by A5; ::_thesis: verum end; hence ex r being real number st ( r > 0 & Ball (p,r) c= K21 ) by A7, XREAL_1:139; ::_thesis: verum end; then K21 is open by TOPMETR:15; then K2 ` is open by A3, PRE_TOPC:30; hence { p where p is Point of (TOP-REAL 2) : F1(p) <= F2(p) } is closed Subset of (TOP-REAL 2) by TOPS_1:3; ::_thesis: verum end; deffunc H1( Point of (TOP-REAL 2)) -> Element of REAL = $1 `1 ; deffunc H2( Point of (TOP-REAL 2)) -> Element of REAL = $1 `2 ; Lm3: for p, q being Point of (TOP-REAL 2) holds ( H1(p - q) = H1(p) - H1(q) & H2(p - q) = H2(p) - H2(q) ) by TOPREAL3:3; Lm4: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H1(p - q) ^2) + (H2(p - q) ^2) by JGRAPH_1:29; Lm5: { p7 where p7 is Point of (TOP-REAL 2) : H1(p7) <= H2(p7) } is closed Subset of (TOP-REAL 2) from JGRAPH_2:sch_3(Lm3, Lm4); Lm6: for p, q being Point of (TOP-REAL 2) holds ( H2(p - q) = H2(p) - H2(q) & H1(p - q) = H1(p) - H1(q) ) by TOPREAL3:3; Lm7: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H2(p - q) ^2) + (H1(p - q) ^2) by JGRAPH_1:29; Lm8: { p7 where p7 is Point of (TOP-REAL 2) : H2(p7) <= H1(p7) } is closed Subset of (TOP-REAL 2) from JGRAPH_2:sch_3(Lm6, Lm7); deffunc H3( Point of (TOP-REAL 2)) -> Element of REAL = - ($1 `1); deffunc H4( Point of (TOP-REAL 2)) -> Element of REAL = - ($1 `2); Lm9: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_ (_H3(p_-_q)_=_H3(p)_-_H3(q)_&_H2(p_-_q)_=_H2(p)_-_H2(q)_) let p, q be Point of (TOP-REAL 2); ::_thesis: ( H3(p - q) = H3(p) - H3(q) & H2(p - q) = H2(p) - H2(q) ) thus H3(p - q) = - ((p `1) - (q `1)) by TOPREAL3:3 .= H3(p) - H3(q) ; ::_thesis: H2(p - q) = H2(p) - H2(q) thus H2(p - q) = H2(p) - H2(q) by TOPREAL3:3; ::_thesis: verum end; Lm10: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_|.(p_-_q).|_^2_=_(H3(p_-_q)_^2)_+_(H2(p_-_q)_^2) let p, q be Point of (TOP-REAL 2); ::_thesis: |.(p - q).| ^2 = (H3(p - q) ^2) + (H2(p - q) ^2) H3(p - q) ^2 = H1(p - q) ^2 ; hence |.(p - q).| ^2 = (H3(p - q) ^2) + (H2(p - q) ^2) by JGRAPH_1:29; ::_thesis: verum end; Lm11: { p7 where p7 is Point of (TOP-REAL 2) : H3(p7) <= H2(p7) } is closed Subset of (TOP-REAL 2) from JGRAPH_2:sch_3(Lm9, Lm10); Lm12: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_ (_H2(p_-_q)_=_H2(p)_-_H2(q)_&_H3(p_-_q)_=_H3(p)_-_H3(q)_) let p, q be Point of (TOP-REAL 2); ::_thesis: ( H2(p - q) = H2(p) - H2(q) & H3(p - q) = H3(p) - H3(q) ) thus H2(p - q) = H2(p) - H2(q) by TOPREAL3:3; ::_thesis: H3(p - q) = H3(p) - H3(q) thus H3(p - q) = - ((p `1) - (q `1)) by TOPREAL3:3 .= H3(p) - H3(q) ; ::_thesis: verum end; Lm13: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_|.(p_-_q).|_^2_=_(H2(p_-_q)_^2)_+_(H3(p_-_q)_^2) let p, q be Point of (TOP-REAL 2); ::_thesis: |.(p - q).| ^2 = (H2(p - q) ^2) + (H3(p - q) ^2) (- ((p - q) `1)) ^2 = ((p - q) `1) ^2 ; hence |.(p - q).| ^2 = (H2(p - q) ^2) + (H3(p - q) ^2) by JGRAPH_1:29; ::_thesis: verum end; Lm14: { p7 where p7 is Point of (TOP-REAL 2) : H2(p7) <= H3(p7) } is closed Subset of (TOP-REAL 2) from JGRAPH_2:sch_3(Lm12, Lm13); Lm15: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_ (_H4(p_-_q)_=_H4(p)_-_H4(q)_&_H1(p_-_q)_=_H1(p)_-_H1(q)_) let p, q be Point of (TOP-REAL 2); ::_thesis: ( H4(p - q) = H4(p) - H4(q) & H1(p - q) = H1(p) - H1(q) ) thus H4(p - q) = - ((p `2) - (q `2)) by TOPREAL3:3 .= H4(p) - H4(q) ; ::_thesis: H1(p - q) = H1(p) - H1(q) thus H1(p - q) = H1(p) - H1(q) by TOPREAL3:3; ::_thesis: verum end; Lm16: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_|.(p_-_q).|_^2_=_(H4(p_-_q)_^2)_+_(H1(p_-_q)_^2) let p, q be Point of (TOP-REAL 2); ::_thesis: |.(p - q).| ^2 = (H4(p - q) ^2) + (H1(p - q) ^2) (- ((p - q) `2)) ^2 = ((p - q) `2) ^2 ; hence |.(p - q).| ^2 = (H4(p - q) ^2) + (H1(p - q) ^2) by JGRAPH_1:29; ::_thesis: verum end; Lm17: { p7 where p7 is Point of (TOP-REAL 2) : H4(p7) <= H1(p7) } is closed Subset of (TOP-REAL 2) from JGRAPH_2:sch_3(Lm15, Lm16); Lm18: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_ (_H1(p_-_q)_=_H1(p)_-_H1(q)_&_H4(p_-_q)_=_H4(p)_-_H4(q)_) let p, q be Point of (TOP-REAL 2); ::_thesis: ( H1(p - q) = H1(p) - H1(q) & H4(p - q) = H4(p) - H4(q) ) thus H1(p - q) = H1(p) - H1(q) by TOPREAL3:3; ::_thesis: H4(p - q) = H4(p) - H4(q) thus H4(p - q) = - ((p `2) - (q `2)) by TOPREAL3:3 .= H4(p) - H4(q) ; ::_thesis: verum end; Lm19: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_|.(p_-_q).|_^2_=_(H1(p_-_q)_^2)_+_(H4(p_-_q)_^2) let p, q be Point of (TOP-REAL 2); ::_thesis: |.(p - q).| ^2 = (H1(p - q) ^2) + (H4(p - q) ^2) H4(p - q) ^2 = H2(p - q) ^2 ; hence |.(p - q).| ^2 = (H1(p - q) ^2) + (H4(p - q) ^2) by JGRAPH_1:29; ::_thesis: verum end; Lm20: { p7 where p7 is Point of (TOP-REAL 2) : H1(p7) <= H4(p7) } is closed Subset of (TOP-REAL 2) from JGRAPH_2:sch_3(Lm18, Lm19); theorem Th38: :: JGRAPH_2:38 for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) proof reconsider K5 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1) } as closed Subset of (TOP-REAL 2) by Lm14; reconsider K4 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } as closed Subset of (TOP-REAL 2) by Lm5; reconsider K3 = { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1) <= p7 `2 } as closed Subset of (TOP-REAL 2) by Lm11; reconsider K2 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } as closed Subset of (TOP-REAL 2) by Lm8; let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } implies ( f is continuous & K0 is closed ) ) defpred S1[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) ) ); the carrier of ((TOP-REAL 2) | B0) = [#] ((TOP-REAL 2) | B0) .= B0 by PRE_TOPC:def_5 ; then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A1: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: ( f is continuous & K0 is closed ) K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then A3: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7; reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A4: K1 /\ B0 c= K0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 /\ B0 or x in K0 ) assume A5: x in K1 /\ B0 ; ::_thesis: x in K0 then x in B0 by XBOOLE_0:def_4; then not x in {(0. (TOP-REAL 2))} by A1, XBOOLE_0:def_5; then A6: not x = 0. (TOP-REAL 2) by TARSKI:def_1; x in K1 by A5, XBOOLE_0:def_4; then ex p7 being Point of (TOP-REAL 2) st ( p7 = x & ( ( p7 `2 <= p7 `1 & - (p7 `1) <= p7 `2 ) or ( p7 `2 >= p7 `1 & p7 `2 <= - (p7 `1) ) ) ) ; hence x in K0 by A1, A6; ::_thesis: verum end; A7: (K2 /\ K3) \/ (K4 /\ K5) c= K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 ) assume A8: x in (K2 /\ K3) \/ (K4 /\ K5) ; ::_thesis: x in K1 now__::_thesis:_(_(_x_in_K2_/\_K3_&_x_in_K1_)_or_(_x_in_K4_/\_K5_&_x_in_K1_)_) percases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A8, XBOOLE_0:def_3; caseA9: x in K2 /\ K3 ; ::_thesis: x in K1 then x in K3 by XBOOLE_0:def_4; then A10: ex p8 being Point of (TOP-REAL 2) st ( p8 = x & - (p8 `1) <= p8 `2 ) ; x in K2 by A9, XBOOLE_0:def_4; then ex p7 being Point of (TOP-REAL 2) st ( p7 = x & p7 `2 <= p7 `1 ) ; hence x in K1 by A10; ::_thesis: verum end; caseA11: x in K4 /\ K5 ; ::_thesis: x in K1 then x in K5 by XBOOLE_0:def_4; then A12: ex p8 being Point of (TOP-REAL 2) st ( p8 = x & p8 `2 <= - (p8 `1) ) ; x in K4 by A11, XBOOLE_0:def_4; then ex p7 being Point of (TOP-REAL 2) st ( p7 = x & p7 `2 >= p7 `1 ) ; hence x in K1 by A12; ::_thesis: verum end; end; end; hence x in K1 ; ::_thesis: verum end; K1 c= (K2 /\ K3) \/ (K4 /\ K5) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) ) assume x in K1 ; ::_thesis: x in (K2 /\ K3) \/ (K4 /\ K5) then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ) ; then ( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) ; then ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def_4; hence x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def_3; ::_thesis: verum end; then K1 = (K2 /\ K3) \/ (K4 /\ K5) by A7, XBOOLE_0:def_10; then A13: K1 is closed ; K0 c= K1 /\ B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in K1 /\ B0 ) assume x in K0 ; ::_thesis: x in K1 /\ B0 then A14: ex p being Point of (TOP-REAL 2) st ( x = p & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; then A15: x in B0 by A1, A14, XBOOLE_0:def_5; x in K1 by A14; hence x in K1 /\ B0 by A15, XBOOLE_0:def_4; ::_thesis: verum end; then K0 = K1 /\ B0 by A4, XBOOLE_0:def_10 .= K1 /\ ([#] ((TOP-REAL 2) | B0)) by PRE_TOPC:def_5 ; hence ( f is continuous & K0 is closed ) by A1, A3, A13, Th36, PRE_TOPC:13; ::_thesis: verum end; theorem Th39: :: JGRAPH_2:39 for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) proof reconsider K5 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= - (p7 `2) } as closed Subset of (TOP-REAL 2) by Lm20; reconsider K4 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } as closed Subset of (TOP-REAL 2) by Lm8; reconsider K3 = { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `2) <= p7 `1 } as closed Subset of (TOP-REAL 2) by Lm17; reconsider K2 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } as closed Subset of (TOP-REAL 2) by Lm5; let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } implies ( f is continuous & K0 is closed ) ) defpred S1[ Point of (TOP-REAL 2)] means ( ( $1 `1 <= $1 `2 & - ($1 `2) <= $1 `1 ) or ( $1 `1 >= $1 `2 & $1 `1 <= - ($1 `2) ) ); the carrier of ((TOP-REAL 2) | B0) = [#] ((TOP-REAL 2) | B0) .= B0 by PRE_TOPC:def_5 ; then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A1: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: ( f is continuous & K0 is closed ) K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then A3: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7; reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A4: K1 /\ B0 c= K0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 /\ B0 or x in K0 ) assume A5: x in K1 /\ B0 ; ::_thesis: x in K0 then x in B0 by XBOOLE_0:def_4; then not x in {(0. (TOP-REAL 2))} by A1, XBOOLE_0:def_5; then A6: not x = 0. (TOP-REAL 2) by TARSKI:def_1; x in K1 by A5, XBOOLE_0:def_4; then ex p7 being Point of (TOP-REAL 2) st ( p7 = x & ( ( p7 `1 <= p7 `2 & - (p7 `2) <= p7 `1 ) or ( p7 `1 >= p7 `2 & p7 `1 <= - (p7 `2) ) ) ) ; hence x in K0 by A1, A6; ::_thesis: verum end; A7: (K2 /\ K3) \/ (K4 /\ K5) c= K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 ) assume A8: x in (K2 /\ K3) \/ (K4 /\ K5) ; ::_thesis: x in K1 now__::_thesis:_(_(_x_in_K2_/\_K3_&_x_in_K1_)_or_(_x_in_K4_/\_K5_&_x_in_K1_)_) percases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A8, XBOOLE_0:def_3; caseA9: x in K2 /\ K3 ; ::_thesis: x in K1 then x in K3 by XBOOLE_0:def_4; then A10: ex p8 being Point of (TOP-REAL 2) st ( p8 = x & - (p8 `2) <= p8 `1 ) ; x in K2 by A9, XBOOLE_0:def_4; then ex p7 being Point of (TOP-REAL 2) st ( p7 = x & p7 `1 <= p7 `2 ) ; hence x in K1 by A10; ::_thesis: verum end; caseA11: x in K4 /\ K5 ; ::_thesis: x in K1 then x in K5 by XBOOLE_0:def_4; then A12: ex p8 being Point of (TOP-REAL 2) st ( p8 = x & p8 `1 <= - (p8 `2) ) ; x in K4 by A11, XBOOLE_0:def_4; then ex p7 being Point of (TOP-REAL 2) st ( p7 = x & p7 `1 >= p7 `2 ) ; hence x in K1 by A12; ::_thesis: verum end; end; end; hence x in K1 ; ::_thesis: verum end; K1 c= (K2 /\ K3) \/ (K4 /\ K5) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) ) assume x in K1 ; ::_thesis: x in (K2 /\ K3) \/ (K4 /\ K5) then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ) ; then ( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) ; then ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def_4; hence x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def_3; ::_thesis: verum end; then K1 = (K2 /\ K3) \/ (K4 /\ K5) by A7, XBOOLE_0:def_10; then A13: K1 is closed ; K0 c= K1 /\ B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in K1 /\ B0 ) assume x in K0 ; ::_thesis: x in K1 /\ B0 then A14: ex p being Point of (TOP-REAL 2) st ( x = p & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; then A15: x in B0 by A1, A14, XBOOLE_0:def_5; x in K1 by A14; hence x in K1 /\ B0 by A15, XBOOLE_0:def_4; ::_thesis: verum end; then K0 = K1 /\ B0 by A4, XBOOLE_0:def_10 .= K1 /\ ([#] ((TOP-REAL 2) | B0)) by PRE_TOPC:def_5 ; hence ( f is continuous & K0 is closed ) by A1, A3, A13, Th37, PRE_TOPC:13; ::_thesis: verum end; theorem Th40: :: JGRAPH_2:40 for D being non empty Subset of (TOP-REAL 2) st D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = Out_In_Sq & h is continuous ) proof set Y1 = |[(- 1),1]|; reconsider B0 = {(0. (TOP-REAL 2))} as Subset of (TOP-REAL 2) ; let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( D ` = {(0. (TOP-REAL 2))} implies ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = Out_In_Sq & h is continuous ) ) assume A1: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = Out_In_Sq & h is continuous ) then A2: D = B0 ` .= NonZero (TOP-REAL 2) by SUBSET_1:def_4 ; A3: { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } or x in the carrier of ((TOP-REAL 2) | D) ) assume x in { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in the carrier of ((TOP-REAL 2) | D) then A4: ex p being Point of (TOP-REAL 2) st ( x = p & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) ; now__::_thesis:_x_in_D assume not x in D ; ::_thesis: contradiction then x in the carrier of (TOP-REAL 2) \ D by A4, XBOOLE_0:def_5; then x in D ` by SUBSET_1:def_4; hence contradiction by A1, A4, TARSKI:def_1; ::_thesis: verum end; then x in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; hence x in the carrier of ((TOP-REAL 2) | D) ; ::_thesis: verum end; A5: NonZero (TOP-REAL 2) <> {} by Th9; A6: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19; ( ( (1.REAL 2) `2 <= (1.REAL 2) `1 & - ((1.REAL 2) `1) <= (1.REAL 2) `2 ) or ( (1.REAL 2) `2 >= (1.REAL 2) `1 & (1.REAL 2) `2 <= - ((1.REAL 2) `1) ) ) by Th5; then 1.REAL 2 in { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } by A6; then reconsider K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A3; A7: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5 .= the carrier of (((TOP-REAL 2) | D) | K0) ; A8: { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } or x in the carrier of ((TOP-REAL 2) | D) ) assume x in { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in the carrier of ((TOP-REAL 2) | D) then A9: ex p being Point of (TOP-REAL 2) st ( x = p & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) ; now__::_thesis:_x_in_D assume not x in D ; ::_thesis: contradiction then x in the carrier of (TOP-REAL 2) \ D by A9, XBOOLE_0:def_5; then x in D ` by SUBSET_1:def_4; hence contradiction by A1, A9, TARSKI:def_1; ::_thesis: verum end; then x in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; hence x in the carrier of ((TOP-REAL 2) | D) ; ::_thesis: verum end; ( |[(- 1),1]| `1 = - 1 & |[(- 1),1]| `2 = 1 ) by EUCLID:52; then |[(- 1),1]| in { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } by Th3; then reconsider K1 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A8; A10: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5 .= the carrier of (((TOP-REAL 2) | D) | K1) ; A11: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D) .= D by PRE_TOPC:def_5 ; A12: rng (Out_In_Sq | K1) c= the carrier of (((TOP-REAL 2) | D) | K1) proof reconsider K10 = K1 as Subset of (TOP-REAL 2) by A11, XBOOLE_1:1; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Out_In_Sq | K1) or y in the carrier of (((TOP-REAL 2) | D) | K1) ) A13: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K10) holds q `2 <> 0 proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K10) implies q `2 <> 0 ) A14: the carrier of ((TOP-REAL 2) | K10) = [#] ((TOP-REAL 2) | K10) .= K1 by PRE_TOPC:def_5 ; assume q in the carrier of ((TOP-REAL 2) | K10) ; ::_thesis: q `2 <> 0 then A15: ex p3 being Point of (TOP-REAL 2) st ( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A14; now__::_thesis:_not_q_`2_=_0 assume A16: q `2 = 0 ; ::_thesis: contradiction then q `1 = 0 by A15; hence contradiction by A15, A16, EUCLID:53, EUCLID:54; ::_thesis: verum end; hence q `2 <> 0 ; ::_thesis: verum end; assume y in rng (Out_In_Sq | K1) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K1) then consider x being set such that A17: x in dom (Out_In_Sq | K1) and A18: y = (Out_In_Sq | K1) . x by FUNCT_1:def_3; A19: x in (dom Out_In_Sq) /\ K1 by A17, RELAT_1:61; then A20: x in K1 by XBOOLE_0:def_4; K1 c= the carrier of (TOP-REAL 2) by A11, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A20; A21: Out_In_Sq . p = y by A18, A20, FUNCT_1:49; set p9 = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|; K10 = [#] ((TOP-REAL 2) | K10) by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | K10) ; then A22: p in the carrier of ((TOP-REAL 2) | K10) by A19, XBOOLE_0:def_4; A23: now__::_thesis:_not_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_=_0._(TOP-REAL_2) assume |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 0 by EUCLID:52, EUCLID:54; then 0 * (p `2) = (1 / (p `2)) * (p `2) by EUCLID:52; hence contradiction by A22, A13, XCMPLX_1:87; ::_thesis: verum end; A24: ex px being Point of (TOP-REAL 2) st ( x = px & ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) by A20; then A25: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by Th14; now__::_thesis:_(_(_p_`2_>=_0_&_y_in_K1_)_or_(_p_`2_<_0_&_y_in_K1_)_) percases ( p `2 >= 0 or p `2 < 0 ) ; caseA26: p `2 >= 0 ; ::_thesis: y in K1 then ( ( (p `1) / (p `2) <= (p `2) / (p `2) & (- (1 * (p `2))) / (p `2) <= (p `1) / (p `2) ) or ( p `1 >= p `2 & p `1 <= - (1 * (p `2)) ) ) by A24, XREAL_1:72; then A27: ( ( (p `1) / (p `2) <= 1 & ((- 1) * (p `2)) / (p `2) <= (p `1) / (p `2) ) or ( p `1 >= p `2 & p `1 <= - (1 * (p `2)) ) ) by A22, A13, XCMPLX_1:60; then A28: ( ( (p `1) / (p `2) <= 1 & - 1 <= (p `1) / (p `2) ) or ( (p `1) / (p `2) >= 1 & (p `1) / (p `2) <= ((- 1) * (p `2)) / (p `2) ) ) by A22, A13, A26, XCMPLX_1:89, XREAL_1:72; A29: ( not (p `1) / (p `2) >= 1 or not (p `1) / (p `2) <= - 1 ) ; ( ( (p `1) / (p `2) <= 1 & - 1 <= (p `1) / (p `2) ) or ( (p `1) / (p `2) >= (p `2) / (p `2) & p `1 <= - (1 * (p `2)) ) ) by A22, A13, A26, A27, XCMPLX_1:89; then (- 1) / (p `2) <= ((p `1) / (p `2)) / (p `2) by A22, A13, A26, A29, XCMPLX_1:60, XREAL_1:72; then A30: ( ( ((p `1) / (p `2)) / (p `2) <= 1 / (p `2) & - (1 / (p `2)) <= ((p `1) / (p `2)) / (p `2) ) or ( ((p `1) / (p `2)) / (p `2) >= 1 / (p `2) & ((p `1) / (p `2)) / (p `2) <= - (1 / (p `2)) ) ) by A26, A28, XREAL_1:72; ( |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) & |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 = ((p `1) / (p `2)) / (p `2) ) by EUCLID:52; hence y in K1 by A21, A23, A25, A30; ::_thesis: verum end; caseA31: p `2 < 0 ; ::_thesis: y in K1 then ( ( p `1 <= p `2 & - (1 * (p `2)) <= p `1 ) or ( (p `1) / (p `2) <= (p `2) / (p `2) & (p `1) / (p `2) >= (- (1 * (p `2))) / (p `2) ) ) by A24, XREAL_1:73; then A32: ( ( p `1 <= p `2 & - (1 * (p `2)) <= p `1 ) or ( (p `1) / (p `2) <= 1 & (p `1) / (p `2) >= ((- 1) * (p `2)) / (p `2) ) ) by A31, XCMPLX_1:60; then ( ( (p `1) / (p `2) >= 1 & ((- 1) * (p `2)) / (p `2) >= (p `1) / (p `2) ) or ( (p `1) / (p `2) <= 1 & (p `1) / (p `2) >= - 1 ) ) by A31, XCMPLX_1:89; then (- 1) / (p `2) >= ((p `1) / (p `2)) / (p `2) by A31, XREAL_1:73; then A33: ( ( ((p `1) / (p `2)) / (p `2) <= 1 / (p `2) & - (1 / (p `2)) <= ((p `1) / (p `2)) / (p `2) ) or ( ((p `1) / (p `2)) / (p `2) >= 1 / (p `2) & ((p `1) / (p `2)) / (p `2) <= - (1 / (p `2)) ) ) by A31, A32, XREAL_1:73; ( |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) & |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 = ((p `1) / (p `2)) / (p `2) ) by EUCLID:52; hence y in K1 by A21, A23, A25, A33; ::_thesis: verum end; end; end; then y in [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5; hence y in the carrier of (((TOP-REAL 2) | D) | K1) ; ::_thesis: verum end; A34: D c= K0 \/ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in K0 \/ K1 ) assume A35: x in D ; ::_thesis: x in K0 \/ K1 then reconsider px = x as Point of (TOP-REAL 2) ; not x in {(0. (TOP-REAL 2))} by A2, A35, XBOOLE_0:def_5; then ( ( ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) or ( ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1, XREAL_1:26; then ( x in K0 or x in K1 ) ; hence x in K0 \/ K1 by XBOOLE_0:def_3; ::_thesis: verum end; A36: NonZero (TOP-REAL 2) <> {} by Th9; A37: K1 c= NonZero (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) ) assume z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2) then A38: ex p8 being Point of (TOP-REAL 2) st ( p8 = z & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) ; then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence z in NonZero (TOP-REAL 2) by A38, XBOOLE_0:def_5; ::_thesis: verum end; A39: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D) .= NonZero (TOP-REAL 2) by A2, PRE_TOPC:def_5 ; A40: rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) proof reconsider K00 = K0 as Subset of (TOP-REAL 2) by A11, XBOOLE_1:1; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Out_In_Sq | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) ) A41: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K00) holds q `1 <> 0 proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K00) implies q `1 <> 0 ) A42: the carrier of ((TOP-REAL 2) | K00) = [#] ((TOP-REAL 2) | K00) .= K0 by PRE_TOPC:def_5 ; assume q in the carrier of ((TOP-REAL 2) | K00) ; ::_thesis: q `1 <> 0 then A43: ex p3 being Point of (TOP-REAL 2) st ( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A42; now__::_thesis:_not_q_`1_=_0 assume A44: q `1 = 0 ; ::_thesis: contradiction then q `2 = 0 by A43; hence contradiction by A43, A44, EUCLID:53, EUCLID:54; ::_thesis: verum end; hence q `1 <> 0 ; ::_thesis: verum end; assume y in rng (Out_In_Sq | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0) then consider x being set such that A45: x in dom (Out_In_Sq | K0) and A46: y = (Out_In_Sq | K0) . x by FUNCT_1:def_3; A47: x in (dom Out_In_Sq) /\ K0 by A45, RELAT_1:61; then A48: x in K0 by XBOOLE_0:def_4; K0 c= the carrier of (TOP-REAL 2) by A11, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A48; A49: Out_In_Sq . p = y by A46, A48, FUNCT_1:49; set p9 = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]|; K00 = [#] ((TOP-REAL 2) | K00) by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | K00) ; then A50: p in the carrier of ((TOP-REAL 2) | K00) by A47, XBOOLE_0:def_4; A51: |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) by EUCLID:52; A52: now__::_thesis:_not_|[(1_/_(p_`1)),(((p_`2)_/_(p_`1))_/_(p_`1))]|_=_0._(TOP-REAL_2) assume |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then 0 * (p `1) = (1 / (p `1)) * (p `1) by A51, EUCLID:52, EUCLID:54; hence contradiction by A50, A41, XCMPLX_1:87; ::_thesis: verum end; A53: ex px being Point of (TOP-REAL 2) st ( x = px & ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) by A48; then A54: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by Def1; A55: p `1 <> 0 by A50, A41; now__::_thesis:_(_(_p_`1_>=_0_&_y_in_K0_)_or_(_p_`1_<_0_&_y_in_K0_)_) percases ( p `1 >= 0 or p `1 < 0 ) ; caseA56: p `1 >= 0 ; ::_thesis: y in K0 A57: ( not (p `2) / (p `1) >= 1 or not (p `2) / (p `1) <= - 1 ) ; ( ( (p `2) / (p `1) <= (p `1) / (p `1) & (- (1 * (p `1))) / (p `1) <= (p `2) / (p `1) ) or ( p `2 >= p `1 & p `2 <= - (1 * (p `1)) ) ) by A53, A56, XREAL_1:72; then A58: ( ( (p `2) / (p `1) <= 1 & ((- 1) * (p `1)) / (p `1) <= (p `2) / (p `1) ) or ( p `2 >= p `1 & p `2 <= - (1 * (p `1)) ) ) by A50, A41, XCMPLX_1:60; then ( ( (p `2) / (p `1) <= 1 & - 1 <= (p `2) / (p `1) ) or ( (p `2) / (p `1) >= (p `1) / (p `1) & p `2 <= - (1 * (p `1)) ) ) by A50, A41, A56, XCMPLX_1:89; then (- 1) / (p `1) <= ((p `2) / (p `1)) / (p `1) by A50, A41, A56, A57, XCMPLX_1:60, XREAL_1:72; then A59: ( ( ((p `2) / (p `1)) / (p `1) <= 1 / (p `1) & - (1 / (p `1)) <= ((p `2) / (p `1)) / (p `1) ) or ( ((p `2) / (p `1)) / (p `1) >= 1 / (p `1) & ((p `2) / (p `1)) / (p `1) <= - (1 / (p `1)) ) ) by A55, A56, A58, XREAL_1:72; ( |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) & |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 = ((p `2) / (p `1)) / (p `1) ) by EUCLID:52; hence y in K0 by A49, A52, A54, A59; ::_thesis: verum end; caseA60: p `1 < 0 ; ::_thesis: y in K0 A61: ( not (p `2) / (p `1) >= 1 or not (p `2) / (p `1) <= - 1 ) ; ( ( p `2 <= p `1 & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= (p `1) / (p `1) & (p `2) / (p `1) >= (- (1 * (p `1))) / (p `1) ) ) by A53, A60, XREAL_1:73; then A62: ( ( p `2 <= p `1 & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= 1 & (p `2) / (p `1) >= ((- 1) * (p `1)) / (p `1) ) ) by A60, XCMPLX_1:60; then ( ( (p `2) / (p `1) >= (p `1) / (p `1) & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= 1 & (p `2) / (p `1) >= - 1 ) ) by A60, XCMPLX_1:89; then (- 1) / (p `1) >= ((p `2) / (p `1)) / (p `1) by A60, A61, XCMPLX_1:60, XREAL_1:73; then A63: ( ( ((p `2) / (p `1)) / (p `1) <= 1 / (p `1) & - (1 / (p `1)) <= ((p `2) / (p `1)) / (p `1) ) or ( ((p `2) / (p `1)) / (p `1) >= 1 / (p `1) & ((p `2) / (p `1)) / (p `1) <= - (1 / (p `1)) ) ) by A60, A62, XREAL_1:73; ( |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) & |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 = ((p `2) / (p `1)) / (p `1) ) by EUCLID:52; hence y in K0 by A49, A52, A54, A63; ::_thesis: verum end; end; end; then y in [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5; hence y in the carrier of (((TOP-REAL 2) | D) | K0) ; ::_thesis: verum end; A64: K0 c= NonZero (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K0 or z in NonZero (TOP-REAL 2) ) assume z in K0 ; ::_thesis: z in NonZero (TOP-REAL 2) then A65: ex p8 being Point of (TOP-REAL 2) st ( p8 = z & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) ; then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence z in NonZero (TOP-REAL 2) by A65, XBOOLE_0:def_5; ::_thesis: verum end; dom (Out_In_Sq | K0) = (dom Out_In_Sq) /\ K0 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K0 by A5, FUNCT_2:def_1 .= K0 by A64, XBOOLE_1:28 ; then reconsider f = Out_In_Sq | K0 as Function of (((TOP-REAL 2) | D) | K0),((TOP-REAL 2) | D) by A7, A40, FUNCT_2:2, XBOOLE_1:1; A66: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5; dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61 .= (NonZero (TOP-REAL 2)) /\ K1 by A36, FUNCT_2:def_1 .= K1 by A37, XBOOLE_1:28 ; then reconsider g = Out_In_Sq | K1 as Function of (((TOP-REAL 2) | D) | K1),((TOP-REAL 2) | D) by A10, A12, FUNCT_2:2, XBOOLE_1:1; A67: dom g = K1 by A10, FUNCT_2:def_1; g = Out_In_Sq | K1 ; then A68: K1 is closed by A2, Th39; A69: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5; A70: for x being set st x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) holds f . x = g . x proof let x be set ; ::_thesis: ( x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) implies f . x = g . x ) assume A71: x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) ; ::_thesis: f . x = g . x then x in K0 by A69, XBOOLE_0:def_4; then f . x = Out_In_Sq . x by FUNCT_1:49; hence f . x = g . x by A66, A71, FUNCT_1:49; ::_thesis: verum end; f = Out_In_Sq | K0 ; then A72: K0 is closed by A2, Th38; A73: dom f = K0 by A7, FUNCT_2:def_1; D = [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; then A74: ([#] (((TOP-REAL 2) | D) | K0)) \/ ([#] (((TOP-REAL 2) | D) | K1)) = [#] ((TOP-REAL 2) | D) by A69, A66, A34, XBOOLE_0:def_10; A75: ( f is continuous & g is continuous ) by A2, Th38, Th39; then consider h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) such that A76: h = f +* g and h is continuous by A69, A66, A74, A72, A68, A70, Th1; ( K0 = [#] (((TOP-REAL 2) | D) | K0) & K1 = [#] (((TOP-REAL 2) | D) | K1) ) by PRE_TOPC:def_5; then A77: f tolerates g by A70, A73, A67, PARTFUN1:def_4; A78: for x being set st x in dom h holds h . x = Out_In_Sq . x proof let x be set ; ::_thesis: ( x in dom h implies h . x = Out_In_Sq . x ) assume A79: x in dom h ; ::_thesis: h . x = Out_In_Sq . x then reconsider p = x as Point of (TOP-REAL 2) by A39, XBOOLE_0:def_5; not x in {(0. (TOP-REAL 2))} by A39, A79, XBOOLE_0:def_5; then A80: x <> 0. (TOP-REAL 2) by TARSKI:def_1; now__::_thesis:_(_(_x_in_K0_&_h_._x_=_Out_In_Sq_._x_)_or_(_not_x_in_K0_&_h_._x_=_Out_In_Sq_._x_)_) percases ( x in K0 or not x in K0 ) ; caseA81: x in K0 ; ::_thesis: h . x = Out_In_Sq . x h . p = (g +* f) . p by A76, A77, FUNCT_4:34 .= f . p by A73, A81, FUNCT_4:13 ; hence h . x = Out_In_Sq . x by A81, FUNCT_1:49; ::_thesis: verum end; case not x in K0 ; ::_thesis: h . x = Out_In_Sq . x then ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A80; then ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by XREAL_1:26; then A82: x in K1 by A80; then Out_In_Sq . p = g . p by FUNCT_1:49; hence h . x = Out_In_Sq . x by A76, A67, A82, FUNCT_4:13; ::_thesis: verum end; end; end; hence h . x = Out_In_Sq . x ; ::_thesis: verum end; ( dom h = the carrier of ((TOP-REAL 2) | D) & dom Out_In_Sq = the carrier of ((TOP-REAL 2) | D) ) by A39, FUNCT_2:def_1; then f +* g = Out_In_Sq by A76, A78, FUNCT_1:2; hence ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = Out_In_Sq & h is continuous ) by A69, A66, A74, A72, A75, A68, A70, Th1; ::_thesis: verum end; theorem Th41: :: JGRAPH_2:41 for B, K0, Kb being Subset of (TOP-REAL 2) st B = {(0. (TOP-REAL 2))} & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } holds ex f being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) st ( f is continuous & f is one-to-one & ( for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds not f . t in K0 \/ Kb ) & ( for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds f . r in K0 ) & ( for s being Point of (TOP-REAL 2) st s in Kb holds f . s = s ) ) proof set K1a = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; set K0a = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; let B, K0, Kb be Subset of (TOP-REAL 2); ::_thesis: ( B = {(0. (TOP-REAL 2))} & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } implies ex f being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) st ( f is continuous & f is one-to-one & ( for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds not f . t in K0 \/ Kb ) & ( for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds f . r in K0 ) & ( for s being Point of (TOP-REAL 2) st s in Kb holds f . s = s ) ) ) assume A1: ( B = {(0. (TOP-REAL 2))} & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } ) ; ::_thesis: ex f being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) st ( f is continuous & f is one-to-one & ( for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds not f . t in K0 \/ Kb ) & ( for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds f . r in K0 ) & ( for s being Point of (TOP-REAL 2) st s in Kb holds f . s = s ) ) then reconsider D = B ` as non empty Subset of (TOP-REAL 2) by Th9; A2: D ` = {(0. (TOP-REAL 2))} by A1; A3: B ` = NonZero (TOP-REAL 2) by A1, SUBSET_1:def_4; A4: for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds not Out_In_Sq . t in K0 \/ Kb proof let t be Point of (TOP-REAL 2); ::_thesis: ( t in K0 & t <> 0. (TOP-REAL 2) implies not Out_In_Sq . t in K0 \/ Kb ) assume that A5: t in K0 and A6: t <> 0. (TOP-REAL 2) ; ::_thesis: not Out_In_Sq . t in K0 \/ Kb A7: ex p3 being Point of (TOP-REAL 2) st ( p3 = t & - 1 < p3 `1 & p3 `1 < 1 & - 1 < p3 `2 & p3 `2 < 1 ) by A1, A5; now__::_thesis:_not_Out_In_Sq_._t_in_K0_\/_Kb assume A8: Out_In_Sq . t in K0 \/ Kb ; ::_thesis: contradiction now__::_thesis:_(_(_Out_In_Sq_._t_in_K0_&_contradiction_)_or_(_Out_In_Sq_._t_in_Kb_&_contradiction_)_) percases ( Out_In_Sq . t in K0 or Out_In_Sq . t in Kb ) by A8, XBOOLE_0:def_3; case Out_In_Sq . t in K0 ; ::_thesis: contradiction then consider p4 being Point of (TOP-REAL 2) such that A9: p4 = Out_In_Sq . t and A10: - 1 < p4 `1 and A11: p4 `1 < 1 and A12: - 1 < p4 `2 and A13: p4 `2 < 1 by A1; now__::_thesis:_(_(_(_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_or_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_)_&_contradiction_)_or_(_not_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_&_not_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_&_contradiction_)_) percases ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) or ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ) ; caseA14: ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: contradiction then Out_In_Sq . t = |[(1 / (t `1)),(((t `2) / (t `1)) / (t `1))]| by A6, Def1; then A15: p4 `1 = 1 / (t `1) by A9, EUCLID:52; now__::_thesis:_(_(_t_`1_>=_0_&_contradiction_)_or_(_t_`1_<_0_&_contradiction_)_) percases ( t `1 >= 0 or t `1 < 0 ) ; caseA16: t `1 >= 0 ; ::_thesis: contradiction now__::_thesis:_(_(_t_`1_>_0_&_contradiction_)_or_(_t_`1_=_0_&_contradiction_)_) percases ( t `1 > 0 or t `1 = 0 ) by A16; caseA17: t `1 > 0 ; ::_thesis: contradiction then (1 / (t `1)) * (t `1) < 1 * (t `1) by A11, A15, XREAL_1:68; hence contradiction by A7, A17, XCMPLX_1:87; ::_thesis: verum end; caseA18: t `1 = 0 ; ::_thesis: contradiction then t `2 = 0 by A14; hence contradiction by A6, A18, EUCLID:53, EUCLID:54; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA19: t `1 < 0 ; ::_thesis: contradiction then (- 1) * (t `1) > (1 / (t `1)) * (t `1) by A10, A15, XREAL_1:69; then (- 1) * (t `1) > 1 by A19, XCMPLX_1:87; then - (- (t `1)) <= - 1 by XREAL_1:24; hence contradiction by A7; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA20: ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: contradiction then Out_In_Sq . t = |[(((t `1) / (t `2)) / (t `2)),(1 / (t `2))]| by A6, Def1; then A21: p4 `2 = 1 / (t `2) by A9, EUCLID:52; now__::_thesis:_(_(_t_`2_>=_0_&_contradiction_)_or_(_t_`2_<_0_&_contradiction_)_) percases ( t `2 >= 0 or t `2 < 0 ) ; caseA22: t `2 >= 0 ; ::_thesis: contradiction now__::_thesis:_(_(_t_`2_>_0_&_contradiction_)_or_(_t_`2_=_0_&_contradiction_)_) percases ( t `2 > 0 or t `2 = 0 ) by A22; caseA23: t `2 > 0 ; ::_thesis: contradiction then (1 / (t `2)) * (t `2) < 1 * (t `2) by A13, A21, XREAL_1:68; hence contradiction by A7, A23, XCMPLX_1:87; ::_thesis: verum end; case t `2 = 0 ; ::_thesis: contradiction hence contradiction by A20; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA24: t `2 < 0 ; ::_thesis: contradiction then (- 1) * (t `2) > (1 / (t `2)) * (t `2) by A12, A21, XREAL_1:69; then (- 1) * (t `2) > 1 by A24, XCMPLX_1:87; then - (- (t `2)) <= - 1 by XREAL_1:24; hence contradiction by A7; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; case Out_In_Sq . t in Kb ; ::_thesis: contradiction then consider p4 being Point of (TOP-REAL 2) such that A25: p4 = Out_In_Sq . t and A26: ( ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( p4 `1 = 1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ) by A1; now__::_thesis:_(_(_(_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_or_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_)_&_contradiction_)_or_(_not_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_&_not_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_&_contradiction_)_) percases ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) or ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ) ; caseA27: ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: contradiction then A28: Out_In_Sq . t = |[(1 / (t `1)),(((t `2) / (t `1)) / (t `1))]| by A6, Def1; then A29: p4 `1 = 1 / (t `1) by A25, EUCLID:52; now__::_thesis:_(_(_-_1_=_p4_`1_&_-_1_<=_p4_`2_&_p4_`2_<=_1_&_contradiction_)_or_(_p4_`1_=_1_&_-_1_<=_p4_`2_&_p4_`2_<=_1_&_contradiction_)_or_(_-_1_=_p4_`2_&_-_1_<=_p4_`1_&_p4_`1_<=_1_&_contradiction_)_or_(_1_=_p4_`2_&_-_1_<=_p4_`1_&_p4_`1_<=_1_&_contradiction_)_) percases ( ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( p4 `1 = 1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ) by A26; case ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ; ::_thesis: contradiction then A30: (t `1) * ((t `1) ") = - (t `1) by A29; now__::_thesis:_(_(_t_`1_<>_0_&_contradiction_)_or_(_t_`1_=_0_&_contradiction_)_) percases ( t `1 <> 0 or t `1 = 0 ) ; case t `1 <> 0 ; ::_thesis: contradiction then - (t `1) = 1 by A30, XCMPLX_0:def_7; hence contradiction by A7; ::_thesis: verum end; caseA31: t `1 = 0 ; ::_thesis: contradiction then t `2 = 0 by A27; hence contradiction by A6, A31, EUCLID:53, EUCLID:54; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; case ( p4 `1 = 1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ; ::_thesis: contradiction then A32: (t `1) * ((t `1) ") = t `1 by A29; now__::_thesis:_(_(_t_`1_<>_0_&_contradiction_)_or_(_t_`1_=_0_&_contradiction_)_) percases ( t `1 <> 0 or t `1 = 0 ) ; case t `1 <> 0 ; ::_thesis: contradiction hence contradiction by A7, A32, XCMPLX_0:def_7; ::_thesis: verum end; caseA33: t `1 = 0 ; ::_thesis: contradiction then t `2 = 0 by A27; hence contradiction by A6, A33, EUCLID:53, EUCLID:54; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA34: ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ; ::_thesis: contradiction reconsider K01 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th17; A35: the carrier of (((TOP-REAL 2) | D) | K01) = [#] (((TOP-REAL 2) | D) | K01) .= K01 by PRE_TOPC:def_5 ; A36: dom (Out_In_Sq | K01) = (dom Out_In_Sq) /\ K01 by RELAT_1:61 .= D /\ K01 by A3, FUNCT_2:def_1 .= ([#] ((TOP-REAL 2) | D)) /\ K01 by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | D) /\ K01 .= K01 by XBOOLE_1:28 ; t in K01 by A6, A27; then A37: (Out_In_Sq | K01) . t in rng (Out_In_Sq | K01) by A36, FUNCT_1:3; rng (Out_In_Sq | K01) c= the carrier of (((TOP-REAL 2) | D) | K01) by Th15; then A38: (Out_In_Sq | K01) . t in the carrier of (((TOP-REAL 2) | D) | K01) by A37; t in K01 by A6, A27; then Out_In_Sq . t in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } by A38, A35, FUNCT_1:49; then A39: ex p5 being Point of (TOP-REAL 2) st ( p5 = p4 & ( ( p5 `2 <= p5 `1 & - (p5 `1) <= p5 `2 ) or ( p5 `2 >= p5 `1 & p5 `2 <= - (p5 `1) ) ) & p5 <> 0. (TOP-REAL 2) ) by A25; now__::_thesis:_(_(_p4_`1_>=_1_&_contradiction_)_or_(_-_1_>=_p4_`1_&_contradiction_)_) percases ( p4 `1 >= 1 or - 1 >= p4 `1 ) by A34, A39, XREAL_1:24; caseA40: p4 `1 >= 1 ; ::_thesis: contradiction then ((t `2) / (t `1)) / (t `1) = ((t `2) / (t `1)) * 1 by A29, A34, XXREAL_0:1 .= (t `2) * 1 by A29, A34, A40, XXREAL_0:1 .= t `2 ; hence contradiction by A7, A25, A28, A34, EUCLID:52; ::_thesis: verum end; caseA41: - 1 >= p4 `1 ; ::_thesis: contradiction then ((t `2) / (t `1)) / (t `1) = ((t `2) / (t `1)) * (- 1) by A29, A34, XXREAL_0:1 .= - ((t `2) / (t `1)) .= - ((t `2) * (- 1)) by A29, A34, A41, XXREAL_0:1 .= t `2 ; hence contradiction by A7, A25, A28, A34, EUCLID:52; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA42: ( 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ; ::_thesis: contradiction reconsider K01 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th17; t in K01 by A6, A27; then A43: Out_In_Sq . t = (Out_In_Sq | K01) . t by FUNCT_1:49; dom (Out_In_Sq | K01) = (dom Out_In_Sq) /\ K01 by RELAT_1:61 .= D /\ K01 by A3, FUNCT_2:def_1 .= ([#] ((TOP-REAL 2) | D)) /\ K01 by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | D) /\ K01 .= K01 by XBOOLE_1:28 ; then t in dom (Out_In_Sq | K01) by A6, A27; then A44: (Out_In_Sq | K01) . t in rng (Out_In_Sq | K01) by FUNCT_1:3; rng (Out_In_Sq | K01) c= the carrier of (((TOP-REAL 2) | D) | K01) by Th15; then A45: (Out_In_Sq | K01) . t in the carrier of (((TOP-REAL 2) | D) | K01) by A44; the carrier of (((TOP-REAL 2) | D) | K01) = [#] (((TOP-REAL 2) | D) | K01) .= K01 by PRE_TOPC:def_5 ; then A46: ex p5 being Point of (TOP-REAL 2) st ( p5 = p4 & ( ( p5 `2 <= p5 `1 & - (p5 `1) <= p5 `2 ) or ( p5 `2 >= p5 `1 & p5 `2 <= - (p5 `1) ) ) & p5 <> 0. (TOP-REAL 2) ) by A25, A45, A43; now__::_thesis:_(_(_p4_`1_>=_1_&_contradiction_)_or_(_-_1_>=_p4_`1_&_contradiction_)_) percases ( p4 `1 >= 1 or - 1 >= p4 `1 ) by A42, A46, XREAL_1:25; caseA47: p4 `1 >= 1 ; ::_thesis: contradiction then ((t `2) / (t `1)) / (t `1) = ((t `2) / (t `1)) * 1 by A29, A42, XXREAL_0:1 .= (t `2) * 1 by A29, A42, A47, XXREAL_0:1 .= t `2 ; hence contradiction by A7, A25, A28, A42, EUCLID:52; ::_thesis: verum end; caseA48: - 1 >= p4 `1 ; ::_thesis: contradiction then ((t `2) / (t `1)) / (t `1) = ((t `2) / (t `1)) * (- 1) by A29, A42, XXREAL_0:1 .= - ((t `2) / (t `1)) .= - ((t `2) * (- 1)) by A29, A42, A48, XXREAL_0:1 .= t `2 ; hence contradiction by A7, A25, A28, A42, EUCLID:52; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA49: ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: contradiction then A50: Out_In_Sq . t = |[(((t `1) / (t `2)) / (t `2)),(1 / (t `2))]| by A6, Def1; then A51: p4 `2 = 1 / (t `2) by A25, EUCLID:52; now__::_thesis:_(_(_-_1_=_p4_`2_&_-_1_<=_p4_`1_&_p4_`1_<=_1_&_contradiction_)_or_(_p4_`2_=_1_&_-_1_<=_p4_`1_&_p4_`1_<=_1_&_contradiction_)_or_(_-_1_=_p4_`1_&_-_1_<=_p4_`2_&_p4_`2_<=_1_&_contradiction_)_or_(_1_=_p4_`1_&_-_1_<=_p4_`2_&_p4_`2_<=_1_&_contradiction_)_) percases ( ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( p4 `2 = 1 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ) by A26; case ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ; ::_thesis: contradiction then A52: (t `2) * ((t `2) ") = - (t `2) by A51; now__::_thesis:_(_(_t_`2_<>_0_&_contradiction_)_or_(_t_`2_=_0_&_contradiction_)_) percases ( t `2 <> 0 or t `2 = 0 ) ; case t `2 <> 0 ; ::_thesis: contradiction then - (t `2) = 1 by A52, XCMPLX_0:def_7; hence contradiction by A7; ::_thesis: verum end; case t `2 = 0 ; ::_thesis: contradiction hence contradiction by A49; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; case ( p4 `2 = 1 & - 1 <= p4 `1 & p4 `1 <= 1 ) ; ::_thesis: contradiction then A53: (t `2) * ((t `2) ") = t `2 by A51; now__::_thesis:_(_(_t_`2_<>_0_&_contradiction_)_or_(_t_`2_=_0_&_contradiction_)_) percases ( t `2 <> 0 or t `2 = 0 ) ; case t `2 <> 0 ; ::_thesis: contradiction hence contradiction by A7, A53, XCMPLX_0:def_7; ::_thesis: verum end; case t `2 = 0 ; ::_thesis: contradiction hence contradiction by A49; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA54: ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ; ::_thesis: contradiction reconsider K11 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th18; A55: dom (Out_In_Sq | K11) = (dom Out_In_Sq) /\ K11 by RELAT_1:61 .= D /\ K11 by A3, FUNCT_2:def_1 .= ([#] ((TOP-REAL 2) | D)) /\ K11 by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | D) /\ K11 .= K11 by XBOOLE_1:28 ; A56: ( ( t `1 <= t `2 & - (t `2) <= t `1 ) or ( t `1 >= t `2 & t `1 <= - (t `2) ) ) by A49, Th13; then t in K11 by A6; then A57: Out_In_Sq . t = (Out_In_Sq | K11) . t by FUNCT_1:49; t in K11 by A6, A56; then A58: (Out_In_Sq | K11) . t in rng (Out_In_Sq | K11) by A55, FUNCT_1:3; rng (Out_In_Sq | K11) c= the carrier of (((TOP-REAL 2) | D) | K11) by Th16; then A59: (Out_In_Sq | K11) . t in the carrier of (((TOP-REAL 2) | D) | K11) by A58; the carrier of (((TOP-REAL 2) | D) | K11) = [#] (((TOP-REAL 2) | D) | K11) .= K11 by PRE_TOPC:def_5 ; then A60: ex p5 being Point of (TOP-REAL 2) st ( p5 = p4 & ( ( p5 `1 <= p5 `2 & - (p5 `2) <= p5 `1 ) or ( p5 `1 >= p5 `2 & p5 `1 <= - (p5 `2) ) ) & p5 <> 0. (TOP-REAL 2) ) by A25, A59, A57; now__::_thesis:_(_(_p4_`2_>=_1_&_contradiction_)_or_(_-_1_>=_p4_`2_&_contradiction_)_) percases ( p4 `2 >= 1 or - 1 >= p4 `2 ) by A54, A60, XREAL_1:24; caseA61: p4 `2 >= 1 ; ::_thesis: contradiction then ((t `1) / (t `2)) / (t `2) = ((t `1) / (t `2)) * 1 by A51, A54, XXREAL_0:1 .= (t `1) * 1 by A51, A54, A61, XXREAL_0:1 .= t `1 ; hence contradiction by A7, A25, A50, A54, EUCLID:52; ::_thesis: verum end; caseA62: - 1 >= p4 `2 ; ::_thesis: contradiction then ((t `1) / (t `2)) / (t `2) = ((t `1) / (t `2)) * (- 1) by A51, A54, XXREAL_0:1 .= - ((t `1) / (t `2)) .= - ((t `1) * (- 1)) by A51, A54, A62, XXREAL_0:1 .= t `1 ; hence contradiction by A7, A25, A50, A54, EUCLID:52; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA63: ( 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ; ::_thesis: contradiction reconsider K11 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th18; A64: the carrier of (((TOP-REAL 2) | D) | K11) = [#] (((TOP-REAL 2) | D) | K11) .= K11 by PRE_TOPC:def_5 ; A65: dom (Out_In_Sq | K11) = (dom Out_In_Sq) /\ K11 by RELAT_1:61 .= D /\ K11 by A3, FUNCT_2:def_1 .= ([#] ((TOP-REAL 2) | D)) /\ K11 by PRE_TOPC:def_5 .= the carrier of ((TOP-REAL 2) | D) /\ K11 .= K11 by XBOOLE_1:28 ; A66: ( ( t `1 <= t `2 & - (t `2) <= t `1 ) or ( t `1 >= t `2 & t `1 <= - (t `2) ) ) by A49, Th13; then t in K11 by A6; then A67: (Out_In_Sq | K11) . t in rng (Out_In_Sq | K11) by A65, FUNCT_1:3; rng (Out_In_Sq | K11) c= the carrier of (((TOP-REAL 2) | D) | K11) by Th16; then A68: (Out_In_Sq | K11) . t in the carrier of (((TOP-REAL 2) | D) | K11) by A67; t in K11 by A6, A66; then Out_In_Sq . t in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } by A68, A64, FUNCT_1:49; then A69: ex p5 being Point of (TOP-REAL 2) st ( p5 = p4 & ( ( p5 `1 <= p5 `2 & - (p5 `2) <= p5 `1 ) or ( p5 `1 >= p5 `2 & p5 `1 <= - (p5 `2) ) ) & p5 <> 0. (TOP-REAL 2) ) by A25; now__::_thesis:_(_(_p4_`2_>=_1_&_contradiction_)_or_(_-_1_>=_p4_`2_&_contradiction_)_) percases ( p4 `2 >= 1 or - 1 >= p4 `2 ) by A63, A69, XREAL_1:25; caseA70: p4 `2 >= 1 ; ::_thesis: contradiction then ((t `1) / (t `2)) / (t `2) = ((t `1) / (t `2)) * 1 by A51, A63, XXREAL_0:1 .= (t `1) * 1 by A51, A63, A70, XXREAL_0:1 .= t `1 ; hence contradiction by A7, A25, A50, A63, EUCLID:52; ::_thesis: verum end; caseA71: - 1 >= p4 `2 ; ::_thesis: contradiction then ((t `1) / (t `2)) / (t `2) = ((t `1) / (t `2)) * (- 1) by A51, A63, XXREAL_0:1 .= - ((t `1) / (t `2)) .= - ((t `1) * (- 1)) by A51, A63, A71, XXREAL_0:1 .= t `1 ; hence contradiction by A7, A25, A50, A63, EUCLID:52; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; hence not Out_In_Sq . t in K0 \/ Kb ; ::_thesis: verum end; A72: for t being Point of (TOP-REAL 2) st not t in K0 \/ Kb holds Out_In_Sq . t in K0 proof let t be Point of (TOP-REAL 2); ::_thesis: ( not t in K0 \/ Kb implies Out_In_Sq . t in K0 ) assume A73: not t in K0 \/ Kb ; ::_thesis: Out_In_Sq . t in K0 then A74: not t in K0 by XBOOLE_0:def_3; then A75: not t = 0. (TOP-REAL 2) by A1, Th3; then not t in {(0. (TOP-REAL 2))} by TARSKI:def_1; then t in NonZero (TOP-REAL 2) by XBOOLE_0:def_5; then Out_In_Sq . t in NonZero (TOP-REAL 2) by FUNCT_2:5; then reconsider p4 = Out_In_Sq . t as Point of (TOP-REAL 2) ; A76: not t in Kb by A73, XBOOLE_0:def_3; now__::_thesis:_(_(_(_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_or_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_)_&_Out_In_Sq_._t_in_K0_)_or_(_not_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_&_not_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_&_Out_In_Sq_._t_in_K0_)_) percases ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) or ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ) ; caseA77: ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: Out_In_Sq . t in K0 A78: now__::_thesis:_(_(_t_`1_>_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_or_(_t_`1_<=_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_) percases ( t `1 > 0 or t `1 <= 0 ) ; caseA79: t `1 > 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) now__::_thesis:_(_(_t_`2_>_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_or_(_t_`2_<=_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_) percases ( t `2 > 0 or t `2 <= 0 ) ; caseA80: t `2 > 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) ( - 1 >= t `1 or t `1 >= 1 or - 1 >= t `2 or t `2 >= 1 ) by A1, A74; then A81: t `1 >= 1 by A77, A79, A80, XXREAL_0:2; not t `1 = 1 by A1, A76, A77; then A82: t `1 > 1 by A81, XXREAL_0:1; then t `1 < (t `1) ^2 by SQUARE_1:14; then t `2 < (t `1) ^2 by A77, A79, XXREAL_0:2; then (t `2) / (t `1) < ((t `1) ^2) / (t `1) by A79, XREAL_1:74; then (t `2) / (t `1) < t `1 by A79, XCMPLX_1:89; then A83: ((t `2) / (t `1)) / (t `1) < (t `1) / (t `1) by A79, XREAL_1:74; 0 < (t `2) / (t `1) by A79, A80, XREAL_1:139; then A84: ((- 1) * (t `1)) / (t `1) < ((t `2) / (t `1)) / (t `1) by A79, XREAL_1:74; (t `1) / (t `1) > 1 / (t `1) by A82, XREAL_1:74; hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) by A79, A84, A83, XCMPLX_1:60, XCMPLX_1:89; ::_thesis: verum end; caseA85: t `2 <= 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) A86: now__::_thesis:_(_t_`1_<_1_implies_t_`1_>=_1_) assume t `1 < 1 ; ::_thesis: t `1 >= 1 then - 1 >= t `2 by A1, A74, A79, A85; then - (t `1) <= - 1 by A77, A79, XXREAL_0:2; hence t `1 >= 1 by XREAL_1:24; ::_thesis: verum end; not t `1 = 1 by A1, A76, A77; then A87: t `1 > 1 by A86, XXREAL_0:1; then A88: t `1 < (t `1) ^2 by SQUARE_1:14; - (- (t `1)) >= - (t `2) by A77, A79, XREAL_1:24; then (t `1) ^2 > - (t `2) by A88, XXREAL_0:2; then ((t `1) ^2) / (t `1) > (- (t `2)) / (t `1) by A79, XREAL_1:74; then t `1 > - ((t `2) / (t `1)) by A79, XCMPLX_1:89; then - (t `1) < - (- ((t `2) / (t `1))) by XREAL_1:24; then A89: ((- 1) * (t `1)) / (t `1) < ((t `2) / (t `1)) / (t `1) by A79, XREAL_1:74; (t `1) / (t `1) > 1 / (t `1) by A87, XREAL_1:74; hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) by A79, A85, A89, XCMPLX_1:60, XCMPLX_1:89; ::_thesis: verum end; end; end; hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) ; ::_thesis: verum end; caseA90: t `1 <= 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) now__::_thesis:_(_(_t_`1_=_0_&_contradiction_)_or_(_t_`1_<_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_) percases ( t `1 = 0 or t `1 < 0 ) by A90; caseA91: t `1 = 0 ; ::_thesis: contradiction then t `2 = 0 by A77; hence contradiction by A1, A74, A91; ::_thesis: verum end; caseA92: t `1 < 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) now__::_thesis:_(_(_t_`2_>_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_or_(_t_`2_<=_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_) percases ( t `2 > 0 or t `2 <= 0 ) ; caseA93: t `2 > 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) ( - 1 >= t `1 or t `1 >= 1 or - 1 >= t `2 or t `2 >= 1 ) by A1, A74; then ( t `1 <= - 1 or 1 <= - (t `1) ) by A77, A92, XXREAL_0:2; then A94: ( t `1 <= - 1 or - 1 >= - (- (t `1)) ) by XREAL_1:24; not t `1 = - 1 by A1, A76, A77; then A95: t `1 < - 1 by A94, XXREAL_0:1; then (t `1) / (t `1) > (- 1) / (t `1) by XREAL_1:75; then A96: - ((t `1) / (t `1)) < - ((- 1) / (t `1)) by XREAL_1:24; - (t `1) < (t `1) ^2 by A95, SQUARE_1:46; then t `2 < (t `1) ^2 by A77, A92, XXREAL_0:2; then (t `2) / (t `1) > ((t `1) ^2) / (t `1) by A92, XREAL_1:75; then (t `2) / (t `1) > t `1 by A92, XCMPLX_1:89; then A97: ((t `2) / (t `1)) / (t `1) < (t `1) / (t `1) by A92, XREAL_1:75; 0 > (t `2) / (t `1) by A92, A93, XREAL_1:142; then ((- 1) * (t `1)) / (t `1) < ((t `2) / (t `1)) / (t `1) by A92, XREAL_1:75; hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) by A92, A96, A97, XCMPLX_1:60; ::_thesis: verum end; caseA98: t `2 <= 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) then ( - 1 >= t `1 or - 1 >= t `2 ) by A1, A74, A92; then A99: t `1 <= - 1 by A77, A92, XXREAL_0:2; not t `1 = - 1 by A1, A76, A77; then A100: t `1 < - 1 by A99, XXREAL_0:1; then A101: - (t `1) < (t `1) ^2 by SQUARE_1:46; - (t `1) >= - (t `2) by A77, A92, XREAL_1:24; then (t `1) ^2 > - (t `2) by A101, XXREAL_0:2; then ((t `1) ^2) / (t `1) < (- (t `2)) / (t `1) by A92, XREAL_1:75; then t `1 < - ((t `2) / (t `1)) by A92, XCMPLX_1:89; then - (t `1) > - (- ((t `2) / (t `1))) by XREAL_1:24; then A102: ((- 1) * (t `1)) / (t `1) < ((t `2) / (t `1)) / (t `1) by A92, XREAL_1:75; (t `1) / (t `1) > (- 1) / (t `1) by A100, XREAL_1:75; then 1 > (- 1) / (t `1) by A92, XCMPLX_1:60; then - 1 < - ((- 1) / (t `1)) by XREAL_1:24; hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) by A92, A98, A102, XCMPLX_1:89; ::_thesis: verum end; end; end; hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) ; ::_thesis: verum end; end; end; hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) ; ::_thesis: verum end; end; end; Out_In_Sq . t = |[(1 / (t `1)),(((t `2) / (t `1)) / (t `1))]| by A75, A77, Def1; then ( p4 `1 = 1 / (t `1) & p4 `2 = ((t `2) / (t `1)) / (t `1) ) by EUCLID:52; hence Out_In_Sq . t in K0 by A1, A78; ::_thesis: verum end; caseA103: ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: Out_In_Sq . t in K0 then A104: ( ( t `1 <= t `2 & - (t `2) <= t `1 ) or ( t `1 >= t `2 & t `1 <= - (t `2) ) ) by Th13; A105: now__::_thesis:_(_(_t_`2_>_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_or_(_t_`2_<=_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_) percases ( t `2 > 0 or t `2 <= 0 ) ; caseA106: t `2 > 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) now__::_thesis:_(_(_t_`1_>_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_or_(_t_`1_<=_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_) percases ( t `1 > 0 or t `1 <= 0 ) ; caseA107: t `1 > 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) A108: ( - 1 >= t `2 or t `2 >= 1 or - 1 >= t `1 or t `1 >= 1 ) by A1, A74; not t `2 = 1 by A1, A76, A103, A107; then A109: t `2 > 1 by A103, A106, A107, A108, XXREAL_0:1, XXREAL_0:2; then t `2 < (t `2) ^2 by SQUARE_1:14; then t `1 < (t `2) ^2 by A103, A106, XXREAL_0:2; then (t `1) / (t `2) < ((t `2) ^2) / (t `2) by A106, XREAL_1:74; then (t `1) / (t `2) < t `2 by A106, XCMPLX_1:89; then A110: ((t `1) / (t `2)) / (t `2) < (t `2) / (t `2) by A106, XREAL_1:74; 0 < (t `1) / (t `2) by A106, A107, XREAL_1:139; then A111: ((- 1) * (t `2)) / (t `2) < ((t `1) / (t `2)) / (t `2) by A106, XREAL_1:74; (t `2) / (t `2) > 1 / (t `2) by A109, XREAL_1:74; hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) by A106, A111, A110, XCMPLX_1:60, XCMPLX_1:89; ::_thesis: verum end; caseA112: t `1 <= 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) A113: now__::_thesis:_(_t_`2_<_1_implies_t_`2_>=_1_) assume t `2 < 1 ; ::_thesis: t `2 >= 1 then - 1 >= t `1 by A1, A74, A106, A112; then - (t `2) <= - 1 by A104, A106, XXREAL_0:2; hence t `2 >= 1 by XREAL_1:24; ::_thesis: verum end; not t `2 = 1 by A1, A76, A104; then A114: t `2 > 1 by A113, XXREAL_0:1; then t `2 < (t `2) ^2 by SQUARE_1:14; then (t `2) ^2 > - (t `1) by A103, A106, XXREAL_0:2; then ((t `2) ^2) / (t `2) > (- (t `1)) / (t `2) by A106, XREAL_1:74; then t `2 > - ((t `1) / (t `2)) by A106, XCMPLX_1:89; then - (t `2) < - (- ((t `1) / (t `2))) by XREAL_1:24; then A115: ((- 1) * (t `2)) / (t `2) < ((t `1) / (t `2)) / (t `2) by A106, XREAL_1:74; (t `2) / (t `2) > 1 / (t `2) by A114, XREAL_1:74; hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) by A106, A112, A115, XCMPLX_1:60, XCMPLX_1:89; ::_thesis: verum end; end; end; hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) ; ::_thesis: verum end; caseA116: t `2 <= 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) then A117: t `2 < 0 by A103; A118: ( t `1 <= t `2 or t `1 <= - (t `2) ) by A103, Th13; now__::_thesis:_(_(_t_`1_>_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_or_(_t_`1_<=_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_) percases ( t `1 > 0 or t `1 <= 0 ) ; caseA119: t `1 > 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) ( - 1 >= t `2 or t `2 >= 1 or - 1 >= t `1 or t `1 >= 1 ) by A1, A74; then ( t `2 <= - 1 or 1 <= - (t `2) ) by A104, A116, XXREAL_0:2; then A120: ( t `2 <= - 1 or - 1 >= - (- (t `2)) ) by XREAL_1:24; not t `2 = - 1 by A1, A76, A104; then A121: t `2 < - 1 by A120, XXREAL_0:1; then (t `2) / (t `2) > (- 1) / (t `2) by XREAL_1:75; then A122: - ((t `2) / (t `2)) < - ((- 1) / (t `2)) by XREAL_1:24; - (t `2) < (t `2) ^2 by A121, SQUARE_1:46; then t `1 < (t `2) ^2 by A116, A118, XXREAL_0:2; then (t `1) / (t `2) > ((t `2) ^2) / (t `2) by A117, XREAL_1:75; then (t `1) / (t `2) > t `2 by A117, XCMPLX_1:89; then A123: ((t `1) / (t `2)) / (t `2) < (t `2) / (t `2) by A117, XREAL_1:75; 0 > (t `1) / (t `2) by A117, A119, XREAL_1:142; then ((- 1) * (t `2)) / (t `2) < ((t `1) / (t `2)) / (t `2) by A117, XREAL_1:75; hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) by A117, A122, A123, XCMPLX_1:60; ::_thesis: verum end; caseA124: t `1 <= 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) A125: not t `2 = - 1 by A1, A76, A104; ( - 1 >= t `2 or - 1 >= t `1 ) by A1, A74, A116, A124; then A126: t `2 < - 1 by A103, A116, A125, XXREAL_0:1, XXREAL_0:2; then A127: - (t `2) < (t `2) ^2 by SQUARE_1:46; - (t `2) >= - (t `1) by A103, A116, XREAL_1:24; then (t `2) ^2 > - (t `1) by A127, XXREAL_0:2; then ((t `2) ^2) / (t `2) < (- (t `1)) / (t `2) by A117, XREAL_1:75; then t `2 < - ((t `1) / (t `2)) by A117, XCMPLX_1:89; then - (t `2) > - (- ((t `1) / (t `2))) by XREAL_1:24; then A128: ((- 1) * (t `2)) / (t `2) < ((t `1) / (t `2)) / (t `2) by A117, XREAL_1:75; (t `2) / (t `2) > (- 1) / (t `2) by A126, XREAL_1:75; then 1 > (- 1) / (t `2) by A117, XCMPLX_1:60; then - 1 < - ((- 1) / (t `2)) by XREAL_1:24; hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) by A103, A116, A124, A128, XCMPLX_1:89; ::_thesis: verum end; end; end; hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) ; ::_thesis: verum end; end; end; Out_In_Sq . t = |[(((t `1) / (t `2)) / (t `2)),(1 / (t `2))]| by A75, A103, Def1; then ( p4 `2 = 1 / (t `2) & p4 `1 = ((t `1) / (t `2)) / (t `2) ) by EUCLID:52; hence Out_In_Sq . t in K0 by A1, A105; ::_thesis: verum end; end; end; hence Out_In_Sq . t in K0 ; ::_thesis: verum end; A129: D = NonZero (TOP-REAL 2) by A1, SUBSET_1:def_4; for x1, x2 being set st x1 in dom Out_In_Sq & x2 in dom Out_In_Sq & Out_In_Sq . x1 = Out_In_Sq . x2 holds x1 = x2 proof A130: { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } or x in D ) assume x in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in D then A131: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) ; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in D by A3, A131, XBOOLE_0:def_5; ::_thesis: verum end; A132: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19; ( ( (1.REAL 2) `1 <= (1.REAL 2) `2 & - ((1.REAL 2) `2) <= (1.REAL 2) `1 ) or ( (1.REAL 2) `1 >= (1.REAL 2) `2 & (1.REAL 2) `1 <= - ((1.REAL 2) `2) ) ) by Th5; then A133: 1.REAL 2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } by A132; the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D) .= D by PRE_TOPC:def_5 ; then reconsider K11 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A133, A130; reconsider K01 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th17; let x1, x2 be set ; ::_thesis: ( x1 in dom Out_In_Sq & x2 in dom Out_In_Sq & Out_In_Sq . x1 = Out_In_Sq . x2 implies x1 = x2 ) assume that A134: x1 in dom Out_In_Sq and A135: x2 in dom Out_In_Sq and A136: Out_In_Sq . x1 = Out_In_Sq . x2 ; ::_thesis: x1 = x2 NonZero (TOP-REAL 2) <> {} by Th9; then A137: dom Out_In_Sq = NonZero (TOP-REAL 2) by FUNCT_2:def_1; then A138: x2 in D by A1, A135, SUBSET_1:def_4; reconsider p1 = x1, p2 = x2 as Point of (TOP-REAL 2) by A134, A135, XBOOLE_0:def_5; A139: D c= K01 \/ K11 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in K01 \/ K11 ) assume A140: x in D ; ::_thesis: x in K01 \/ K11 then reconsider px = x as Point of (TOP-REAL 2) ; not x in {(0. (TOP-REAL 2))} by A129, A140, XBOOLE_0:def_5; then ( ( ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) or ( ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1, XREAL_1:25; then ( x in K01 or x in K11 ) ; hence x in K01 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum end; A141: x1 in D by A1, A134, A137, SUBSET_1:def_4; now__::_thesis:_(_(_x1_in_K01_&_x1_=_x2_)_or_(_x1_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`1_<=_p8_`2_&_-_(p8_`2)_<=_p8_`1_)_or_(_p8_`1_>=_p8_`2_&_p8_`1_<=_-_(p8_`2)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_x1_=_x2_)_) percases ( x1 in K01 or x1 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) by A139, A141, XBOOLE_0:def_3; case x1 in K01 ; ::_thesis: x1 = x2 then A142: ex p7 being Point of (TOP-REAL 2) st ( p1 = p7 & ( ( p7 `2 <= p7 `1 & - (p7 `1) <= p7 `2 ) or ( p7 `2 >= p7 `1 & p7 `2 <= - (p7 `1) ) ) & p7 <> 0. (TOP-REAL 2) ) ; then A143: Out_In_Sq . p1 = |[(1 / (p1 `1)),(((p1 `2) / (p1 `1)) / (p1 `1))]| by Def1; now__::_thesis:_(_(_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`2_<=_p8_`1_&_-_(p8_`1)_<=_p8_`2_)_or_(_p8_`2_>=_p8_`1_&_p8_`2_<=_-_(p8_`1)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_x1_=_x2_)_or_(_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`1_<=_p8_`2_&_-_(p8_`2)_<=_p8_`1_)_or_(_p8_`1_>=_p8_`2_&_p8_`1_<=_-_(p8_`2)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_not_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`2_<=_p8_`1_&_-_(p8_`1)_<=_p8_`2_)_or_(_p8_`2_>=_p8_`1_&_p8_`2_<=_-_(p8_`1)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_contradiction_)_) percases ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } or ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } & not x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) ) by A139, A138, XBOOLE_0:def_3; case x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; ::_thesis: x1 = x2 then ex p8 being Point of (TOP-REAL 2) st ( p2 = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) ; then A144: |[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]| = |[(1 / (p1 `1)),(((p1 `2) / (p1 `1)) / (p1 `1))]| by A136, A143, Def1; A145: p1 = |[(p1 `1),(p1 `2)]| by EUCLID:53; set qq = |[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]|; A146: (1 / (p1 `1)) " = ((p1 `1) ") " .= p1 `1 ; A147: now__::_thesis:_not_p1_`1_=_0 assume A148: p1 `1 = 0 ; ::_thesis: contradiction then p1 `2 = 0 by A142; hence contradiction by A142, A148, EUCLID:53, EUCLID:54; ::_thesis: verum end; |[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]| `1 = 1 / (p2 `1) by EUCLID:52; then A149: 1 / (p1 `1) = 1 / (p2 `1) by A144, EUCLID:52; |[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]| `2 = ((p2 `2) / (p2 `1)) / (p2 `1) by EUCLID:52; then (p1 `2) / (p1 `1) = (p2 `2) / (p1 `1) by A144, A149, A146, A147, EUCLID:52, XCMPLX_1:53; then p1 `2 = p2 `2 by A147, XCMPLX_1:53; hence x1 = x2 by A149, A146, A145, EUCLID:53; ::_thesis: verum end; caseA150: ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } & not x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: contradiction A151: now__::_thesis:_not_p1_`1_=_0 assume A152: p1 `1 = 0 ; ::_thesis: contradiction then p1 `2 = 0 by A142; hence contradiction by A142, A152, EUCLID:53, EUCLID:54; ::_thesis: verum end; A153: now__::_thesis:_(_(_p1_`2_<=_p1_`1_&_-_(p1_`1)_<=_p1_`2_&_(p1_`2)_/_(p1_`1)_<=_1_)_or_(_p1_`2_>=_p1_`1_&_p1_`2_<=_-_(p1_`1)_&_(p1_`2)_/_(p1_`1)_<=_1_)_) percases ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) by A142; caseA154: ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) ; ::_thesis: (p1 `2) / (p1 `1) <= 1 then p1 `1 >= 0 ; then (p1 `2) / (p1 `1) <= (p1 `1) / (p1 `1) by A154, XREAL_1:72; hence (p1 `2) / (p1 `1) <= 1 by A151, XCMPLX_1:60; ::_thesis: verum end; caseA155: ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ; ::_thesis: (p1 `2) / (p1 `1) <= 1 then p1 `1 <= 0 ; then (p1 `2) / (p1 `1) <= (p1 `1) / (p1 `1) by A155, XREAL_1:73; hence (p1 `2) / (p1 `1) <= 1 by A151, XCMPLX_1:60; ::_thesis: verum end; end; end; A156: now__::_thesis:_(_(_p1_`2_<=_p1_`1_&_-_(p1_`1)_<=_p1_`2_&_-_1_<=_(p1_`2)_/_(p1_`1)_)_or_(_p1_`2_>=_p1_`1_&_p1_`2_<=_-_(p1_`1)_&_-_1_<=_(p1_`2)_/_(p1_`1)_)_) percases ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) by A142; caseA157: ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) ; ::_thesis: - 1 <= (p1 `2) / (p1 `1) then p1 `1 >= 0 ; then (- (p1 `1)) / (p1 `1) <= (p1 `2) / (p1 `1) by A157, XREAL_1:72; hence - 1 <= (p1 `2) / (p1 `1) by A151, XCMPLX_1:197; ::_thesis: verum end; case ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ; ::_thesis: - 1 <= (p1 `2) / (p1 `1) then ( - (p1 `2) >= - (- (p1 `1)) & p1 `1 <= 0 ) by XREAL_1:24; then (- (p1 `2)) / (- (p1 `1)) >= (p1 `1) / (- (p1 `1)) by XREAL_1:72; then (- (p1 `2)) / (- (p1 `1)) >= - 1 by A151, XCMPLX_1:198; hence - 1 <= (p1 `2) / (p1 `1) by XCMPLX_1:191; ::_thesis: verum end; end; end; A158: ex p8 being Point of (TOP-REAL 2) st ( p2 = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A150; A159: now__::_thesis:_not_p2_`2_=_0 assume A160: p2 `2 = 0 ; ::_thesis: contradiction then p2 `1 = 0 by A158; hence contradiction by A158, A160, EUCLID:53, EUCLID:54; ::_thesis: verum end; ( ( not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) or not p2 <> 0. (TOP-REAL 2) ) by A150; then A161: Out_In_Sq . p2 = |[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]| by A158, Def1; then ((p1 `2) / (p1 `1)) / (p1 `1) = 1 / (p2 `2) by A136, A143, SPPOL_2:1; then A162: (p1 `2) / (p1 `1) = (1 / (p2 `2)) * (p1 `1) by A151, XCMPLX_1:87 .= (p1 `1) / (p2 `2) ; 1 / (p1 `1) = ((p2 `1) / (p2 `2)) / (p2 `2) by A136, A143, A161, SPPOL_2:1; then A163: (p2 `1) / (p2 `2) = (1 / (p1 `1)) * (p2 `2) by A159, XCMPLX_1:87 .= (p2 `2) / (p1 `1) ; then A164: ((p2 `1) / (p2 `2)) * ((p1 `2) / (p1 `1)) = 1 by A159, A151, A162, XCMPLX_1:112; A165: (((p2 `1) / (p2 `2)) * ((p1 `2) / (p1 `1))) * (p1 `1) = 1 * (p1 `1) by A159, A151, A163, A162, XCMPLX_1:112; then A166: p1 `2 <> 0 by A151; A167: ex p9 being Point of (TOP-REAL 2) st ( p2 = p9 & ( ( p9 `1 <= p9 `2 & - (p9 `2) <= p9 `1 ) or ( p9 `1 >= p9 `2 & p9 `1 <= - (p9 `2) ) ) & p9 <> 0. (TOP-REAL 2) ) by A150; A168: now__::_thesis:_(_(_p2_`1_<=_p2_`2_&_-_(p2_`2)_<=_p2_`1_&_-_1_<=_(p2_`1)_/_(p2_`2)_)_or_(_p2_`1_>=_p2_`2_&_p2_`1_<=_-_(p2_`2)_&_-_1_<=_(p2_`1)_/_(p2_`2)_)_) percases ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A167; caseA169: ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) ; ::_thesis: - 1 <= (p2 `1) / (p2 `2) then p2 `2 >= 0 ; then (- (p2 `2)) / (p2 `2) <= (p2 `1) / (p2 `2) by A169, XREAL_1:72; hence - 1 <= (p2 `1) / (p2 `2) by A159, XCMPLX_1:197; ::_thesis: verum end; case ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ; ::_thesis: - 1 <= (p2 `1) / (p2 `2) then ( - (p2 `1) >= - (- (p2 `2)) & p2 `2 <= 0 ) by XREAL_1:24; then (- (p2 `1)) / (- (p2 `2)) >= (p2 `2) / (- (p2 `2)) by XREAL_1:72; then (- (p2 `1)) / (- (p2 `2)) >= - 1 by A159, XCMPLX_1:198; hence - 1 <= (p2 `1) / (p2 `2) by XCMPLX_1:191; ::_thesis: verum end; end; end; ((p2 `1) / (p2 `2)) * (((p1 `2) / (p1 `1)) * (p1 `1)) = p1 `1 by A165; then A170: ((p2 `1) / (p2 `2)) * (p1 `2) = p1 `1 by A151, XCMPLX_1:87; then A171: (p2 `1) / (p2 `2) = (p1 `1) / (p1 `2) by A166, XCMPLX_1:89; A172: now__::_thesis:_(_(_p2_`1_<=_p2_`2_&_-_(p2_`2)_<=_p2_`1_&_(p2_`1)_/_(p2_`2)_<=_1_)_or_(_p2_`1_>=_p2_`2_&_p2_`1_<=_-_(p2_`2)_&_(p2_`1)_/_(p2_`2)_<=_1_)_) percases ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A167; caseA173: ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) ; ::_thesis: (p2 `1) / (p2 `2) <= 1 then p2 `2 >= 0 ; then (p2 `1) / (p2 `2) <= (p2 `2) / (p2 `2) by A173, XREAL_1:72; hence (p2 `1) / (p2 `2) <= 1 by A159, XCMPLX_1:60; ::_thesis: verum end; caseA174: ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ; ::_thesis: (p2 `1) / (p2 `2) <= 1 then p2 `2 <= 0 ; then (p2 `1) / (p2 `2) <= (p2 `2) / (p2 `2) by A174, XREAL_1:73; hence (p2 `1) / (p2 `2) <= 1 by A159, XCMPLX_1:60; ::_thesis: verum end; end; end; now__::_thesis:_(_(_0_<=_(p2_`1)_/_(p2_`2)_&_contradiction_)_or_(_0_>_(p2_`1)_/_(p2_`2)_&_contradiction_)_) percases ( 0 <= (p2 `1) / (p2 `2) or 0 > (p2 `1) / (p2 `2) ) ; case 0 <= (p2 `1) / (p2 `2) ; ::_thesis: contradiction then A175: ( ( p1 `2 > 0 & p1 `1 >= 0 ) or ( p1 `2 < 0 & p1 `1 <= 0 ) ) by A151, A170; now__::_thesis:_not_(p1_`2)_/_(p1_`1)_<>_1 assume (p1 `2) / (p1 `1) <> 1 ; ::_thesis: contradiction then (p1 `2) / (p1 `1) < 1 by A153, XXREAL_0:1; hence contradiction by A164, A172, A175, XREAL_1:162; ::_thesis: verum end; then p1 `2 = 1 * (p1 `1) by A151, XCMPLX_1:87; then ((p2 `1) / (p2 `2)) * (p2 `2) = 1 * (p2 `2) by A151, A171, XCMPLX_1:60 .= p2 `2 ; then p2 `1 = p2 `2 by A159, XCMPLX_1:87; hence contradiction by A150, A167; ::_thesis: verum end; case 0 > (p2 `1) / (p2 `2) ; ::_thesis: contradiction then A176: ( ( p1 `2 < 0 & p1 `1 > 0 ) or ( p1 `2 > 0 & p1 `1 < 0 ) ) by A171, XREAL_1:143; now__::_thesis:_not_(p1_`2)_/_(p1_`1)_<>_-_1 assume (p1 `2) / (p1 `1) <> - 1 ; ::_thesis: contradiction then - 1 < (p1 `2) / (p1 `1) by A156, XXREAL_0:1; hence contradiction by A164, A168, A176, XREAL_1:166; ::_thesis: verum end; then p1 `2 = (- 1) * (p1 `1) by A151, XCMPLX_1:87 .= - (p1 `1) ; then - (p1 `2) = p1 `1 ; then (p2 `1) / (p2 `2) = - 1 by A166, A171, XCMPLX_1:197; then p2 `1 = (- 1) * (p2 `2) by A159, XCMPLX_1:87; then - (p2 `1) = p2 `2 ; hence contradiction by A150, A167; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; case x1 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; ::_thesis: x1 = x2 then A177: ex p7 being Point of (TOP-REAL 2) st ( p1 = p7 & ( ( p7 `1 <= p7 `2 & - (p7 `2) <= p7 `1 ) or ( p7 `1 >= p7 `2 & p7 `1 <= - (p7 `2) ) ) & p7 <> 0. (TOP-REAL 2) ) ; then A178: Out_In_Sq . p1 = |[(((p1 `1) / (p1 `2)) / (p1 `2)),(1 / (p1 `2))]| by Th14; now__::_thesis:_(_(_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`1_<=_p8_`2_&_-_(p8_`2)_<=_p8_`1_)_or_(_p8_`1_>=_p8_`2_&_p8_`1_<=_-_(p8_`2)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_x1_=_x2_)_or_(_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`2_<=_p8_`1_&_-_(p8_`1)_<=_p8_`2_)_or_(_p8_`2_>=_p8_`1_&_p8_`2_<=_-_(p8_`1)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_not_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`1_<=_p8_`2_&_-_(p8_`2)_<=_p8_`1_)_or_(_p8_`1_>=_p8_`2_&_p8_`1_<=_-_(p8_`2)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_contradiction_)_) percases ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } or ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } & not x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) ) by A139, A138, XBOOLE_0:def_3; case x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; ::_thesis: x1 = x2 then ex p8 being Point of (TOP-REAL 2) st ( p2 = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) ; then A179: |[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]| = |[(((p1 `1) / (p1 `2)) / (p1 `2)),(1 / (p1 `2))]| by A136, A178, Th14; A180: p1 = |[(p1 `1),(p1 `2)]| by EUCLID:53; set qq = |[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]|; A181: (1 / (p1 `2)) " = ((p1 `2) ") " .= p1 `2 ; A182: now__::_thesis:_not_p1_`2_=_0 assume A183: p1 `2 = 0 ; ::_thesis: contradiction then p1 `1 = 0 by A177; hence contradiction by A177, A183, EUCLID:53, EUCLID:54; ::_thesis: verum end; |[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]| `2 = 1 / (p2 `2) by EUCLID:52; then A184: 1 / (p1 `2) = 1 / (p2 `2) by A179, EUCLID:52; |[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]| `1 = ((p2 `1) / (p2 `2)) / (p2 `2) by EUCLID:52; then (p1 `1) / (p1 `2) = (p2 `1) / (p1 `2) by A179, A184, A181, A182, EUCLID:52, XCMPLX_1:53; then p1 `1 = p2 `1 by A182, XCMPLX_1:53; hence x1 = x2 by A184, A181, A180, EUCLID:53; ::_thesis: verum end; caseA185: ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } & not x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: contradiction A186: now__::_thesis:_not_p1_`2_=_0 assume A187: p1 `2 = 0 ; ::_thesis: contradiction then p1 `1 = 0 by A177; hence contradiction by A177, A187, EUCLID:53, EUCLID:54; ::_thesis: verum end; A188: now__::_thesis:_(_(_p1_`1_<=_p1_`2_&_-_(p1_`2)_<=_p1_`1_&_(p1_`1)_/_(p1_`2)_<=_1_)_or_(_p1_`1_>=_p1_`2_&_p1_`1_<=_-_(p1_`2)_&_(p1_`1)_/_(p1_`2)_<=_1_)_) percases ( ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) or ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ) by A177; caseA189: ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) ; ::_thesis: (p1 `1) / (p1 `2) <= 1 then p1 `2 >= 0 ; then (p1 `1) / (p1 `2) <= (p1 `2) / (p1 `2) by A189, XREAL_1:72; hence (p1 `1) / (p1 `2) <= 1 by A186, XCMPLX_1:60; ::_thesis: verum end; caseA190: ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ; ::_thesis: (p1 `1) / (p1 `2) <= 1 then p1 `2 <= 0 ; then (p1 `1) / (p1 `2) <= (p1 `2) / (p1 `2) by A190, XREAL_1:73; hence (p1 `1) / (p1 `2) <= 1 by A186, XCMPLX_1:60; ::_thesis: verum end; end; end; A191: now__::_thesis:_(_(_p1_`1_<=_p1_`2_&_-_(p1_`2)_<=_p1_`1_&_-_1_<=_(p1_`1)_/_(p1_`2)_)_or_(_p1_`1_>=_p1_`2_&_p1_`1_<=_-_(p1_`2)_&_-_1_<=_(p1_`1)_/_(p1_`2)_)_) percases ( ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) or ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ) by A177; caseA192: ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) ; ::_thesis: - 1 <= (p1 `1) / (p1 `2) then p1 `2 >= 0 ; then (- (p1 `2)) / (p1 `2) <= (p1 `1) / (p1 `2) by A192, XREAL_1:72; hence - 1 <= (p1 `1) / (p1 `2) by A186, XCMPLX_1:197; ::_thesis: verum end; case ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ; ::_thesis: - 1 <= (p1 `1) / (p1 `2) then ( - (p1 `1) >= - (- (p1 `2)) & p1 `2 <= 0 ) by XREAL_1:24; then (- (p1 `1)) / (- (p1 `2)) >= (p1 `2) / (- (p1 `2)) by XREAL_1:72; then (- (p1 `1)) / (- (p1 `2)) >= - 1 by A186, XCMPLX_1:198; hence - 1 <= (p1 `1) / (p1 `2) by XCMPLX_1:191; ::_thesis: verum end; end; end; A193: ex p8 being Point of (TOP-REAL 2) st ( p2 = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A185; A194: now__::_thesis:_not_p2_`1_=_0 assume A195: p2 `1 = 0 ; ::_thesis: contradiction then p2 `2 = 0 by A193; hence contradiction by A193, A195, EUCLID:53, EUCLID:54; ::_thesis: verum end; A196: ex p9 being Point of (TOP-REAL 2) st ( p2 = p9 & ( ( p9 `2 <= p9 `1 & - (p9 `1) <= p9 `2 ) or ( p9 `2 >= p9 `1 & p9 `2 <= - (p9 `1) ) ) & p9 <> 0. (TOP-REAL 2) ) by A185; A197: now__::_thesis:_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_&_-_1_<=_(p2_`2)_/_(p2_`1)_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_&_-_1_<=_(p2_`2)_/_(p2_`1)_)_) percases ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) by A196; caseA198: ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) ; ::_thesis: - 1 <= (p2 `2) / (p2 `1) then p2 `1 >= 0 ; then (- (p2 `1)) / (p2 `1) <= (p2 `2) / (p2 `1) by A198, XREAL_1:72; hence - 1 <= (p2 `2) / (p2 `1) by A194, XCMPLX_1:197; ::_thesis: verum end; case ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ; ::_thesis: - 1 <= (p2 `2) / (p2 `1) then ( - (p2 `2) >= - (- (p2 `1)) & p2 `1 <= 0 ) by XREAL_1:24; then (- (p2 `2)) / (- (p2 `1)) >= (p2 `1) / (- (p2 `1)) by XREAL_1:72; then (- (p2 `2)) / (- (p2 `1)) >= - 1 by A194, XCMPLX_1:198; hence - 1 <= (p2 `2) / (p2 `1) by XCMPLX_1:191; ::_thesis: verum end; end; end; A199: Out_In_Sq . p2 = |[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]| by A193, Def1; then 1 / (p1 `2) = ((p2 `2) / (p2 `1)) / (p2 `1) by A136, A178, SPPOL_2:1; then A200: (p2 `2) / (p2 `1) = (1 / (p1 `2)) * (p2 `1) by A194, XCMPLX_1:87 .= (p2 `1) / (p1 `2) ; ((p1 `1) / (p1 `2)) / (p1 `2) = 1 / (p2 `1) by A136, A178, A199, SPPOL_2:1; then (p1 `1) / (p1 `2) = (1 / (p2 `1)) * (p1 `2) by A186, XCMPLX_1:87 .= (p1 `2) / (p2 `1) ; then A201: ((p2 `2) / (p2 `1)) * ((p1 `1) / (p1 `2)) = 1 by A194, A186, A200, XCMPLX_1:112; then A202: p1 `1 <> 0 ; (((p2 `2) / (p2 `1)) * ((p1 `1) / (p1 `2))) * (p1 `2) = p1 `2 by A201; then ((p2 `2) / (p2 `1)) * (((p1 `1) / (p1 `2)) * (p1 `2)) = p1 `2 ; then ((p2 `2) / (p2 `1)) * (p1 `1) = p1 `2 by A186, XCMPLX_1:87; then A203: (p2 `2) / (p2 `1) = (p1 `2) / (p1 `1) by A202, XCMPLX_1:89; A204: now__::_thesis:_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_&_(p2_`2)_/_(p2_`1)_<=_1_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_&_(p2_`2)_/_(p2_`1)_<=_1_)_) percases ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) by A196; caseA205: ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) ; ::_thesis: (p2 `2) / (p2 `1) <= 1 then p2 `1 >= 0 ; then (p2 `2) / (p2 `1) <= (p2 `1) / (p2 `1) by A205, XREAL_1:72; hence (p2 `2) / (p2 `1) <= 1 by A194, XCMPLX_1:60; ::_thesis: verum end; caseA206: ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ; ::_thesis: (p2 `2) / (p2 `1) <= 1 then p2 `1 <= 0 ; then (p2 `2) / (p2 `1) <= (p2 `1) / (p2 `1) by A206, XREAL_1:73; hence (p2 `2) / (p2 `1) <= 1 by A194, XCMPLX_1:60; ::_thesis: verum end; end; end; now__::_thesis:_(_(_0_<=_(p2_`2)_/_(p2_`1)_&_contradiction_)_or_(_0_>_(p2_`2)_/_(p2_`1)_&_contradiction_)_) percases ( 0 <= (p2 `2) / (p2 `1) or 0 > (p2 `2) / (p2 `1) ) ; case 0 <= (p2 `2) / (p2 `1) ; ::_thesis: contradiction then A207: ( ( p1 `1 > 0 & p1 `2 >= 0 ) or ( p1 `1 < 0 & p1 `2 <= 0 ) ) by A201, A202; now__::_thesis:_not_(p1_`1)_/_(p1_`2)_<>_1 assume (p1 `1) / (p1 `2) <> 1 ; ::_thesis: contradiction then (p1 `1) / (p1 `2) < 1 by A188, XXREAL_0:1; hence contradiction by A201, A204, A207, XREAL_1:162; ::_thesis: verum end; then p1 `1 = 1 * (p1 `2) by A186, XCMPLX_1:87; then ((p2 `2) / (p2 `1)) * (p2 `1) = 1 * (p2 `1) by A186, A203, XCMPLX_1:60 .= p2 `1 ; then p2 `2 = p2 `1 by A194, XCMPLX_1:87; hence contradiction by A185, A196; ::_thesis: verum end; case 0 > (p2 `2) / (p2 `1) ; ::_thesis: contradiction then A208: ( ( p1 `1 < 0 & p1 `2 > 0 ) or ( p1 `1 > 0 & p1 `2 < 0 ) ) by A203, XREAL_1:143; now__::_thesis:_not_(p1_`1)_/_(p1_`2)_<>_-_1 assume (p1 `1) / (p1 `2) <> - 1 ; ::_thesis: contradiction then - 1 < (p1 `1) / (p1 `2) by A191, XXREAL_0:1; hence contradiction by A201, A197, A208, XREAL_1:166; ::_thesis: verum end; then p1 `1 = (- 1) * (p1 `2) by A186, XCMPLX_1:87 .= - (p1 `2) ; then - (p1 `1) = p1 `2 ; then (p2 `2) / (p2 `1) = - 1 by A202, A203, XCMPLX_1:197; then p2 `2 = (- 1) * (p2 `1) by A194, XCMPLX_1:87; then - (p2 `2) = p2 `1 ; hence contradiction by A185, A196; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; then A209: Out_In_Sq is one-to-one by FUNCT_1:def_4; A210: for s being Point of (TOP-REAL 2) st s in Kb holds Out_In_Sq . s = s proof let t be Point of (TOP-REAL 2); ::_thesis: ( t in Kb implies Out_In_Sq . t = t ) assume t in Kb ; ::_thesis: Out_In_Sq . t = t then A211: ex p4 being Point of (TOP-REAL 2) st ( p4 = t & ( ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( p4 `1 = 1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ) ) by A1; then A212: t <> 0. (TOP-REAL 2) by EUCLID:52, EUCLID:54; A213: not t = 0. (TOP-REAL 2) by A211, EUCLID:52, EUCLID:54; now__::_thesis:_(_(_(_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_or_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_)_&_Out_In_Sq_._t_=_t_)_or_(_not_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_&_not_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_&_Out_In_Sq_._t_=_t_)_) percases ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) or ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ) ; caseA214: ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: Out_In_Sq . t = t then A215: Out_In_Sq . t = |[(1 / (t `1)),(((t `2) / (t `1)) / (t `1))]| by A213, Def1; A216: ( ( 1 <= t `1 & t `1 >= - 1 ) or ( 1 >= t `1 & - 1 >= - (- (t `1)) ) ) by A211, A214, XREAL_1:24; now__::_thesis:_(_(_t_`1_=_1_&_Out_In_Sq_._t_=_t_)_or_(_t_`1_=_-_1_&_Out_In_Sq_._t_=_t_)_) percases ( t `1 = 1 or t `1 = - 1 ) by A211, A216, XXREAL_0:1; case t `1 = 1 ; ::_thesis: Out_In_Sq . t = t hence Out_In_Sq . t = t by A215, EUCLID:53; ::_thesis: verum end; case t `1 = - 1 ; ::_thesis: Out_In_Sq . t = t hence Out_In_Sq . t = t by A215, EUCLID:53; ::_thesis: verum end; end; end; hence Out_In_Sq . t = t ; ::_thesis: verum end; caseA217: ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: Out_In_Sq . t = t then A218: Out_In_Sq . t = |[(((t `1) / (t `2)) / (t `2)),(1 / (t `2))]| by A212, Def1; now__::_thesis:_(_(_t_`2_=_1_&_Out_In_Sq_._t_=_t_)_or_(_t_`2_=_-_1_&_Out_In_Sq_._t_=_t_)_) percases ( t `2 = 1 or t `2 = - 1 ) by A211, A217; case t `2 = 1 ; ::_thesis: Out_In_Sq . t = t hence Out_In_Sq . t = t by A218, EUCLID:53; ::_thesis: verum end; case t `2 = - 1 ; ::_thesis: Out_In_Sq . t = t hence Out_In_Sq . t = t by A218, EUCLID:53; ::_thesis: verum end; end; end; hence Out_In_Sq . t = t ; ::_thesis: verum end; end; end; hence Out_In_Sq . t = t ; ::_thesis: verum end; ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = Out_In_Sq & h is continuous ) by A2, Th40; hence ex f being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) st ( f is continuous & f is one-to-one & ( for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds not f . t in K0 \/ Kb ) & ( for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds f . r in K0 ) & ( for s being Point of (TOP-REAL 2) st s in Kb holds f . s = s ) ) by A209, A4, A72, A210; ::_thesis: verum end; theorem Th42: :: JGRAPH_2:42 for f, g being Function of I[01],(TOP-REAL 2) for K0 being Subset of (TOP-REAL 2) for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & rng f misses K0 & rng g misses K0 holds rng f meets rng g proof reconsider B = {(0. (TOP-REAL 2))} as Subset of (TOP-REAL 2) ; A1: B ` <> {} by Th9; reconsider W = B ` as non empty Subset of (TOP-REAL 2) by Th9; defpred S1[ Point of (TOP-REAL 2)] means ( ( - 1 = $1 `1 & - 1 <= $1 `2 & $1 `2 <= 1 ) or ( $1 `1 = 1 & - 1 <= $1 `2 & $1 `2 <= 1 ) or ( - 1 = $1 `2 & - 1 <= $1 `1 & $1 `1 <= 1 ) or ( 1 = $1 `2 & - 1 <= $1 `1 & $1 `1 <= 1 ) ); A2: the carrier of ((TOP-REAL 2) | (B `)) = [#] ((TOP-REAL 2) | (B `)) .= B ` by PRE_TOPC:def_5 ; reconsider Kb = { q where q is Point of (TOP-REAL 2) : S1[q] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for K0 being Subset of (TOP-REAL 2) for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & rng f misses K0 & rng g misses K0 holds rng f meets rng g let K0 be Subset of (TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & rng f misses K0 & rng g misses K0 holds rng f meets rng g let O, I be Point of I[01]; ::_thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & rng f misses K0 & rng g misses K0 implies rng f meets rng g ) A3: dom f = the carrier of I[01] by FUNCT_2:def_1; assume A4: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & (rng f) /\ K0 = {} & (rng g) /\ K0 = {} ) ; :: according to XBOOLE_0:def_7 ::_thesis: rng f meets rng g then consider h being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) such that A5: h is continuous and A6: h is one-to-one and for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds not h . t in K0 \/ Kb and A7: for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds h . r in K0 and A8: for s being Point of (TOP-REAL 2) st s in Kb holds h . s = s by Th41; rng f c= B ` proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng f or x in B ` ) assume A9: x in rng f ; ::_thesis: x in B ` now__::_thesis:_not_x_in_B assume x in B ; ::_thesis: contradiction then A10: x = 0. (TOP-REAL 2) by TARSKI:def_1; ( (0. (TOP-REAL 2)) `1 = 0 & (0. (TOP-REAL 2)) `2 = 0 ) by EUCLID:52, EUCLID:54; then 0. (TOP-REAL 2) in K0 by A4; hence contradiction by A4, A9, A10, XBOOLE_0:def_4; ::_thesis: verum end; then x in the carrier of (TOP-REAL 2) \ B by A9, XBOOLE_0:def_5; hence x in B ` by SUBSET_1:def_4; ::_thesis: verum end; then A11: ex w being Function of I[01],(TOP-REAL 2) st ( w is continuous & w = h * f ) by A4, A5, A1, Th12; then reconsider d1 = h * f as Function of I[01],(TOP-REAL 2) ; the carrier of ((TOP-REAL 2) | W) <> {} ; then A12: dom h = the carrier of ((TOP-REAL 2) | (B `)) by FUNCT_2:def_1; rng g c= B ` proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g or x in B ` ) assume A13: x in rng g ; ::_thesis: x in B ` now__::_thesis:_not_x_in_B assume x in B ; ::_thesis: contradiction then A14: x = 0. (TOP-REAL 2) by TARSKI:def_1; 0. (TOP-REAL 2) in K0 by A4, Th3; hence contradiction by A4, A13, A14, XBOOLE_0:def_4; ::_thesis: verum end; then x in the carrier of (TOP-REAL 2) \ B by A13, XBOOLE_0:def_5; hence x in B ` by SUBSET_1:def_4; ::_thesis: verum end; then A15: ex w2 being Function of I[01],(TOP-REAL 2) st ( w2 is continuous & w2 = h * g ) by A4, A5, A1, Th12; then reconsider d2 = h * g as Function of I[01],(TOP-REAL 2) ; A16: dom g = the carrier of I[01] by FUNCT_2:def_1; A17: for r being Point of I[01] holds ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) proof let r be Point of I[01]; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) A18: ( g . r in Kb implies d2 . r in K0 \/ Kb ) proof A19: d2 . r = h . (g . r) by A16, FUNCT_1:13; assume A20: g . r in Kb ; ::_thesis: d2 . r in K0 \/ Kb then h . (g . r) = g . r by A8; hence d2 . r in K0 \/ Kb by A20, A19, XBOOLE_0:def_3; ::_thesis: verum end; f . r in rng f by A3, FUNCT_1:3; then A21: not f . r in K0 by A4, XBOOLE_0:def_4; A22: ( not f . r in Kb implies d1 . r in K0 \/ Kb ) proof assume not f . r in Kb ; ::_thesis: d1 . r in K0 \/ Kb then not f . r in K0 \/ Kb by A21, XBOOLE_0:def_3; then A23: h . (f . r) in K0 by A7; d1 . r = h . (f . r) by A3, FUNCT_1:13; hence d1 . r in K0 \/ Kb by A23, XBOOLE_0:def_3; ::_thesis: verum end; g . r in rng g by A16, FUNCT_1:3; then A24: not g . r in K0 by A4, XBOOLE_0:def_4; A25: ( not g . r in Kb implies d2 . r in K0 \/ Kb ) proof assume not g . r in Kb ; ::_thesis: d2 . r in K0 \/ Kb then not g . r in K0 \/ Kb by A24, XBOOLE_0:def_3; then A26: h . (g . r) in K0 by A7; d2 . r = h . (g . r) by A16, FUNCT_1:13; hence d2 . r in K0 \/ Kb by A26, XBOOLE_0:def_3; ::_thesis: verum end; A27: ( f . r in Kb implies d1 . r in K0 \/ Kb ) proof A28: d1 . r = h . (f . r) by A3, FUNCT_1:13; assume A29: f . r in Kb ; ::_thesis: d1 . r in K0 \/ Kb then h . (f . r) = f . r by A8; hence d1 . r in K0 \/ Kb by A29, A28, XBOOLE_0:def_3; ::_thesis: verum end; now__::_thesis:_(_(_d1_._r_in_K0_&_d2_._r_in_K0_&_-_1_<=_(d1_._r)_`1_&_(d1_._r)_`1_<=_1_&_-_1_<=_(d2_._r)_`1_&_(d2_._r)_`1_<=_1_&_-_1_<=_(d1_._r)_`2_&_(d1_._r)_`2_<=_1_&_-_1_<=_(d2_._r)_`2_&_(d2_._r)_`2_<=_1_)_or_(_d1_._r_in_K0_&_d2_._r_in_Kb_&_-_1_<=_(d1_._r)_`1_&_(d1_._r)_`1_<=_1_&_-_1_<=_(d2_._r)_`1_&_(d2_._r)_`1_<=_1_&_-_1_<=_(d1_._r)_`2_&_(d1_._r)_`2_<=_1_&_-_1_<=_(d2_._r)_`2_&_(d2_._r)_`2_<=_1_)_or_(_d1_._r_in_Kb_&_d2_._r_in_K0_&_-_1_<=_(d1_._r)_`1_&_(d1_._r)_`1_<=_1_&_-_1_<=_(d2_._r)_`1_&_(d2_._r)_`1_<=_1_&_-_1_<=_(d1_._r)_`2_&_(d1_._r)_`2_<=_1_&_-_1_<=_(d2_._r)_`2_&_(d2_._r)_`2_<=_1_)_or_(_d1_._r_in_Kb_&_d2_._r_in_Kb_&_-_1_<=_(d1_._r)_`1_&_(d1_._r)_`1_<=_1_&_-_1_<=_(d2_._r)_`1_&_(d2_._r)_`1_<=_1_&_-_1_<=_(d1_._r)_`2_&_(d1_._r)_`2_<=_1_&_-_1_<=_(d2_._r)_`2_&_(d2_._r)_`2_<=_1_)_) percases ( ( d1 . r in K0 & d2 . r in K0 ) or ( d1 . r in K0 & d2 . r in Kb ) or ( d1 . r in Kb & d2 . r in K0 ) or ( d1 . r in Kb & d2 . r in Kb ) ) by A22, A27, A25, A18, XBOOLE_0:def_3; case ( d1 . r in K0 & d2 . r in K0 ) ; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) then ( ex p being Point of (TOP-REAL 2) st ( p = d1 . r & - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) & ex q being Point of (TOP-REAL 2) st ( q = d2 . r & - 1 < q `1 & q `1 < 1 & - 1 < q `2 & q `2 < 1 ) ) by A4; hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum end; case ( d1 . r in K0 & d2 . r in Kb ) ; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) then ( ex p being Point of (TOP-REAL 2) st ( p = d1 . r & - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) & ex q being Point of (TOP-REAL 2) st ( q = d2 . r & ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) ) ) by A4; hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum end; case ( d1 . r in Kb & d2 . r in K0 ) ; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) then ( ex p being Point of (TOP-REAL 2) st ( p = d2 . r & - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) & ex q being Point of (TOP-REAL 2) st ( q = d1 . r & ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) ) ) by A4; hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum end; case ( d1 . r in Kb & d2 . r in Kb ) ; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) then ( ex p being Point of (TOP-REAL 2) st ( p = d2 . r & ( ( - 1 = p `1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) ) & ex q being Point of (TOP-REAL 2) st ( q = d1 . r & ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) ) ) ; hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum end; end; end; hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum end; f . I in Kb by A4; then h . (f . I) = f . I by A8; then A30: (d1 . I) `1 = 1 by A4, A3, FUNCT_1:13; f . O in Kb by A4; then h . (f . O) = f . O by A8; then A31: (d1 . O) `1 = - 1 by A4, A3, FUNCT_1:13; g . I in Kb by A4; then h . (g . I) = g . I by A8; then A32: (d2 . I) `2 = 1 by A4, A16, FUNCT_1:13; g . O in Kb by A4; then h . (g . O) = g . O by A8; then A33: (d2 . O) `2 = - 1 by A4, A16, FUNCT_1:13; set s = the Element of (rng d1) /\ (rng d2); ( d1 is one-to-one & d2 is one-to-one ) by A4, A6, FUNCT_1:24; then rng d1 meets rng d2 by A4, A11, A15, A31, A30, A33, A32, A17, JGRAPH_1:47; then A34: (rng d1) /\ (rng d2) <> {} by XBOOLE_0:def_7; then the Element of (rng d1) /\ (rng d2) in rng d1 by XBOOLE_0:def_4; then consider t1 being set such that A35: t1 in dom d1 and A36: the Element of (rng d1) /\ (rng d2) = d1 . t1 by FUNCT_1:def_3; A37: f . t1 in rng f by A3, A35, FUNCT_1:3; the Element of (rng d1) /\ (rng d2) in rng d2 by A34, XBOOLE_0:def_4; then consider t2 being set such that A38: t2 in dom d2 and A39: the Element of (rng d1) /\ (rng d2) = d2 . t2 by FUNCT_1:def_3; h . (f . t1) = d1 . t1 by A35, FUNCT_1:12; then A40: h . (f . t1) = h . (g . t2) by A36, A38, A39, FUNCT_1:12; rng g c= the carrier of (TOP-REAL 2) \ B proof let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in rng g or e in the carrier of (TOP-REAL 2) \ B ) assume A41: e in rng g ; ::_thesis: e in the carrier of (TOP-REAL 2) \ B now__::_thesis:_not_e_in_B assume e in B ; ::_thesis: contradiction then A42: e = 0. (TOP-REAL 2) by TARSKI:def_1; 0. (TOP-REAL 2) in { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } by Th3; hence contradiction by A4, A41, A42, XBOOLE_0:def_4; ::_thesis: verum end; hence e in the carrier of (TOP-REAL 2) \ B by A41, XBOOLE_0:def_5; ::_thesis: verum end; then A43: rng g c= the carrier of ((TOP-REAL 2) | (B `)) by A2, SUBSET_1:def_4; dom g = the carrier of I[01] by FUNCT_2:def_1; then A44: g . t2 in rng g by A38, FUNCT_1:3; rng f c= the carrier of (TOP-REAL 2) \ B proof let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in rng f or e in the carrier of (TOP-REAL 2) \ B ) assume A45: e in rng f ; ::_thesis: e in the carrier of (TOP-REAL 2) \ B now__::_thesis:_not_e_in_B assume e in B ; ::_thesis: contradiction then A46: e = 0. (TOP-REAL 2) by TARSKI:def_1; 0. (TOP-REAL 2) in { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } by Th3; hence contradiction by A4, A45, A46, XBOOLE_0:def_4; ::_thesis: verum end; hence e in the carrier of (TOP-REAL 2) \ B by A45, XBOOLE_0:def_5; ::_thesis: verum end; then rng f c= the carrier of ((TOP-REAL 2) | (B `)) by A2, SUBSET_1:def_4; then f . t1 = g . t2 by A6, A43, A40, A12, A37, A44, FUNCT_1:def_4; then (rng f) /\ (rng g) <> {} by A37, A44, XBOOLE_0:def_4; hence rng f meets rng g by XBOOLE_0:def_7; ::_thesis: verum end; theorem Th43: :: JGRAPH_2:43 for A, B, C, D being real number for f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for t being Point of (TOP-REAL 2) holds f . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) holds f is continuous proof reconsider h11 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; set K0 = [#] (TOP-REAL 2); let A, B, C, D be real number ; ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for t being Point of (TOP-REAL 2) holds f . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) holds f is continuous let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( ( for t being Point of (TOP-REAL 2) holds f . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) implies f is continuous ) A1: (TOP-REAL 2) | ([#] (TOP-REAL 2)) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) by TSEP_1:93; then reconsider h1 = h11 as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 ; h11 is continuous by JORDAN5A:27; then h1 is continuous by A1, PRE_TOPC:32; then consider g1 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A2: for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1 being real number st h1 . p = r1 holds g1 . p = A * r1 and A3: g1 is continuous by Th23; reconsider f1 = proj1 * f as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by A1, TOPMETR:17; consider g11 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A4: for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1 being real number st g1 . p = r1 holds g11 . p = r1 + B and A5: g11 is continuous by A3, Th24; reconsider f2 = proj2 * f as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by A1, TOPMETR:17; reconsider h11 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h1 = h11 as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by A1; dom f1 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A6: dom f1 = dom g11 by A1, FUNCT_2:def_1; assume A7: for t being Point of (TOP-REAL 2) holds f . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ; ::_thesis: f is continuous A8: for x being set st x in dom f1 holds f1 . x = g11 . x proof let x be set ; ::_thesis: ( x in dom f1 implies f1 . x = g11 . x ) assume A9: x in dom f1 ; ::_thesis: f1 . x = g11 . x then reconsider p = x as Point of (TOP-REAL 2) by FUNCT_2:def_1; f1 . x = proj1 . (f . x) by A9, FUNCT_1:12; then A10: f1 . x = proj1 . |[((A * (p `1)) + B),((C * (p `2)) + D)]| by A7 .= (A * (p `1)) + B by PSCOMP_1:65 .= (A * (proj1 . p)) + B by PSCOMP_1:def_5 ; A * (proj1 . p) = g1 . p by A1, A2; hence f1 . x = g11 . x by A1, A4, A10; ::_thesis: verum end; h11 is continuous by JORDAN5A:27; then h1 is continuous by A1, PRE_TOPC:32; then consider g1 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A11: for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1 being real number st h1 . p = r1 holds g1 . p = C * r1 and A12: g1 is continuous by Th23; consider g11 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A13: for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1 being real number st g1 . p = r1 holds g11 . p = r1 + D and A14: g11 is continuous by A12, Th24; A15: for x being set st x in dom f2 holds f2 . x = g11 . x proof let x be set ; ::_thesis: ( x in dom f2 implies f2 . x = g11 . x ) assume A16: x in dom f2 ; ::_thesis: f2 . x = g11 . x then reconsider p = x as Point of (TOP-REAL 2) by FUNCT_2:def_1; f2 . x = proj2 . (f . x) by A16, FUNCT_1:12; then A17: f2 . x = proj2 . |[((A * (p `1)) + B),((C * (p `2)) + D)]| by A7 .= (C * (p `2)) + D by PSCOMP_1:65 .= (C * (proj2 . p)) + D by PSCOMP_1:def_6 ; C * (proj2 . p) = g1 . p by A1, A11; hence f2 . x = g11 . x by A1, A13, A17; ::_thesis: verum end; reconsider f0 = f as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),((TOP-REAL 2) | ([#] (TOP-REAL 2))) by A1; A18: for x, y, r, s being real number st |[x,y]| in [#] (TOP-REAL 2) & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f0 . |[x,y]| = |[r,s]| proof let x, y, r, s be real number ; ::_thesis: ( |[x,y]| in [#] (TOP-REAL 2) & r = f1 . |[x,y]| & s = f2 . |[x,y]| implies f0 . |[x,y]| = |[r,s]| ) assume that |[x,y]| in [#] (TOP-REAL 2) and A19: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f0 . |[x,y]| = |[r,s]| A20: f . |[x,y]| is Point of (TOP-REAL 2) ; dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then ( proj1 . (f0 . |[x,y]|) = r & proj2 . (f0 . |[x,y]|) = s ) by A19, FUNCT_1:13; hence f0 . |[x,y]| = |[r,s]| by A20, Th8; ::_thesis: verum end; dom f2 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then dom f2 = dom g11 by A1, FUNCT_2:def_1; then A21: f2 is continuous by A14, A15, FUNCT_1:2; f1 is continuous by A5, A6, A8, FUNCT_1:2; then f0 is continuous by A21, A18, Th35; hence f is continuous by A1, PRE_TOPC:34; ::_thesis: verum end; definition let A, B, C, D be real number ; func AffineMap (A,B,C,D) -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def2: :: JGRAPH_2:def 2 for t being Point of (TOP-REAL 2) holds it . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]|; existence ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st for t being Point of (TOP-REAL 2) holds b1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| proof defpred S1[ set , set ] means for t being Point of (TOP-REAL 2) st t = $1 holds $2 = |[((A * (t `1)) + B),((C * (t `2)) + D)]|; A1: for x being set st x in the carrier of (TOP-REAL 2) holds ex y being set st S1[x,y] proof let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL 2) implies ex y being set st S1[x,y] ) assume x in the carrier of (TOP-REAL 2) ; ::_thesis: ex y being set st S1[x,y] then reconsider t2 = x as Point of (TOP-REAL 2) ; reconsider y2 = |[((A * (t2 `1)) + B),((C * (t2 `2)) + D)]| as set ; for t being Point of (TOP-REAL 2) st t = x holds y2 = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ; hence ex y being set st S1[x,y] ; ::_thesis: verum end; ex ff being Function st ( dom ff = the carrier of (TOP-REAL 2) & ( for x being set st x in the carrier of (TOP-REAL 2) holds S1[x,ff . x] ) ) from CLASSES1:sch_1(A1); then consider ff being Function such that A2: dom ff = the carrier of (TOP-REAL 2) and A3: for x being set st x in the carrier of (TOP-REAL 2) holds for t being Point of (TOP-REAL 2) st t = x holds ff . x = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ; for x being set st x in the carrier of (TOP-REAL 2) holds ff . x in the carrier of (TOP-REAL 2) proof let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL 2) implies ff . x in the carrier of (TOP-REAL 2) ) assume x in the carrier of (TOP-REAL 2) ; ::_thesis: ff . x in the carrier of (TOP-REAL 2) then reconsider t = x as Point of (TOP-REAL 2) ; ff . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| by A3; hence ff . x in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; then reconsider ff = ff as Function of (TOP-REAL 2),(TOP-REAL 2) by A2, FUNCT_2:3; take ff ; ::_thesis: for t being Point of (TOP-REAL 2) holds ff . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| thus for t being Point of (TOP-REAL 2) holds ff . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| by A3; ::_thesis: verum end; uniqueness for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for t being Point of (TOP-REAL 2) holds b1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) & ( for t being Point of (TOP-REAL 2) holds b2 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) holds b1 = b2 proof let m1, m2 be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( ( for t being Point of (TOP-REAL 2) holds m1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) & ( for t being Point of (TOP-REAL 2) holds m2 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) implies m1 = m2 ) assume that A4: for t being Point of (TOP-REAL 2) holds m1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| and A5: for t being Point of (TOP-REAL 2) holds m2 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ; ::_thesis: m1 = m2 for x being Point of (TOP-REAL 2) holds m1 . x = m2 . x proof let t be Point of (TOP-REAL 2); ::_thesis: m1 . t = m2 . t thus m1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| by A4 .= m2 . t by A5 ; ::_thesis: verum end; hence m1 = m2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def2 defines AffineMap JGRAPH_2:def_2_:_ for A, B, C, D being real number for b5 being Function of (TOP-REAL 2),(TOP-REAL 2) holds ( b5 = AffineMap (A,B,C,D) iff for t being Point of (TOP-REAL 2) holds b5 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ); registration let a, b, c, d be real number ; cluster AffineMap (a,b,c,d) -> continuous ; coherence AffineMap (a,b,c,d) is continuous proof for t being Point of (TOP-REAL 2) holds (AffineMap (a,b,c,d)) . t = |[((a * (t `1)) + b),((c * (t `2)) + d)]| by Def2; hence AffineMap (a,b,c,d) is continuous by Th43; ::_thesis: verum end; end; theorem Th44: :: JGRAPH_2:44 for A, B, C, D being real number st A > 0 & C > 0 holds AffineMap (A,B,C,D) is one-to-one proof let A, B, C, D be real number ; ::_thesis: ( A > 0 & C > 0 implies AffineMap (A,B,C,D) is one-to-one ) assume that A1: A > 0 and A2: C > 0 ; ::_thesis: AffineMap (A,B,C,D) is one-to-one set ff = AffineMap (A,B,C,D); for x1, x2 being set st x1 in dom (AffineMap (A,B,C,D)) & x2 in dom (AffineMap (A,B,C,D)) & (AffineMap (A,B,C,D)) . x1 = (AffineMap (A,B,C,D)) . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom (AffineMap (A,B,C,D)) & x2 in dom (AffineMap (A,B,C,D)) & (AffineMap (A,B,C,D)) . x1 = (AffineMap (A,B,C,D)) . x2 implies x1 = x2 ) assume that A3: x1 in dom (AffineMap (A,B,C,D)) and A4: x2 in dom (AffineMap (A,B,C,D)) and A5: (AffineMap (A,B,C,D)) . x1 = (AffineMap (A,B,C,D)) . x2 ; ::_thesis: x1 = x2 reconsider p2 = x2 as Point of (TOP-REAL 2) by A4; reconsider p1 = x1 as Point of (TOP-REAL 2) by A3; A6: ( (AffineMap (A,B,C,D)) . x1 = |[((A * (p1 `1)) + B),((C * (p1 `2)) + D)]| & (AffineMap (A,B,C,D)) . x2 = |[((A * (p2 `1)) + B),((C * (p2 `2)) + D)]| ) by Def2; then (A * (p1 `1)) + B = (A * (p2 `1)) + B by A5, SPPOL_2:1; then p1 `1 = (A * (p2 `1)) / A by A1, XCMPLX_1:89; then A7: p1 `1 = p2 `1 by A1, XCMPLX_1:89; (C * (p1 `2)) + D = (C * (p2 `2)) + D by A5, A6, SPPOL_2:1; then p1 `2 = (C * (p2 `2)) / C by A2, XCMPLX_1:89; hence x1 = x2 by A2, A7, TOPREAL3:6, XCMPLX_1:89; ::_thesis: verum end; hence AffineMap (A,B,C,D) is one-to-one by FUNCT_1:def_4; ::_thesis: verum end; theorem :: JGRAPH_2:45 for f, g being Function of I[01],(TOP-REAL 2) for a, b, c, d being real number for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & (f . O) `1 = a & (f . I) `1 = b & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & (g . I) `2 = d & a <= (g . O) `1 & (g . O) `1 <= b & a <= (g . I) `1 & (g . I) `1 <= b & a < b & c < d & ( for r being Point of I[01] holds ( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) ) & ( for r being Point of I[01] holds ( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ) holds rng f meets rng g proof defpred S1[ Point of (TOP-REAL 2)] means ( - 1 < $1 `1 & $1 `1 < 1 & - 1 < $1 `2 & $1 `2 < 1 ); reconsider K0 = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for a, b, c, d being real number for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & (f . O) `1 = a & (f . I) `1 = b & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & (g . I) `2 = d & a <= (g . O) `1 & (g . O) `1 <= b & a <= (g . I) `1 & (g . I) `1 <= b & a < b & c < d & ( for r being Point of I[01] holds ( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) ) & ( for r being Point of I[01] holds ( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ) holds rng f meets rng g let a, b, c, d be real number ; ::_thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & (f . O) `1 = a & (f . I) `1 = b & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & (g . I) `2 = d & a <= (g . O) `1 & (g . O) `1 <= b & a <= (g . I) `1 & (g . I) `1 <= b & a < b & c < d & ( for r being Point of I[01] holds ( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) ) & ( for r being Point of I[01] holds ( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ) holds rng f meets rng g let O, I be Point of I[01]; ::_thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & (f . O) `1 = a & (f . I) `1 = b & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & (g . I) `2 = d & a <= (g . O) `1 & (g . O) `1 <= b & a <= (g . I) `1 & (g . I) `1 <= b & a < b & c < d & ( for r being Point of I[01] holds ( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) ) & ( for r being Point of I[01] holds ( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ) implies rng f meets rng g ) assume that A1: ( O = 0 & I = 1 ) and A2: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one ) and A3: (f . O) `1 = a and A4: (f . I) `1 = b and A5: c <= (f . O) `2 and A6: (f . O) `2 <= d and A7: c <= (f . I) `2 and A8: (f . I) `2 <= d and A9: (g . O) `2 = c and A10: (g . I) `2 = d and A11: a <= (g . O) `1 and A12: (g . O) `1 <= b and A13: a <= (g . I) `1 and A14: (g . I) `1 <= b and A15: a < b and A16: c < d and A17: for r being Point of I[01] holds ( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) and A18: for r being Point of I[01] holds ( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ; ::_thesis: rng f meets rng g set A = 2 / (b - a); set B = 1 - ((2 * b) / (b - a)); set C = 2 / (d - c); set D = 1 - ((2 * d) / (d - c)); set ff = AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c)))); reconsider f2 = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) * f, g2 = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) * g as Function of I[01],(TOP-REAL 2) ; A19: d - c > 0 by A16, XREAL_1:50; then A20: 2 / (d - c) > 0 by XREAL_1:139; A21: dom g = the carrier of I[01] by FUNCT_2:def_1; then A22: g2 . I = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (g . I) by FUNCT_1:13 .= |[(((2 / (b - a)) * ((g . I) `1)) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * d) + (1 - ((2 * d) / (d - c))))]| by A10, Def2 ; then A23: (g2 . I) `2 = ((2 / (d - c)) * d) + (1 - ((2 * d) / (d - c))) by EUCLID:52 .= ((d * 2) / (d - c)) + (1 - ((2 * d) / (d - c))) .= 1 ; A24: g2 . O = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (g . O) by A21, FUNCT_1:13 .= |[(((2 / (b - a)) * ((g . O) `1)) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * c) + (1 - ((2 * d) / (d - c))))]| by A9, Def2 ; then A25: (g2 . O) `2 = ((2 / (d - c)) * c) + (1 - ((2 * d) / (d - c))) by EUCLID:52 .= ((c * 2) / (d - c)) + (1 - ((2 * d) / (d - c))) .= ((c * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60 .= ((c * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)) .= ((c * 2) + ((d - c) - (2 * d))) / (d - c) .= (- (d - c)) / (d - c) .= - ((d - c) / (d - c)) .= - 1 by A19, XCMPLX_1:60 ; A26: b - a > 0 by A15, XREAL_1:50; A27: ( - 1 <= (g2 . O) `1 & (g2 . O) `1 <= 1 & - 1 <= (g2 . I) `1 & (g2 . I) `1 <= 1 ) proof reconsider s1 = (g . I) `1 as Real ; reconsider s0 = (g . O) `1 as Real ; A28: (a - b) / (b - a) = (- (b - a)) / (b - a) .= - ((b - a) / (b - a)) .= - 1 by A26, XCMPLX_1:60 ; A29: (g2 . I) `1 = ((2 / (b - a)) * s1) + (1 - ((2 * b) / (b - a))) by A22, EUCLID:52 .= ((s1 * 2) / (b - a)) + (1 - ((2 * b) / (b - a))) .= ((s1 * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60 .= ((s1 * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)) .= ((s1 * 2) + ((b - a) - (2 * b))) / (b - a) .= (((s1 - b) + (s1 - b)) - (a - b)) / (b - a) ; b - b >= s0 - b by A12, XREAL_1:9; then (0 + (b - b)) - (a - b) >= ((s0 - b) + (s0 - b)) - (a - b) by XREAL_1:9; then A30: (b - a) / (b - a) >= (((s0 - b) + (s0 - b)) - (a - b)) / (b - a) by A26, XREAL_1:72; b - b >= s1 - b by A14, XREAL_1:9; then A31: (0 + (b - b)) - (a - b) >= ((s1 - b) + (s1 - b)) - (a - b) by XREAL_1:9; a - b <= s1 - b by A13, XREAL_1:9; then (a - b) + (a - b) <= (s1 - b) + (s1 - b) by XREAL_1:7; then A32: ((a - b) + (a - b)) - (a - b) <= ((s1 - b) + (s1 - b)) - (a - b) by XREAL_1:9; a - b <= s0 - b by A11, XREAL_1:9; then (a - b) + (a - b) <= (s0 - b) + (s0 - b) by XREAL_1:7; then A33: ((a - b) + (a - b)) - (a - b) <= ((s0 - b) + (s0 - b)) - (a - b) by XREAL_1:9; (g2 . O) `1 = ((2 / (b - a)) * s0) + (1 - ((2 * b) / (b - a))) by A24, EUCLID:52 .= ((s0 * 2) / (b - a)) + (1 - ((2 * b) / (b - a))) .= ((s0 * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60 .= ((s0 * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)) .= ((s0 * 2) + ((b - a) - (2 * b))) / (b - a) .= (((s0 - b) + (s0 - b)) - (a - b)) / (b - a) ; hence ( - 1 <= (g2 . O) `1 & (g2 . O) `1 <= 1 & - 1 <= (g2 . I) `1 & (g2 . I) `1 <= 1 ) by A26, A33, A28, A30, A29, A32, A31, XREAL_1:72; ::_thesis: verum end; A34: now__::_thesis:_not_rng_f2_meets_K0 assume rng f2 meets K0 ; ::_thesis: contradiction then consider x being set such that A35: x in rng f2 and A36: x in K0 by XBOOLE_0:3; reconsider q = x as Point of (TOP-REAL 2) by A35; consider p being Point of (TOP-REAL 2) such that A37: p = q and A38: - 1 < p `1 and A39: p `1 < 1 and A40: - 1 < p `2 and A41: p `2 < 1 by A36; consider z being set such that A42: z in dom f2 and A43: x = f2 . z by A35, FUNCT_1:def_3; reconsider u = z as Point of I[01] by A42; reconsider t = f . u as Point of (TOP-REAL 2) ; A44: ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) = (((t `1) * 2) / (b - a)) + (1 - ((2 * b) / (b - a))) .= (((t `1) * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60 .= (((t `1) * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)) .= (((t `1) * 2) + ((b - a) - (2 * b))) / (b - a) .= ((2 * ((t `1) - b)) - (a - b)) / (b - a) ; A45: (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . t = p by A37, A42, A43, FUNCT_1:12; A46: ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) = (((t `2) * 2) / (d - c)) + (1 - ((2 * d) / (d - c))) .= (((t `2) * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60 .= (((t `2) * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)) .= (((t `2) * 2) + ((d - c) - (2 * d))) / (d - c) .= ((2 * ((t `2) - d)) - (c - d)) / (d - c) ; A47: (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . t = |[(((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))))]| by Def2; then - 1 < ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) by A40, A45, EUCLID:52; then (- 1) * (d - c) < (((2 * ((t `2) - d)) - (c - d)) / (d - c)) * (d - c) by A19, A46, XREAL_1:68; then (- 1) * (d - c) < (2 * ((t `2) - d)) - (c - d) by A19, XCMPLX_1:87; then ((- 1) * (d - c)) + (c - d) < ((2 * ((t `2) - d)) - (c - d)) + (c - d) by XREAL_1:8; then (2 * (c - d)) / 2 < (2 * ((t `2) - d)) / 2 by XREAL_1:74; then A48: c < t `2 by XREAL_1:9; ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) < 1 by A41, A47, A45, EUCLID:52; then 1 * (d - c) > (((2 * ((t `2) - d)) - (c - d)) / (d - c)) * (d - c) by A19, A46, XREAL_1:68; then 1 * (d - c) > (2 * ((t `2) - d)) - (c - d) by A19, XCMPLX_1:87; then (1 * (d - c)) + (c - d) > ((2 * ((t `2) - d)) - (c - d)) + (c - d) by XREAL_1:8; then 0 / 2 > (((t `2) - d) * 2) / 2 ; then A49: 0 + d > t `2 by XREAL_1:19; ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) < 1 by A39, A47, A45, EUCLID:52; then 1 * (b - a) > (((2 * ((t `1) - b)) - (a - b)) / (b - a)) * (b - a) by A26, A44, XREAL_1:68; then 1 * (b - a) > (2 * ((t `1) - b)) - (a - b) by A26, XCMPLX_1:87; then (1 * (b - a)) + (a - b) > ((2 * ((t `1) - b)) - (a - b)) + (a - b) by XREAL_1:8; then 0 / 2 > (((t `1) - b) * 2) / 2 ; then A50: 0 + b > t `1 by XREAL_1:19; - 1 < ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) by A38, A47, A45, EUCLID:52; then (- 1) * (b - a) < (((2 * ((t `1) - b)) - (a - b)) / (b - a)) * (b - a) by A26, A44, XREAL_1:68; then (- 1) * (b - a) < (2 * ((t `1) - b)) - (a - b) by A26, XCMPLX_1:87; then ((- 1) * (b - a)) + (a - b) < ((2 * ((t `1) - b)) - (a - b)) + (a - b) by XREAL_1:8; then (2 * (a - b)) / 2 < (2 * ((t `1) - b)) / 2 by XREAL_1:74; then a < t `1 by XREAL_1:9; hence contradiction by A17, A50, A48, A49; ::_thesis: verum end; A51: dom f = the carrier of I[01] by FUNCT_2:def_1; then A52: f2 . I = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (f . I) by FUNCT_1:13 .= |[(((2 / (b - a)) * b) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (1 - ((2 * d) / (d - c))))]| by A4, Def2 ; then A53: (f2 . I) `1 = ((2 / (b - a)) * b) + (1 - ((2 * b) / (b - a))) by EUCLID:52 .= ((b * 2) / (b - a)) + (1 - ((2 * b) / (b - a))) .= 1 ; A54: f2 . O = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (f . O) by A51, FUNCT_1:13 .= |[(((2 / (b - a)) * a) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (1 - ((2 * d) / (d - c))))]| by A3, Def2 ; then A55: (f2 . O) `1 = ((2 / (b - a)) * a) + (1 - ((2 * b) / (b - a))) by EUCLID:52 .= ((a * 2) / (b - a)) + (1 - ((2 * b) / (b - a))) .= ((a * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60 .= ((a * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)) .= ((a * 2) + ((b - a) - (2 * b))) / (b - a) .= (- (b - a)) / (b - a) .= - ((b - a) / (b - a)) .= - 1 by A26, XCMPLX_1:60 ; A56: now__::_thesis:_not_rng_g2_meets_K0 assume rng g2 meets K0 ; ::_thesis: contradiction then consider x being set such that A57: x in rng g2 and A58: x in K0 by XBOOLE_0:3; reconsider q = x as Point of (TOP-REAL 2) by A57; consider p being Point of (TOP-REAL 2) such that A59: p = q and A60: - 1 < p `1 and A61: p `1 < 1 and A62: - 1 < p `2 and A63: p `2 < 1 by A58; consider z being set such that A64: z in dom g2 and A65: x = g2 . z by A57, FUNCT_1:def_3; reconsider u = z as Point of I[01] by A64; reconsider t = g . u as Point of (TOP-REAL 2) ; A66: ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) = (((t `1) * 2) / (b - a)) + (1 - ((2 * b) / (b - a))) .= (((t `1) * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60 .= (((t `1) * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a)) .= (((t `1) * 2) + ((b - a) - (2 * b))) / (b - a) .= ((2 * ((t `1) - b)) - (a - b)) / (b - a) ; A67: (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . t = p by A59, A64, A65, FUNCT_1:12; A68: ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) = (((t `2) * 2) / (d - c)) + (1 - ((2 * d) / (d - c))) .= (((t `2) * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60 .= (((t `2) * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)) .= (((t `2) * 2) + ((d - c) - (2 * d))) / (d - c) .= ((2 * ((t `2) - d)) - (c - d)) / (d - c) ; A69: (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . t = |[(((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))))]| by Def2; then - 1 < ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) by A62, A67, EUCLID:52; then (- 1) * (d - c) < (((2 * ((t `2) - d)) - (c - d)) / (d - c)) * (d - c) by A19, A68, XREAL_1:68; then (- 1) * (d - c) < (2 * ((t `2) - d)) - (c - d) by A19, XCMPLX_1:87; then ((- 1) * (d - c)) + (c - d) < ((2 * ((t `2) - d)) - (c - d)) + (c - d) by XREAL_1:8; then (2 * (c - d)) / 2 < (2 * ((t `2) - d)) / 2 by XREAL_1:74; then A70: c < t `2 by XREAL_1:9; ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) < 1 by A63, A69, A67, EUCLID:52; then 1 * (d - c) > (((2 * ((t `2) - d)) - (c - d)) / (d - c)) * (d - c) by A19, A68, XREAL_1:68; then 1 * (d - c) > (2 * ((t `2) - d)) - (c - d) by A19, XCMPLX_1:87; then (1 * (d - c)) + (c - d) > ((2 * ((t `2) - d)) - (c - d)) + (c - d) by XREAL_1:8; then 0 / 2 > (((t `2) - d) * 2) / 2 ; then A71: 0 + d > t `2 by XREAL_1:19; ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) < 1 by A61, A69, A67, EUCLID:52; then 1 * (b - a) > (((2 * ((t `1) - b)) - (a - b)) / (b - a)) * (b - a) by A26, A66, XREAL_1:68; then 1 * (b - a) > (2 * ((t `1) - b)) - (a - b) by A26, XCMPLX_1:87; then (1 * (b - a)) + (a - b) > ((2 * ((t `1) - b)) - (a - b)) + (a - b) by XREAL_1:8; then 0 / 2 > (((t `1) - b) * 2) / 2 ; then A72: 0 + b > t `1 by XREAL_1:19; - 1 < ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) by A60, A69, A67, EUCLID:52; then (- 1) * (b - a) < (((2 * ((t `1) - b)) - (a - b)) / (b - a)) * (b - a) by A26, A66, XREAL_1:68; then (- 1) * (b - a) < (2 * ((t `1) - b)) - (a - b) by A26, XCMPLX_1:87; then ((- 1) * (b - a)) + (a - b) < ((2 * ((t `1) - b)) - (a - b)) + (a - b) by XREAL_1:8; then (2 * (a - b)) / 2 < (2 * ((t `1) - b)) / 2 by XREAL_1:74; then a < t `1 by XREAL_1:9; hence contradiction by A18, A72, A70, A71; ::_thesis: verum end; A73: ( - 1 <= (f2 . O) `2 & (f2 . O) `2 <= 1 & - 1 <= (f2 . I) `2 & (f2 . I) `2 <= 1 ) proof reconsider s1 = (f . I) `2 as Real ; reconsider s0 = (f . O) `2 as Real ; A74: (c - d) / (d - c) = (- (d - c)) / (d - c) .= - ((d - c) / (d - c)) .= - 1 by A19, XCMPLX_1:60 ; A75: (f2 . I) `2 = ((2 / (d - c)) * s1) + (1 - ((2 * d) / (d - c))) by A52, EUCLID:52 .= ((s1 * 2) / (d - c)) + (1 - ((2 * d) / (d - c))) .= ((s1 * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60 .= ((s1 * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)) .= ((s1 * 2) + ((d - c) - (2 * d))) / (d - c) .= (((s1 - d) + (s1 - d)) - (c - d)) / (d - c) ; d - d >= s0 - d by A6, XREAL_1:9; then (0 + (d - d)) - (c - d) >= ((s0 - d) + (s0 - d)) - (c - d) by XREAL_1:9; then A76: (d - c) / (d - c) >= (((s0 - d) + (s0 - d)) - (c - d)) / (d - c) by A19, XREAL_1:72; d - d >= s1 - d by A8, XREAL_1:9; then A77: (0 + (d - d)) - (c - d) >= ((s1 - d) + (s1 - d)) - (c - d) by XREAL_1:9; c - d <= s1 - d by A7, XREAL_1:9; then (c - d) + (c - d) <= (s1 - d) + (s1 - d) by XREAL_1:7; then A78: ((c - d) + (c - d)) - (c - d) <= ((s1 - d) + (s1 - d)) - (c - d) by XREAL_1:9; c - d <= s0 - d by A5, XREAL_1:9; then (c - d) + (c - d) <= (s0 - d) + (s0 - d) by XREAL_1:7; then A79: ((c - d) + (c - d)) - (c - d) <= ((s0 - d) + (s0 - d)) - (c - d) by XREAL_1:9; (f2 . O) `2 = ((2 / (d - c)) * s0) + (1 - ((2 * d) / (d - c))) by A54, EUCLID:52 .= ((s0 * 2) / (d - c)) + (1 - ((2 * d) / (d - c))) .= ((s0 * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60 .= ((s0 * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c)) .= ((s0 * 2) + ((d - c) - (2 * d))) / (d - c) .= (((s0 - d) + (s0 - d)) - (c - d)) / (d - c) ; hence ( - 1 <= (f2 . O) `2 & (f2 . O) `2 <= 1 & - 1 <= (f2 . I) `2 & (f2 . I) `2 <= 1 ) by A19, A79, A74, A76, A75, A78, A77, XREAL_1:72; ::_thesis: verum end; set y = the Element of (rng f2) /\ (rng g2); 2 / (b - a) > 0 by A26, XREAL_1:139; then A80: AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c)))) is one-to-one by A20, Th44; then ( f2 is one-to-one & g2 is one-to-one ) by A2, FUNCT_1:24; then rng f2 meets rng g2 by A1, A2, A55, A53, A25, A23, A73, A27, A34, A56, Th42; then A81: (rng f2) /\ (rng g2) <> {} by XBOOLE_0:def_7; then the Element of (rng f2) /\ (rng g2) in rng f2 by XBOOLE_0:def_4; then consider x being set such that A82: x in dom f2 and A83: the Element of (rng f2) /\ (rng g2) = f2 . x by FUNCT_1:def_3; dom f2 c= dom f by RELAT_1:25; then A84: f . x in rng f by A82, FUNCT_1:3; the Element of (rng f2) /\ (rng g2) in rng g2 by A81, XBOOLE_0:def_4; then consider x2 being set such that A85: x2 in dom g2 and A86: the Element of (rng f2) /\ (rng g2) = g2 . x2 by FUNCT_1:def_3; A87: the Element of (rng f2) /\ (rng g2) = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (g . x2) by A85, A86, FUNCT_1:12; dom g2 c= dom g by RELAT_1:25; then A88: g . x2 in rng g by A85, FUNCT_1:3; ( dom (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) = the carrier of (TOP-REAL 2) & the Element of (rng f2) /\ (rng g2) = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (f . x) ) by A82, A83, FUNCT_1:12, FUNCT_2:def_1; then f . x = g . x2 by A80, A87, A84, A88, FUNCT_1:def_4; then (rng f) /\ (rng g) <> {} by A84, A88, XBOOLE_0:def_4; hence rng f meets rng g by XBOOLE_0:def_7; ::_thesis: verum end; theorem :: JGRAPH_2:46 ( { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } is closed Subset of (TOP-REAL 2) & { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } is closed Subset of (TOP-REAL 2) ) by Lm5, Lm8; theorem :: JGRAPH_2:47 ( { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1) <= p7 `2 } is closed Subset of (TOP-REAL 2) & { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1) } is closed Subset of (TOP-REAL 2) ) by Lm11, Lm14; theorem :: JGRAPH_2:48 ( { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `2) <= p7 `1 } is closed Subset of (TOP-REAL 2) & { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= - (p7 `2) } is closed Subset of (TOP-REAL 2) ) by Lm17, Lm20;