:: NFCONT_3 semantic presentation begin theorem Th1: :: NFCONT_3:1 for n being Element of NAT for S being RealNormSpace for seq being Real_Sequence for h being PartFunc of REAL, the carrier of S st rng seq c= dom h holds seq . n in dom h proof let n be Element of NAT ; ::_thesis: for S being RealNormSpace for seq being Real_Sequence for h being PartFunc of REAL, the carrier of S st rng seq c= dom h holds seq . n in dom h let S be RealNormSpace; ::_thesis: for seq being Real_Sequence for h being PartFunc of REAL, the carrier of S st rng seq c= dom h holds seq . n in dom h let seq be Real_Sequence; ::_thesis: for h being PartFunc of REAL, the carrier of S st rng seq c= dom h holds seq . n in dom h let h be PartFunc of REAL, the carrier of S; ::_thesis: ( rng seq c= dom h implies seq . n in dom h ) n in NAT ; then A1: n in dom seq by FUNCT_2:def_1; assume rng seq c= dom h ; ::_thesis: seq . n in dom h then n in dom (h * seq) by A1, RELAT_1:27; hence seq . n in dom h by FUNCT_1:11; ::_thesis: verum end; theorem Th2: :: NFCONT_3:2 for S being RealNormSpace for h1, h2 being PartFunc of REAL, the carrier of S for seq being Real_Sequence st rng seq c= (dom h1) /\ (dom h2) holds ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) proof let S be RealNormSpace; ::_thesis: for h1, h2 being PartFunc of REAL, the carrier of S for seq being Real_Sequence st rng seq c= (dom h1) /\ (dom h2) holds ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) let h1, h2 be PartFunc of REAL, the carrier of S; ::_thesis: for seq being Real_Sequence st rng seq c= (dom h1) /\ (dom h2) holds ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) let seq be Real_Sequence; ::_thesis: ( rng seq c= (dom h1) /\ (dom h2) implies ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) ) A1: ( (dom h1) /\ (dom h2) c= dom h1 & (dom h1) /\ (dom h2) c= dom h2 ) by XBOOLE_1:17; assume A2: rng seq c= (dom h1) /\ (dom h2) ; ::_thesis: ( (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) & (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) ) now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_+_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_+_((h2_/*_seq)_._n) let n be Element of NAT ; ::_thesis: ((h1 + h2) /* seq) . n = ((h1 /* seq) . n) + ((h2 /* seq) . n) A3: ( h1 /. (seq . n) = (h1 /* seq) . n & h2 /. (seq . n) = (h2 /* seq) . n ) by A1, A2, FUNCT_2:109, XBOOLE_1:1; A4: rng seq c= dom (h1 + h2) by A2, VFUNCT_1:def_1; then A5: seq . n in dom (h1 + h2) by Th1; thus ((h1 + h2) /* seq) . n = (h1 + h2) /. (seq . n) by A4, FUNCT_2:109 .= ((h1 /* seq) . n) + ((h2 /* seq) . n) by A3, A5, VFUNCT_1:def_1 ; ::_thesis: verum end; hence (h1 + h2) /* seq = (h1 /* seq) + (h2 /* seq) by NORMSP_1:def_2; ::_thesis: (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) now__::_thesis:_for_n_being_Element_of_NAT_holds_((h1_-_h2)_/*_seq)_._n_=_((h1_/*_seq)_._n)_-_((h2_/*_seq)_._n) let n be Element of NAT ; ::_thesis: ((h1 - h2) /* seq) . n = ((h1 /* seq) . n) - ((h2 /* seq) . n) A6: ( h1 /. (seq . n) = (h1 /* seq) . n & h2 /. (seq . n) = (h2 /* seq) . n ) by A1, A2, FUNCT_2:109, XBOOLE_1:1; A7: rng seq c= dom (h1 - h2) by A2, VFUNCT_1:def_2; then A8: seq . n in dom (h1 - h2) by Th1; thus ((h1 - h2) /* seq) . n = (h1 - h2) /. (seq . n) by A7, FUNCT_2:109 .= ((h1 /* seq) . n) - ((h2 /* seq) . n) by A6, A8, VFUNCT_1:def_2 ; ::_thesis: verum end; hence (h1 - h2) /* seq = (h1 /* seq) - (h2 /* seq) by NORMSP_1:def_3; ::_thesis: verum end; theorem :: NFCONT_3:3 for S being RealNormSpace for h being sequence of S for r being Real holds r (#) h = r * h proof let S be RealNormSpace; ::_thesis: for h being sequence of S for r being Real holds r (#) h = r * h let h be sequence of S; ::_thesis: for r being Real holds r (#) h = r * h let r be Real; ::_thesis: r (#) h = r * h dom h = NAT by FUNCT_2:def_1; then A1: dom (r (#) h) = NAT by VFUNCT_1:def_4; now__::_thesis:_for_n_being_Element_of_NAT_holds_(r_(#)_h)_._n_=_r_*_(h_._n) let n be Element of NAT ; ::_thesis: (r (#) h) . n = r * (h . n) (r (#) h) . n = (r (#) h) /. n ; then (r (#) h) . n = r * (h /. n) by A1, VFUNCT_1:def_4; hence (r (#) h) . n = r * (h . n) ; ::_thesis: verum end; hence r (#) h = r * h by NORMSP_1:def_5; ::_thesis: verum end; theorem Th4: :: NFCONT_3:4 for S being RealNormSpace for h being PartFunc of REAL, the carrier of S for seq being Real_Sequence for r being Real st rng seq c= dom h holds (r (#) h) /* seq = r * (h /* seq) proof let S be RealNormSpace; ::_thesis: for h being PartFunc of REAL, the carrier of S for seq being Real_Sequence for r being Real st rng seq c= dom h holds (r (#) h) /* seq = r * (h /* seq) let h be PartFunc of REAL, the carrier of S; ::_thesis: for seq being Real_Sequence for r being Real st rng seq c= dom h holds (r (#) h) /* seq = r * (h /* seq) let seq be Real_Sequence; ::_thesis: for r being Real st rng seq c= dom h holds (r (#) h) /* seq = r * (h /* seq) let r be Real; ::_thesis: ( rng seq c= dom h implies (r (#) h) /* seq = r * (h /* seq) ) assume A1: rng seq c= dom h ; ::_thesis: (r (#) h) /* seq = r * (h /* seq) then A2: rng seq c= dom (r (#) h) by VFUNCT_1:def_4; now__::_thesis:_for_n_being_Element_of_NAT_holds_((r_(#)_h)_/*_seq)_._n_=_r_*_((h_/*_seq)_._n) let n be Element of NAT ; ::_thesis: ((r (#) h) /* seq) . n = r * ((h /* seq) . n) A3: seq . n in dom (r (#) h) by A2, Th1; thus ((r (#) h) /* seq) . n = (r (#) h) /. (seq . n) by A2, FUNCT_2:109 .= r * (h /. (seq . n)) by A3, VFUNCT_1:def_4 .= r * ((h /* seq) . n) by A1, FUNCT_2:109 ; ::_thesis: verum end; hence (r (#) h) /* seq = r * (h /* seq) by NORMSP_1:def_5; ::_thesis: verum end; theorem Th5: :: NFCONT_3:5 for S being RealNormSpace for h being PartFunc of REAL, the carrier of S for seq being Real_Sequence st rng seq c= dom h holds ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq ) proof let S be RealNormSpace; ::_thesis: for h being PartFunc of REAL, the carrier of S for seq being Real_Sequence st rng seq c= dom h holds ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq ) let h be PartFunc of REAL, the carrier of S; ::_thesis: for seq being Real_Sequence st rng seq c= dom h holds ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq ) let seq be Real_Sequence; ::_thesis: ( rng seq c= dom h implies ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq ) ) assume A1: rng seq c= dom h ; ::_thesis: ( ||.(h /* seq).|| = ||.h.|| /* seq & - (h /* seq) = (- h) /* seq ) then A2: rng seq c= dom ||.h.|| by NORMSP_0:def_3; now__::_thesis:_for_n_being_Element_of_NAT_holds_||.(h_/*_seq).||_._n_=_(||.h.||_/*_seq)_._n let n be Element of NAT ; ::_thesis: ||.(h /* seq).|| . n = (||.h.|| /* seq) . n seq . n in rng seq by FUNCT_2:4; then seq . n in dom h by A1; then A3: seq . n in dom ||.h.|| by NORMSP_0:def_3; thus ||.(h /* seq).|| . n = ||.((h /* seq) . n).|| by NORMSP_0:def_4 .= ||.(h /. (seq . n)).|| by A1, FUNCT_2:109 .= ||.h.|| . (seq . n) by A3, NORMSP_0:def_3 .= ||.h.|| /. (seq . n) by A3, PARTFUN1:def_6 .= (||.h.|| /* seq) . n by A2, FUNCT_2:109 ; ::_thesis: verum end; hence ||.(h /* seq).|| = ||.h.|| /* seq by FUNCT_2:63; ::_thesis: - (h /* seq) = (- h) /* seq now__::_thesis:_for_n_being_Element_of_NAT_holds_(-_(h_/*_seq))_._n_=_((-_h)_/*_seq)_._n let n be Element of NAT ; ::_thesis: (- (h /* seq)) . n = ((- h) /* seq) . n thus (- (h /* seq)) . n = - ((h /* seq) . n) by BHSP_1:44 .= (- 1) * ((h /* seq) . n) by RLVECT_1:16 .= ((- 1) * (h /* seq)) . n by NORMSP_1:def_5 .= (((- 1) (#) h) /* seq) . n by A1, Th4 .= ((- h) /* seq) . n by VFUNCT_1:23 ; ::_thesis: verum end; hence - (h /* seq) = (- h) /* seq by FUNCT_2:63; ::_thesis: verum end; begin definition let S be RealNormSpace; let f be PartFunc of REAL, the carrier of S; let x0 be real number ; predf is_continuous_in x0 means :Def1: :: NFCONT_3:def 1 ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ); end; :: deftheorem Def1 defines is_continuous_in NFCONT_3:def_1_:_ for S being RealNormSpace for f being PartFunc of REAL, the carrier of S for x0 being real number holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) ); theorem Th6: :: NFCONT_3:6 for X being set for x0 being real number for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st x0 in X & f is_continuous_in x0 holds f | X is_continuous_in x0 proof let X be set ; ::_thesis: for x0 being real number for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st x0 in X & f is_continuous_in x0 holds f | X is_continuous_in x0 let x0 be real number ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st x0 in X & f is_continuous_in x0 holds f | X is_continuous_in x0 let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st x0 in X & f is_continuous_in x0 holds f | X is_continuous_in x0 let f be PartFunc of REAL, the carrier of S; ::_thesis: ( x0 in X & f is_continuous_in x0 implies f | X is_continuous_in x0 ) assume that A1: x0 in X and A2: f is_continuous_in x0 ; ::_thesis: f | X is_continuous_in x0 A3: x0 in dom f by Def1, A2; A4: dom (f | X) = X /\ (dom f) by RELAT_1:61; hence A5: x0 in dom (f | X) by A1, A3, XBOOLE_0:def_4; :: according to NFCONT_3:def_1 ::_thesis: for s1 being Real_Sequence st rng s1 c= dom (f | X) & s1 is convergent & lim s1 = x0 holds ( (f | X) /* s1 is convergent & (f | X) /. x0 = lim ((f | X) /* s1) ) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom (f | X) & s1 is convergent & lim s1 = x0 implies ( (f | X) /* s1 is convergent & (f | X) /. x0 = lim ((f | X) /* s1) ) ) assume that A6: rng s1 c= dom (f | X) and A7: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f | X) /* s1 is convergent & (f | X) /. x0 = lim ((f | X) /* s1) ) A8: rng s1 c= dom f by A6, A4, XBOOLE_1:18; A9: (f | X) /* s1 = f /* s1 by A6, FUNCT_2:117; hence (f | X) /* s1 is convergent by A2, A7, A8, Def1; ::_thesis: (f | X) /. x0 = lim ((f | X) /* s1) x0 in REAL by XREAL_0:def_1; hence (f | X) /. x0 = f /. x0 by A5, PARTFUN2:15 .= lim ((f | X) /* s1) by A2, A7, A8, A9, Def1 ; ::_thesis: verum end; theorem :: NFCONT_3:7 for x0 being real number for S being RealNormSpace for f being PartFunc of REAL, the carrier of S holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) ) proof let x0 be real number ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) ) let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) ) let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) ) thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) ) by Def1; ::_thesis: ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) implies f is_continuous_in x0 ) assume A1: ( x0 in dom f & ( for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 & ( for n being Element of NAT holds s1 . n <> x0 ) holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) ) ; ::_thesis: f is_continuous_in x0 thus x0 in dom f by A1; :: according to NFCONT_3:def_1 ::_thesis: for s1 being Real_Sequence st rng s1 c= dom f & s1 is convergent & lim s1 = x0 holds ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) let s2 be Real_Sequence; ::_thesis: ( rng s2 c= dom f & s2 is convergent & lim s2 = x0 implies ( f /* s2 is convergent & f /. x0 = lim (f /* s2) ) ) assume that A2: rng s2 c= dom f and A3: ( s2 is convergent & lim s2 = x0 ) ; ::_thesis: ( f /* s2 is convergent & f /. x0 = lim (f /* s2) ) percases ( ex n being Element of NAT st for m being Element of NAT st n <= m holds s2 . m = x0 or for n being Element of NAT ex m being Element of NAT st ( n <= m & s2 . m <> x0 ) ) ; suppose ex n being Element of NAT st for m being Element of NAT st n <= m holds s2 . m = x0 ; ::_thesis: ( f /* s2 is convergent & f /. x0 = lim (f /* s2) ) then consider N being Element of NAT such that A4: for m being Element of NAT st N <= m holds s2 . m = x0 ; A5: f /* (s2 ^\ N) = (f /* s2) ^\ N by A2, VALUED_0:27; A6: now__::_thesis:_for_p_being_Real_st_p_>_0_holds_ ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ ||.(((f_/*_(s2_^\_N))_._m)_-_(f_/._x0)).||_<_p let p be Real; ::_thesis: ( p > 0 implies ex n being Element of NAT st for m being Element of NAT st n <= m holds ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p ) assume A7: p > 0 ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p take n = 0 ; ::_thesis: for m being Element of NAT st n <= m holds ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p let m be Element of NAT ; ::_thesis: ( n <= m implies ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p ) assume n <= m ; ::_thesis: ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p A8: s2 . (m + N) = x0 by A4, NAT_1:12; rng (s2 ^\ N) c= rng s2 by VALUED_0:21; then ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| = ||.((f /. ((s2 ^\ N) . m)) - (f /. x0)).|| by A2, FUNCT_2:109, XBOOLE_1:1 .= ||.((f /. x0) - (f /. x0)).|| by A8, NAT_1:def_3 .= 0 by NORMSP_1:6 ; hence ||.(((f /* (s2 ^\ N)) . m) - (f /. x0)).|| < p by A7; ::_thesis: verum end; then A9: f /* (s2 ^\ N) is convergent by NORMSP_1:def_6; then f /. x0 = lim ((f /* s2) ^\ N) by A6, A5, NORMSP_1:def_7; hence ( f /* s2 is convergent & f /. x0 = lim (f /* s2) ) by A9, A5, LOPBAN_3:10, LOPBAN_3:11; ::_thesis: verum end; supposeA10: for n being Element of NAT ex m being Element of NAT st ( n <= m & s2 . m <> x0 ) ; ::_thesis: ( f /* s2 is convergent & f /. x0 = lim (f /* s2) ) defpred S1[ Element of NAT , set , set ] means for n, m being Element of NAT st \$2 = n & \$3 = m holds ( n < m & s2 . m <> x0 & ( for k being Element of NAT st n < k & s2 . k <> x0 holds m <= k ) ); defpred S2[ set ] means s2 . \$1 <> x0; ex m1 being Element of NAT st ( 0 <= m1 & s2 . m1 <> x0 ) by A10; then A11: ex m being Nat st S2[m] ; consider M being Nat such that A12: ( S2[M] & ( for n being Nat st S2[n] holds M <= n ) ) from NAT_1:sch_5(A11); reconsider M9 = M as Element of NAT by ORDINAL1:def_12; A13: now__::_thesis:_for_n_being_Element_of_NAT_ex_m_being_Element_of_NAT_st_ (_n_<_m_&_s2_._m_<>_x0_) let n be Element of NAT ; ::_thesis: ex m being Element of NAT st ( n < m & s2 . m <> x0 ) consider m being Element of NAT such that A14: ( n + 1 <= m & s2 . m <> x0 ) by A10; take m = m; ::_thesis: ( n < m & s2 . m <> x0 ) thus ( n < m & s2 . m <> x0 ) by A14, NAT_1:13; ::_thesis: verum end; A15: for n, x being Element of NAT ex y being Element of NAT st S1[n,x,y] proof let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S1[n,x,y] defpred S3[ Nat] means ( x < \$1 & s2 . \$1 <> x0 ); ex m being Element of NAT st S3[m] by A13; then A16: ex m being Nat st S3[m] ; consider l being Nat such that A17: ( S3[l] & ( for k being Nat st S3[k] holds l <= k ) ) from NAT_1:sch_5(A16); take l ; ::_thesis: ( l is Element of REAL & l is Element of NAT & S1[n,x,l] ) l in NAT by ORDINAL1:def_12; hence ( l is Element of REAL & l is Element of NAT & S1[n,x,l] ) by A17; ::_thesis: verum end; consider F being Function of NAT,NAT such that A18: ( F . 0 = M9 & ( for n being Element of NAT holds S1[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch_2(A15); A19: ( dom F = NAT & rng F c= REAL ) by FUNCT_2:def_1, XBOOLE_1:1; then reconsider F = F as Real_Sequence by RELSET_1:4; A20: now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_is_Element_of_NAT let n be Element of NAT ; ::_thesis: F . n is Element of NAT ( F . n in rng F & rng F c= NAT ) by A19, FUNCT_1:def_3; hence F . n is Element of NAT ; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_holds_F_._n_<_F_._(n_+_1) let n be Element of NAT ; ::_thesis: F . n < F . (n + 1) ( F . n is Element of NAT & F . (n + 1) is Element of NAT ) by A20; hence F . n < F . (n + 1) by A18; ::_thesis: verum end; then reconsider F = F as increasing sequence of NAT by SEQM_3:def_6; A21: for n being Element of NAT st s2 . n <> x0 holds ex m being Element of NAT st F . m = n proof defpred S3[ set ] means ( s2 . \$1 <> x0 & ( for m being Element of NAT holds F . m <> \$1 ) ); assume ex n being Element of NAT st S3[n] ; ::_thesis: contradiction then A22: ex n being Nat st S3[n] ; consider M1 being Nat such that A23: ( S3[M1] & ( for n being Nat st S3[n] holds M1 <= n ) ) from NAT_1:sch_5(A22); reconsider M1 = M1 as Element of NAT by ORDINAL1:def_12; defpred S4[ Nat] means ( \$1 < M1 & s2 . \$1 <> x0 & ex m being Element of NAT st F . m = \$1 ); A24: ex n being Nat st S4[n] proof take M ; ::_thesis: S4[M] ( M <= M1 & M <> M1 ) by A12, A18, A23; hence M < M1 by XXREAL_0:1; ::_thesis: ( s2 . M <> x0 & ex m being Element of NAT st F . m = M ) thus s2 . M <> x0 by A12; ::_thesis: ex m being Element of NAT st F . m = M take 0 ; ::_thesis: F . 0 = M thus F . 0 = M by A18; ::_thesis: verum end; A25: for n being Nat st S4[n] holds n <= M1 ; consider MX being Nat such that A26: ( S4[MX] & ( for n being Nat st S4[n] holds n <= MX ) ) from NAT_1:sch_6(A25, A24); consider m being Element of NAT such that A27: F . m = MX by A26; A28: ( F . m is Element of NAT & F . (m + 1) is Element of NAT ) by A20; then A29: ( MX < F . (m + 1) & s2 . (F . (m + 1)) <> x0 ) by A18, A27; ( F . (m + 1) <> M1 & F . (m + 1) <= M1 ) by A18, A23, A26, A27, A28; then F . (m + 1) < M1 by XXREAL_0:1; hence contradiction by A26, A29; ::_thesis: verum end; A30: for n being Element of NAT holds (s2 * F) . n <> x0 proof defpred S3[ Element of NAT ] means (s2 * F) . \$1 <> x0; A31: for k being Element of NAT st S3[k] holds S3[k + 1] proof let k be Element of NAT ; ::_thesis: ( S3[k] implies S3[k + 1] ) assume (s2 * F) . k <> x0 ; ::_thesis: S3[k + 1] ( F . k is Element of NAT & F . (k + 1) is Element of NAT ) by A20; then s2 . (F . (k + 1)) <> x0 by A18; hence S3[k + 1] by FUNCT_2:15; ::_thesis: verum end; A32: S3[ 0 ] by A12, A18, FUNCT_2:15; thus for n being Element of NAT holds S3[n] from NAT_1:sch_1(A32, A31); ::_thesis: verum end; A33: ( s2 * F is convergent & lim (s2 * F) = x0 ) by A3, SEQ_4:16, SEQ_4:17; A34: rng (s2 * F) c= rng s2 by VALUED_0:21; then rng (s2 * F) c= dom f by A2, XBOOLE_1:1; then A35: ( f /* (s2 * F) is convergent & f /. x0 = lim (f /* (s2 * F)) ) by A1, A30, A33; A36: now__::_thesis:_for_p_being_Real_st_0_<_p_holds_ ex_k_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_k_<=_m_holds_ ||.(((f_/*_s2)_._m)_-_(f_/._x0)).||_<_p let p be Real; ::_thesis: ( 0 < p implies ex k being Element of NAT st for m being Element of NAT st k <= m holds ||.(((f /* s2) . b5) - (f /. x0)).|| < b3 ) assume A37: 0 < p ; ::_thesis: ex k being Element of NAT st for m being Element of NAT st k <= m holds ||.(((f /* s2) . b5) - (f /. x0)).|| < b3 then consider n being Element of NAT such that A38: for m being Element of NAT st n <= m holds ||.(((f /* (s2 * F)) . m) - (f /. x0)).|| < p by A35, NORMSP_1:def_7; reconsider k = F . n as Element of NAT by A20; take k = k; ::_thesis: for m being Element of NAT st k <= m holds ||.(((f /* s2) . b4) - (f /. x0)).|| < b2 let m be Element of NAT ; ::_thesis: ( k <= m implies ||.(((f /* s2) . b3) - (f /. x0)).|| < b1 ) assume A39: k <= m ; ::_thesis: ||.(((f /* s2) . b3) - (f /. x0)).|| < b1 percases ( s2 . m = x0 or s2 . m <> x0 ) ; suppose s2 . m = x0 ; ::_thesis: ||.(((f /* s2) . b3) - (f /. x0)).|| < b1 then ||.(((f /* s2) . m) - (f /. x0)).|| = ||.((f /. x0) - (f /. x0)).|| by A2, FUNCT_2:109 .= 0 by NORMSP_1:6 ; hence ||.(((f /* s2) . m) - (f /. x0)).|| < p by A37; ::_thesis: verum end; suppose s2 . m <> x0 ; ::_thesis: ||.(((f /* s2) . b3) - (f /. x0)).|| < b1 then consider l being Element of NAT such that A40: m = F . l by A21; n <= l by A39, A40, SEQM_3:1; then ||.(((f /* (s2 * F)) . l) - (f /. x0)).|| < p by A38; then ||.((f /. ((s2 * F) . l)) - (f /. x0)).|| < p by A2, A34, FUNCT_2:109, XBOOLE_1:1; then ||.((f /. (s2 . m)) - (f /. x0)).|| < p by A40, FUNCT_2:15; hence ||.(((f /* s2) . m) - (f /. x0)).|| < p by A2, FUNCT_2:109; ::_thesis: verum end; end; end; hence f /* s2 is convergent by NORMSP_1:def_6; ::_thesis: f /. x0 = lim (f /* s2) hence f /. x0 = lim (f /* s2) by A36, NORMSP_1:def_7; ::_thesis: verum end; end; end; theorem Th8: :: NFCONT_3:8 for x0 being real number for S being RealNormSpace for f being PartFunc of REAL, the carrier of S holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) proof let x0 be real number ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ) ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f is_continuous_in x0 ) proof assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) hence x0 in dom f by Def1; ::_thesis: for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) given r being Real such that A2: 0 < r and A3: for s being real number holds ( not 0 < s or ex x1 being real number st ( x1 in dom f & abs (x1 - x0) < s & not ||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: contradiction defpred S1[ Element of NAT , real number ] means ( \$2 in dom f & abs (\$2 - x0) < 1 / (\$1 + 1) & not ||.((f /. \$2) - (f /. x0)).|| < r ); A4: for n being Element of NAT ex p being Real st S1[n,p] proof let n be Element of NAT ; ::_thesis: ex p being Real st S1[n,p] 0 < (n + 1) " ; then 0 < 1 / (n + 1) by XCMPLX_1:215; then consider x1 being real number such that A5: ( x1 in dom f & abs (x1 - x0) < 1 / (n + 1) & not ||.((f /. x1) - (f /. x0)).|| < r ) by A3; take x1 ; ::_thesis: ( x1 is Real & S1[n,x1] ) thus ( x1 is Real & S1[n,x1] ) by A5; ::_thesis: verum end; consider s1 being Real_Sequence such that A6: for n being Element of NAT holds S1[n,s1 . n] from FUNCT_2:sch_3(A4); now__::_thesis:_for_x_being_set_st_x_in_rng_s1_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng s1 implies x in dom f ) assume x in rng s1 ; ::_thesis: x in dom f then ex n being Element of NAT st x = s1 . n by FUNCT_2:113; hence x in dom f by A6; ::_thesis: verum end; then A7: rng s1 c= dom f by TARSKI:def_3; A8: now__::_thesis:_for_s_being_real_number_st_0_<_s_holds_ ex_k_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_k_<=_m_holds_ abs_((s1_._m)_-_x0)_<_s let s be real number ; ::_thesis: ( 0 < s implies ex k being Element of NAT st for m being Element of NAT st k <= m holds abs ((s1 . m) - x0) < s ) assume A9: 0 < s ; ::_thesis: ex k being Element of NAT st for m being Element of NAT st k <= m holds abs ((s1 . m) - x0) < s consider n being Element of NAT such that A10: s " < n by SEQ_4:3; (s ") + 0 < n + 1 by A10, XREAL_1:8; then 1 / (n + 1) < 1 / (s ") by A9, XREAL_1:76; then A11: 1 / (n + 1) < s by XCMPLX_1:216; take k = n; ::_thesis: for m being Element of NAT st k <= m holds abs ((s1 . m) - x0) < s let m be Element of NAT ; ::_thesis: ( k <= m implies abs ((s1 . m) - x0) < s ) assume k <= m ; ::_thesis: abs ((s1 . m) - x0) < s then k + 1 <= m + 1 by XREAL_1:6; then 1 / (m + 1) <= 1 / (k + 1) by XREAL_1:118; then 1 / (m + 1) < s by A11, XXREAL_0:2; hence abs ((s1 . m) - x0) < s by A6, XXREAL_0:2; ::_thesis: verum end; then A12: s1 is convergent by SEQ_2:def_6; then lim s1 = x0 by A8, SEQ_2:def_7; then ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A1, A7, A12, Def1; then consider n being Element of NAT such that A13: for m being Element of NAT st n <= m holds ||.(((f /* s1) . m) - (f /. x0)).|| < r by A2, NORMSP_1:def_7; ||.(((f /* s1) . n) - (f /. x0)).|| < r by A13; then ||.((f /. (s1 . n)) - (f /. x0)).|| < r by A7, FUNCT_2:109; hence contradiction by A6; ::_thesis: verum end; assume A14: ( x0 in dom f & ( for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) ; ::_thesis: f is_continuous_in x0 now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_=_x0_holds_ (_f_/*_s1_is_convergent_&_f_/._x0_=_lim_(f_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 = x0 implies ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) ) assume that A15: rng s1 c= dom f and A16: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) A17: now__::_thesis:_for_p_being_Real_st_0_<_p_holds_ ex_k_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_k_<=_m_holds_ ||.(((f_/*_s1)_._m)_-_(f_/._x0)).||_<_p let p be Real; ::_thesis: ( 0 < p implies ex k being Element of NAT st for m being Element of NAT st k <= m holds ||.(((f /* s1) . m) - (f /. x0)).|| < p ) assume 0 < p ; ::_thesis: ex k being Element of NAT st for m being Element of NAT st k <= m holds ||.(((f /* s1) . m) - (f /. x0)).|| < p then consider s being real number such that A18: 0 < s and A19: for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < p by A14; consider n being Element of NAT such that A20: for m being Element of NAT st n <= m holds abs ((s1 . m) - x0) < s by A16, A18, SEQ_2:def_7; take k = n; ::_thesis: for m being Element of NAT st k <= m holds ||.(((f /* s1) . m) - (f /. x0)).|| < p let m be Element of NAT ; ::_thesis: ( k <= m implies ||.(((f /* s1) . m) - (f /. x0)).|| < p ) assume k <= m ; ::_thesis: ||.(((f /* s1) . m) - (f /. x0)).|| < p then ( s1 . m in rng s1 & abs ((s1 . m) - x0) < s ) by A20, VALUED_0:28; then ||.((f /. (s1 . m)) - (f /. x0)).|| < p by A15, A19; hence ||.(((f /* s1) . m) - (f /. x0)).|| < p by A15, FUNCT_2:109; ::_thesis: verum end; then f /* s1 is convergent by NORMSP_1:def_6; hence ( f /* s1 is convergent & f /. x0 = lim (f /* s1) ) by A17, NORMSP_1:def_7; ::_thesis: verum end; hence f is_continuous_in x0 by A14, Def1; ::_thesis: verum end; theorem Th9: :: NFCONT_3:9 for S being RealNormSpace for f being PartFunc of REAL, the carrier of S for x0 being real number holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 ) ) ) proof let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S for x0 being real number holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 ) ) ) let f be PartFunc of REAL, the carrier of S; ::_thesis: for x0 being real number holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 ) ) ) let x0 be real number ; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 ) ) ) thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 ) implies f is_continuous_in x0 ) proof assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 ) ) hence x0 in dom f by Def1; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 let N01 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N01 consider r being Real such that A2: 0 < r and A3: { p where p is Point of S : ||.(p - (f /. x0)).|| < r } c= N01 by NFCONT_1:def_1; set N1 = { p where p is Point of S : ||.(p - (f /. x0)).|| < r } ; consider s being real number such that A4: 0 < s and A5: for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r by A1, A2, Th8; reconsider N = ].(x0 - s),(x0 + s).[ as Neighbourhood of x0 by A4, RCOMP_1:def_6; take N ; ::_thesis: for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N01 let x1 be real number ; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N01 ) assume that A6: x1 in dom f and A7: x1 in N ; ::_thesis: f /. x1 in N01 abs (x1 - x0) < s by A7, RCOMP_1:1; then ||.((f /. x1) - (f /. x0)).|| < r by A5, A6; then f /. x1 in { p where p is Point of S : ||.(p - (f /. x0)).|| < r } ; hence f /. x1 in N01 by A3; ::_thesis: verum end; assume A8: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 ) ) ; ::_thesis: f is_continuous_in x0 now__::_thesis:_for_r_being_Real_st_0_<_r_holds_ ex_s_being_real_number_st_ (_0_<_s_&_(_for_x1_being_real_number_st_x1_in_dom_f_&_abs_(x1_-_x0)_<_s_holds_ ||.((f_/._x1)_-_(f_/._x0)).||_<_r_)_) let r be Real; ::_thesis: ( 0 < r implies ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) assume 0 < r ; ::_thesis: ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) then reconsider N1 = { p where p is Point of S : ||.(p - (f /. x0)).|| < r } as Neighbourhood of f /. x0 by NFCONT_1:3; consider N2 being Neighbourhood of x0 such that A9: for x1 being real number st x1 in dom f & x1 in N2 holds f /. x1 in N1 by A8; consider s being real number such that A10: 0 < s and A11: N2 = ].(x0 - s),(x0 + s).[ by RCOMP_1:def_6; take s = s; ::_thesis: ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r proof let x1 be real number ; ::_thesis: ( x1 in dom f & abs (x1 - x0) < s implies ||.((f /. x1) - (f /. x0)).|| < r ) assume that A12: x1 in dom f and A13: abs (x1 - x0) < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r x1 in N2 by A11, A13, RCOMP_1:1; then f /. x1 in N1 by A9, A12; then ex p being Point of S st ( p = f /. x1 & ||.(p - (f /. x0)).|| < r ) ; hence ||.((f /. x1) - (f /. x0)).|| < r ; ::_thesis: verum end; hence ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) by A10; ::_thesis: verum end; hence f is_continuous_in x0 by A8, Th8; ::_thesis: verum end; theorem Th10: :: NFCONT_3:10 for S being RealNormSpace for f being PartFunc of REAL, the carrier of S for x0 being real number holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ) proof let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S for x0 being real number holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ) let f be PartFunc of REAL, the carrier of S; ::_thesis: for x0 being real number holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ) let x0 be real number ; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ) thus ( f is_continuous_in x0 implies ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ) ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) implies f is_continuous_in x0 ) proof assume A1: f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) hence x0 in dom f by Def1; ::_thesis: for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1 consider N being Neighbourhood of x0 such that A2: for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 by A1, Th9; take N ; ::_thesis: f .: N c= N1 now__::_thesis:_for_r_being_set_st_r_in_f_.:_N_holds_ r_in_N1 let r be set ; ::_thesis: ( r in f .: N implies r in N1 ) assume r in f .: N ; ::_thesis: r in N1 then consider x being Element of REAL such that A3: ( x in dom f & x in N & r = f . x ) by PARTFUN2:59; r = f /. x by A3, PARTFUN1:def_6; hence r in N1 by A2, A3; ::_thesis: verum end; hence f .: N c= N1 by TARSKI:def_3; ::_thesis: verum end; assume A4: ( x0 in dom f & ( for N1 being Neighbourhood of f /. x0 ex N being Neighbourhood of x0 st f .: N c= N1 ) ) ; ::_thesis: f is_continuous_in x0 now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_ for_x1_being_real_number_st_x1_in_dom_f_&_x1_in_N_holds_ f_/._x1_in_N1 let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 consider N being Neighbourhood of x0 such that A5: f .: N c= N1 by A4; take N = N; ::_thesis: for x1 being real number st x1 in dom f & x1 in N holds f /. x1 in N1 let x1 be real number ; ::_thesis: ( x1 in dom f & x1 in N implies f /. x1 in N1 ) assume A6: ( x1 in dom f & x1 in N ) ; ::_thesis: f /. x1 in N1 then f . x1 in f .: N by FUNCT_1:def_6; then f /. x1 in f .: N by A6, PARTFUN1:def_6; hence f /. x1 in N1 by A5; ::_thesis: verum end; hence f is_continuous_in x0 by A4, Th9; ::_thesis: verum end; theorem :: NFCONT_3:11 for x0 being real number for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds f is_continuous_in x0 proof let x0 be real number ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds f is_continuous_in x0 let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} holds f is_continuous_in x0 let f be PartFunc of REAL, the carrier of S; ::_thesis: ( ex N being Neighbourhood of x0 st (dom f) /\ N = {x0} implies f is_continuous_in x0 ) given N being Neighbourhood of x0 such that A1: (dom f) /\ N = {x0} ; ::_thesis: f is_continuous_in x0 x0 in (dom f) /\ N by A1, TARSKI:def_1; then A2: x0 in dom f by XBOOLE_0:def_4; now__::_thesis:_for_N1_being_Neighbourhood_of_f_/._x0_ex_N_being_Neighbourhood_of_x0_st_f_.:_N_c=_N1 let N1 be Neighbourhood of f /. x0; ::_thesis: ex N being Neighbourhood of x0 st f .: N c= N1 take N = N; ::_thesis: f .: N c= N1 A3: f /. x0 in N1 by NFCONT_1:4; f .: N = Im (f,x0) by A1, RELAT_1:112 .= {(f . x0)} by A2, FUNCT_1:59 .= {(f /. x0)} by A2, PARTFUN1:def_6 ; hence f .: N c= N1 by A3, ZFMISC_1:31; ::_thesis: verum end; hence f is_continuous_in x0 by Th10, A2; ::_thesis: verum end; theorem :: NFCONT_3:12 for x0 being real number for S being RealNormSpace for f1, f2 being PartFunc of REAL, the carrier of S st x0 in (dom f1) /\ (dom f2) & f1 is_continuous_in x0 & f2 is_continuous_in x0 holds ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) proof let x0 be real number ; ::_thesis: for S being RealNormSpace for f1, f2 being PartFunc of REAL, the carrier of S st x0 in (dom f1) /\ (dom f2) & f1 is_continuous_in x0 & f2 is_continuous_in x0 holds ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) let S be RealNormSpace; ::_thesis: for f1, f2 being PartFunc of REAL, the carrier of S st x0 in (dom f1) /\ (dom f2) & f1 is_continuous_in x0 & f2 is_continuous_in x0 holds ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) let f1, f2 be PartFunc of REAL, the carrier of S; ::_thesis: ( x0 in (dom f1) /\ (dom f2) & f1 is_continuous_in x0 & f2 is_continuous_in x0 implies ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) ) assume A1: x0 in (dom f1) /\ (dom f2) ; ::_thesis: ( not f1 is_continuous_in x0 or not f2 is_continuous_in x0 or ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) ) assume A2: ( f1 is_continuous_in x0 & f2 is_continuous_in x0 ) ; ::_thesis: ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) A3: dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_1; A4: x0 in dom (f1 + f2) by A1, VFUNCT_1:def_1; now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_dom_(f1_+_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_ (_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._x0_=_lim_((f1_+_f2)_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) ) ) assume that A5: rng s1 c= dom (f1 + f2) and A6: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) ) A7: rng s1 c= (dom f1) /\ (dom f2) by A5, VFUNCT_1:def_1; ( dom (f1 + f2) c= dom f1 & dom (f1 + f2) c= dom f2 ) by A3, XBOOLE_1:17; then A8: ( rng s1 c= dom f1 & rng s1 c= dom f2 ) by A5, XBOOLE_1:1; then A9: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A2, A6, Def1; then (f1 /* s1) + (f2 /* s1) is convergent by NORMSP_1:19; hence (f1 + f2) /* s1 is convergent by A7, Th2; ::_thesis: (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) A10: ( f1 /. x0 = lim (f1 /* s1) & f2 /. x0 = lim (f2 /* s1) ) by A2, A6, A8, Def1; thus (f1 + f2) /. x0 = (f1 /. x0) + (f2 /. x0) by A4, VFUNCT_1:def_1 .= lim ((f1 /* s1) + (f2 /* s1)) by A9, A10, NORMSP_1:25 .= lim ((f1 + f2) /* s1) by A7, Th2 ; ::_thesis: verum end; hence f1 + f2 is_continuous_in x0 by A4, Def1; ::_thesis: f1 - f2 is_continuous_in x0 A11: dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def_2; A12: x0 in dom (f1 - f2) by A1, VFUNCT_1:def_2; now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_dom_(f1_-_f2)_&_s1_is_convergent_&_lim_s1_=_x0_holds_ (_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._x0_=_lim_((f1_-_f2)_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) ) ) assume that A13: rng s1 c= dom (f1 - f2) and A14: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) ) A15: rng s1 c= (dom f1) /\ (dom f2) by A13, VFUNCT_1:def_2; ( dom (f1 - f2) c= dom f1 & dom (f1 - f2) c= dom f2 ) by A11, XBOOLE_1:17; then A16: ( rng s1 c= dom f1 & rng s1 c= dom f2 ) by A13, XBOOLE_1:1; then A17: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A2, A14, Def1; then (f1 /* s1) - (f2 /* s1) is convergent by NORMSP_1:20; hence (f1 - f2) /* s1 is convergent by A15, Th2; ::_thesis: (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) A18: ( f1 /. x0 = lim (f1 /* s1) & f2 /. x0 = lim (f2 /* s1) ) by A2, A14, A16, Def1; thus (f1 - f2) /. x0 = (f1 /. x0) - (f2 /. x0) by A12, VFUNCT_1:def_2 .= lim ((f1 /* s1) - (f2 /* s1)) by A18, A17, NORMSP_1:26 .= lim ((f1 - f2) /* s1) by A15, Th2 ; ::_thesis: verum end; hence f1 - f2 is_continuous_in x0 by A12, Def1; ::_thesis: verum end; theorem Th13: :: NFCONT_3:13 for r being Real for x0 being real number for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f is_continuous_in x0 holds r (#) f is_continuous_in x0 proof let r be Real; ::_thesis: for x0 being real number for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f is_continuous_in x0 holds r (#) f is_continuous_in x0 let x0 be real number ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f is_continuous_in x0 holds r (#) f is_continuous_in x0 let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st f is_continuous_in x0 holds r (#) f is_continuous_in x0 let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f is_continuous_in x0 implies r (#) f is_continuous_in x0 ) assume A1: f is_continuous_in x0 ; ::_thesis: r (#) f is_continuous_in x0 then x0 in dom f by Def1; hence A2: x0 in dom (r (#) f) by VFUNCT_1:def_4; :: according to NFCONT_3:def_1 ::_thesis: for s1 being Real_Sequence st rng s1 c= dom (r (#) f) & s1 is convergent & lim s1 = x0 holds ( (r (#) f) /* s1 is convergent & (r (#) f) /. x0 = lim ((r (#) f) /* s1) ) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom (r (#) f) & s1 is convergent & lim s1 = x0 implies ( (r (#) f) /* s1 is convergent & (r (#) f) /. x0 = lim ((r (#) f) /* s1) ) ) assume that A3: rng s1 c= dom (r (#) f) and A4: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) /. x0 = lim ((r (#) f) /* s1) ) A5: rng s1 c= dom f by A3, VFUNCT_1:def_4; then A6: f /. x0 = lim (f /* s1) by A1, A4, Def1; A7: f /* s1 is convergent by A1, A4, A5, Def1; then r * (f /* s1) is convergent by NORMSP_1:22; hence (r (#) f) /* s1 is convergent by A5, Th4; ::_thesis: (r (#) f) /. x0 = lim ((r (#) f) /* s1) thus (r (#) f) /. x0 = r * (f /. x0) by A2, VFUNCT_1:def_4 .= lim (r * (f /* s1)) by A7, A6, NORMSP_1:28 .= lim ((r (#) f) /* s1) by A5, Th4 ; ::_thesis: verum end; theorem :: NFCONT_3:14 for x0 being real number for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st x0 in dom f & f is_continuous_in x0 holds ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) proof let x0 be real number ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st x0 in dom f & f is_continuous_in x0 holds ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st x0 in dom f & f is_continuous_in x0 holds ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) let f be PartFunc of REAL, the carrier of S; ::_thesis: ( x0 in dom f & f is_continuous_in x0 implies ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) ) assume A1: x0 in dom f ; ::_thesis: ( not f is_continuous_in x0 or ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) ) assume A2: f is_continuous_in x0 ; ::_thesis: ( ||.f.|| is_continuous_in x0 & - f is_continuous_in x0 ) A3: x0 in dom ||.f.|| by A1, NORMSP_0:def_3; now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_dom_||.f.||_&_s1_is_convergent_&_lim_s1_=_x0_holds_ (_||.f.||_/*_s1_is_convergent_&_||.f.||_._x0_=_lim_(||.f.||_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom ||.f.|| & s1 is convergent & lim s1 = x0 implies ( ||.f.|| /* s1 is convergent & ||.f.|| . x0 = lim (||.f.|| /* s1) ) ) assume that A4: rng s1 c= dom ||.f.|| and A5: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( ||.f.|| /* s1 is convergent & ||.f.|| . x0 = lim (||.f.|| /* s1) ) A6: rng s1 c= dom f by A4, NORMSP_0:def_3; then A7: f /. x0 = lim (f /* s1) by A2, A5, Def1; A8: f /* s1 is convergent by A2, A5, A6, Def1; then ||.(f /* s1).|| is convergent by NORMSP_1:23; hence ||.f.|| /* s1 is convergent by A6, Th5; ::_thesis: ||.f.|| . x0 = lim (||.f.|| /* s1) thus ||.f.|| . x0 = ||.(f /. x0).|| by A3, NORMSP_0:def_3 .= lim ||.(f /* s1).|| by A8, A7, LOPBAN_1:20 .= lim (||.f.|| /* s1) by A6, Th5 ; ::_thesis: verum end; hence ||.f.|| is_continuous_in x0 by FCONT_1:def_1; ::_thesis: - f is_continuous_in x0 (- 1) (#) f is_continuous_in x0 by A2, Th13; hence - f is_continuous_in x0 by VFUNCT_1:23; ::_thesis: verum end; theorem :: NFCONT_3:15 for x0 being real number for S, T being RealNormSpace for f1 being PartFunc of REAL, the carrier of S for f2 being PartFunc of the carrier of S, the carrier of T st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 holds f2 * f1 is_continuous_in x0 proof let x0 be real number ; ::_thesis: for S, T being RealNormSpace for f1 being PartFunc of REAL, the carrier of S for f2 being PartFunc of the carrier of S, the carrier of T st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 holds f2 * f1 is_continuous_in x0 let S, T be RealNormSpace; ::_thesis: for f1 being PartFunc of REAL, the carrier of S for f2 being PartFunc of the carrier of S, the carrier of T st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 holds f2 * f1 is_continuous_in x0 let f1 be PartFunc of REAL, the carrier of S; ::_thesis: for f2 being PartFunc of the carrier of S, the carrier of T st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 holds f2 * f1 is_continuous_in x0 let f2 be PartFunc of the carrier of S, the carrier of T; ::_thesis: ( x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 implies f2 * f1 is_continuous_in x0 ) assume A1: x0 in dom (f2 * f1) ; ::_thesis: ( not f1 is_continuous_in x0 or not f2 is_continuous_in f1 /. x0 or f2 * f1 is_continuous_in x0 ) assume that A2: f1 is_continuous_in x0 and A3: f2 is_continuous_in f1 /. x0 ; ::_thesis: f2 * f1 is_continuous_in x0 thus x0 in dom (f2 * f1) by A1; :: according to NFCONT_3:def_1 ::_thesis: for s1 being Real_Sequence st rng s1 c= dom (f2 * f1) & s1 is convergent & lim s1 = x0 holds ( (f2 * f1) /* s1 is convergent & (f2 * f1) /. x0 = lim ((f2 * f1) /* s1) ) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom (f2 * f1) & s1 is convergent & lim s1 = x0 implies ( (f2 * f1) /* s1 is convergent & (f2 * f1) /. x0 = lim ((f2 * f1) /* s1) ) ) assume that A4: rng s1 c= dom (f2 * f1) and A5: ( s1 is convergent & lim s1 = x0 ) ; ::_thesis: ( (f2 * f1) /* s1 is convergent & (f2 * f1) /. x0 = lim ((f2 * f1) /* s1) ) A6: dom (f2 * f1) c= dom f1 by RELAT_1:25; now__::_thesis:_for_x_being_set_st_x_in_rng_(f1_/*_s1)_holds_ x_in_dom_f2 let x be set ; ::_thesis: ( x in rng (f1 /* s1) implies x in dom f2 ) assume x in rng (f1 /* s1) ; ::_thesis: x in dom f2 then consider n being Element of NAT such that A7: x = (f1 /* s1) . n by FUNCT_2:113; s1 . n in rng s1 by VALUED_0:28; then f1 . (s1 . n) in dom f2 by A4, FUNCT_1:11; hence x in dom f2 by A4, A6, A7, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then A8: rng (f1 /* s1) c= dom f2 by TARSKI:def_3; A9: now__::_thesis:_for_n_being_Element_of_NAT_holds_((f2_*_f1)_/*_s1)_._n_=_(f2_/*_(f1_/*_s1))_._n let n be Element of NAT ; ::_thesis: ((f2 * f1) /* s1) . n = (f2 /* (f1 /* s1)) . n s1 . n in rng s1 by VALUED_0:28; then A10: s1 . n in dom f1 by A4, FUNCT_1:11; thus ((f2 * f1) /* s1) . n = (f2 * f1) . (s1 . n) by A4, FUNCT_2:108 .= f2 . (f1 . (s1 . n)) by A10, FUNCT_1:13 .= f2 . ((f1 /* s1) . n) by A4, A6, FUNCT_2:108, XBOOLE_1:1 .= (f2 /* (f1 /* s1)) . n by A8, FUNCT_2:108 ; ::_thesis: verum end; then A11: f2 /* (f1 /* s1) = (f2 * f1) /* s1 by FUNCT_2:63; rng s1 c= dom f1 by A4, A6, XBOOLE_1:1; then A12: ( f1 /* s1 is convergent & f1 /. x0 = lim (f1 /* s1) ) by A2, A5, Def1; (f2 * f1) /. x0 = f2 /. (f1 /. x0) by A1, PARTFUN2:3 .= lim (f2 /* (f1 /* s1)) by A12, A3, A8, NFCONT_1:def_5 .= lim ((f2 * f1) /* s1) by A9, FUNCT_2:63 ; hence ( (f2 * f1) /* s1 is convergent & (f2 * f1) /. x0 = lim ((f2 * f1) /* s1) ) by A3, A12, A8, A11, NFCONT_1:def_5; ::_thesis: verum end; definition let S be RealNormSpace; let f be PartFunc of REAL, the carrier of S; attrf is continuous means :Def2: :: NFCONT_3:def 2 for x0 being real number st x0 in dom f holds f is_continuous_in x0; end; :: deftheorem Def2 defines continuous NFCONT_3:def_2_:_ for S being RealNormSpace for f being PartFunc of REAL, the carrier of S holds ( f is continuous iff for x0 being real number st x0 in dom f holds f is_continuous_in x0 ); theorem Th16: :: NFCONT_3:16 for S being RealNormSpace for X being set for f being PartFunc of REAL, the carrier of S st X c= dom f holds ( f | X is continuous iff for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) proof let S be RealNormSpace; ::_thesis: for X being set for f being PartFunc of REAL, the carrier of S st X c= dom f holds ( f | X is continuous iff for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) let X be set ; ::_thesis: for f being PartFunc of REAL, the carrier of S st X c= dom f holds ( f | X is continuous iff for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) let f be PartFunc of REAL, the carrier of S; ::_thesis: ( X c= dom f implies ( f | X is continuous iff for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) assume A1: X c= dom f ; ::_thesis: ( f | X is continuous iff for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) thus ( f | X is continuous implies for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ::_thesis: ( ( for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) implies f | X is continuous ) proof assume A2: f | X is continuous ; ::_thesis: for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) hereby ::_thesis: verum let s1 be Real_Sequence; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) assume that A3: rng s1 c= X and A4: s1 is convergent and A5: lim s1 in X ; ::_thesis: ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) A6: dom (f | X) = (dom f) /\ X by RELAT_1:61 .= X by A1, XBOOLE_1:28 ; then A7: f | X is_continuous_in lim s1 by A2, A5, Def2; now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n let n be Element of NAT ; ::_thesis: ((f | X) /* s1) . n = (f /* s1) . n A8: s1 . n in rng s1 by VALUED_0:28; thus ((f | X) /* s1) . n = (f | X) /. (s1 . n) by A3, A6, FUNCT_2:109 .= f /. (s1 . n) by A3, A6, A8, PARTFUN2:15 .= (f /* s1) . n by A1, A3, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum end; then A9: (f | X) /* s1 = f /* s1 by FUNCT_2:63; f /. (lim s1) = (f | X) /. (lim s1) by A5, A6, PARTFUN2:15 .= lim (f /* s1) by A3, A4, A6, A7, A9, Def1 ; hence ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) by A3, A4, A6, A7, A9, Def1; ::_thesis: verum end; end; assume A10: for s1 being Real_Sequence st rng s1 c= X & s1 is convergent & lim s1 in X holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ; ::_thesis: f | X is continuous now__::_thesis:_for_x1_being_real_number_st_x1_in_dom_(f_|_X)_holds_ f_|_X_is_continuous_in_x1 A11: dom (f | X) = (dom f) /\ X by RELAT_1:61 .= X by A1, XBOOLE_1:28 ; let x1 be real number ; ::_thesis: ( x1 in dom (f | X) implies f | X is_continuous_in x1 ) assume A12: x1 in dom (f | X) ; ::_thesis: f | X is_continuous_in x1 now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_dom_(f_|_X)_&_s1_is_convergent_&_lim_s1_=_x1_holds_ (_(f_|_X)_/*_s1_is_convergent_&_(f_|_X)_/._x1_=_lim_((f_|_X)_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom (f | X) & s1 is convergent & lim s1 = x1 implies ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) ) assume that A13: rng s1 c= dom (f | X) and A14: s1 is convergent and A15: lim s1 = x1 ; ::_thesis: ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_X)_/*_s1)_._n_=_(f_/*_s1)_._n let n be Element of NAT ; ::_thesis: ((f | X) /* s1) . n = (f /* s1) . n A16: s1 . n in rng s1 by VALUED_0:28; thus ((f | X) /* s1) . n = (f | X) /. (s1 . n) by A13, FUNCT_2:109 .= f /. (s1 . n) by A13, A16, PARTFUN2:15 .= (f /* s1) . n by A1, A11, A13, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum end; then A17: (f | X) /* s1 = f /* s1 by FUNCT_2:63; (f | X) /. (lim s1) = f /. (lim s1) by A12, A15, PARTFUN2:15 .= lim ((f | X) /* s1) by A10, A12, A11, A13, A14, A15, A17 ; hence ( (f | X) /* s1 is convergent & (f | X) /. x1 = lim ((f | X) /* s1) ) by A10, A12, A11, A13, A14, A15, A17; ::_thesis: verum end; hence f | X is_continuous_in x1 by Def1, A12; ::_thesis: verum end; hence f | X is continuous by Def2; ::_thesis: verum end; theorem Th17: :: NFCONT_3:17 for X being set for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st X c= dom f holds ( f | X is continuous iff for x0 being real number for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) proof let X be set ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st X c= dom f holds ( f | X is continuous iff for x0 being real number for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st X c= dom f holds ( f | X is continuous iff for x0 being real number for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) let f be PartFunc of REAL, the carrier of S; ::_thesis: ( X c= dom f implies ( f | X is continuous iff for x0 being real number for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) ) assume A1: X c= dom f ; ::_thesis: ( f | X is continuous iff for x0 being real number for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) thus ( f | X is continuous implies for x0 being real number for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) ::_thesis: ( ( for x0 being real number for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) implies f | X is continuous ) proof assume A2: f | X is continuous ; ::_thesis: for x0 being real number for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) let x0 be real number ; ::_thesis: for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) assume that A3: x0 in X and A4: 0 < r ; ::_thesis: ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) x0 in dom (f | X) by A1, A3, RELAT_1:62; then f | X is_continuous_in x0 by A2, Def2; then consider s being real number such that A5: 0 < s and A6: for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A4, Th8; take s ; ::_thesis: ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) thus 0 < s by A5; ::_thesis: for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r let x1 be real number ; ::_thesis: ( x1 in X & abs (x1 - x0) < s implies ||.((f /. x1) - (f /. x0)).|| < r ) assume that A7: x1 in X and A8: abs (x1 - x0) < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r A9: x0 in REAL by XREAL_0:def_1; A10: x1 in REAL by XREAL_0:def_1; A11: dom (f | X) = (dom f) /\ X by RELAT_1:61 .= X by A1, XBOOLE_1:28 ; then ||.((f /. x1) - (f /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A7, PARTFUN2:15, A10 .= ||.(((f | X) /. x1) - ((f | X) /. x0)).|| by A3, A11, PARTFUN2:15, A9 ; hence ||.((f /. x1) - (f /. x0)).|| < r by A6, A11, A7, A8; ::_thesis: verum end; assume A12: for x0 being real number for r being Real st x0 in X & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ; ::_thesis: f | X is continuous now__::_thesis:_for_x0_being_real_number_st_x0_in_dom_(f_|_X)_holds_ f_|_X_is_continuous_in_x0 let x0 be real number ; ::_thesis: ( x0 in dom (f | X) implies f | X is_continuous_in x0 ) assume A13: x0 in dom (f | X) ; ::_thesis: f | X is_continuous_in x0 then A14: x0 in X ; for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) ) proof let r be Real; ::_thesis: ( 0 < r implies ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) ) ) assume 0 < r ; ::_thesis: ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) ) then consider s being real number such that A15: 0 < s and A16: for x1 being real number st x1 in X & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r by A12, A14; take s ; ::_thesis: ( 0 < s & ( for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) ) thus 0 < s by A15; ::_thesis: for x1 being real number st x1 in dom (f | X) & abs (x1 - x0) < s holds ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r let x1 be real number ; ::_thesis: ( x1 in dom (f | X) & abs (x1 - x0) < s implies ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r ) assume that A17: x1 in dom (f | X) and A18: abs (x1 - x0) < s ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r A19: x0 in REAL by XREAL_0:def_1; A20: x1 in REAL by XREAL_0:def_1; ||.(((f | X) /. x1) - ((f | X) /. x0)).|| = ||.(((f | X) /. x1) - (f /. x0)).|| by A13, PARTFUN2:15, A19 .= ||.((f /. x1) - (f /. x0)).|| by A17, PARTFUN2:15, A20 ; hence ||.(((f | X) /. x1) - ((f | X) /. x0)).|| < r by A16, A17, A18; ::_thesis: verum end; hence f | X is_continuous_in x0 by Th8, A13; ::_thesis: verum end; hence f | X is continuous by Def2; ::_thesis: verum end; registration let S be RealNormSpace; cluster Function-like constant -> continuous for Element of bool [:REAL, the carrier of S:]; coherence for b1 being PartFunc of REAL, the carrier of S st b1 is constant holds b1 is continuous proof let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f is constant implies f is continuous ) assume A1: f is constant ; ::_thesis: f is continuous now__::_thesis:_for_x0_being_real_number_ for_r_being_Real_st_x0_in_dom_f_&_0_<_r_holds_ ex_s_being_real_number_st_ (_0_<_s_&_(_for_x1_being_real_number_st_x1_in_dom_f_&_abs_(x1_-_x0)_<_s_holds_ ||.((f_/._x1)_-_(f_/._x0)).||_<_r_)_) reconsider s = 1 as real number ; let x0 be real number ; ::_thesis: for r being Real st x0 in dom f & 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) let r be Real; ::_thesis: ( x0 in dom f & 0 < r implies ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) ) assume that A2: x0 in dom f and A3: 0 < r ; ::_thesis: ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) take s = s; ::_thesis: ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) thus 0 < s ; ::_thesis: for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r let x1 be real number ; ::_thesis: ( x1 in dom f & abs (x1 - x0) < s implies ||.((f /. x1) - (f /. x0)).|| < r ) assume A4: x1 in dom f ; ::_thesis: ( abs (x1 - x0) < s implies ||.((f /. x1) - (f /. x0)).|| < r ) assume abs (x1 - x0) < s ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < r f /. x1 = f . x1 by A4, PARTFUN1:def_6 .= f . x0 by A1, A2, A4, FUNCT_1:def_10 .= f /. x0 by A2, PARTFUN1:def_6 ; hence ||.((f /. x1) - (f /. x0)).|| < r by A3, NORMSP_1:6; ::_thesis: verum end; then f | (dom f) is continuous by Th17; hence f is continuous ; ::_thesis: verum end; end; registration let S be RealNormSpace; cluster Relation-like REAL -defined the carrier of S -valued Function-like continuous for Element of bool [:REAL, the carrier of S:]; existence ex b1 being PartFunc of REAL, the carrier of S st b1 is continuous proof set f = the constant PartFunc of REAL, the carrier of S; take the constant PartFunc of REAL, the carrier of S ; ::_thesis: the constant PartFunc of REAL, the carrier of S is continuous thus the constant PartFunc of REAL, the carrier of S is continuous ; ::_thesis: verum end; end; registration let S be RealNormSpace; let f be continuous PartFunc of REAL, the carrier of S; let X be set ; clusterf | X -> continuous for PartFunc of REAL, the carrier of S; coherence for b1 being PartFunc of REAL, the carrier of S st b1 = f | X holds b1 is continuous proof now__::_thesis:_for_x0_being_real_number_st_x0_in_dom_(f_|_X)_holds_ f_|_X_is_continuous_in_x0 let x0 be real number ; ::_thesis: ( x0 in dom (f | X) implies f | X is_continuous_in x0 ) assume A1: x0 in dom (f | X) ; ::_thesis: f | X is_continuous_in x0 then x0 in dom f by RELAT_1:57; then A2: f is_continuous_in x0 by Def2; x0 in X by A1; hence f | X is_continuous_in x0 by A2, Th6; ::_thesis: verum end; hence for b1 being PartFunc of REAL, the carrier of S st b1 = f | X holds b1 is continuous by Def2; ::_thesis: verum end; end; theorem Th18: :: NFCONT_3:18 for X, X1 being set for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f | X is continuous & X1 c= X holds f | X1 is continuous proof let X, X1 be set ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f | X is continuous & X1 c= X holds f | X1 is continuous let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st f | X is continuous & X1 c= X holds f | X1 is continuous let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f | X is continuous & X1 c= X implies f | X1 is continuous ) assume A1: f | X is continuous ; ::_thesis: ( not X1 c= X or f | X1 is continuous ) assume X1 c= X ; ::_thesis: f | X1 is continuous then f | X1 = (f | X) | X1 by RELAT_1:74; hence f | X1 is continuous by A1; ::_thesis: verum end; registration let S be RealNormSpace; cluster empty Function-like -> continuous for Element of bool [:REAL, the carrier of S:]; coherence for b1 being PartFunc of REAL, the carrier of S st b1 is empty holds b1 is continuous ; end; registration let S be RealNormSpace; let f be PartFunc of REAL, the carrier of S; let X be trivial set ; clusterf | X -> continuous for PartFunc of REAL, the carrier of S; coherence for b1 being PartFunc of REAL, the carrier of S st b1 = f | X holds b1 is continuous proof now__::_thesis:_(_not_f_|_X_is_empty_implies_for_b1_being_PartFunc_of_REAL,_the_carrier_of_S_st_b1_=_f_|_X_holds_ b1_is_continuous_) assume not f | X is empty ; ::_thesis: for b1 being PartFunc of REAL, the carrier of S st b1 = f | X holds b1 is continuous then consider x0 being real number such that A1: x0 in dom (f | X) by MEMBERED:9; x0 in X by A1, RELAT_1:57; then A2: X = {x0} by ZFMISC_1:132; now__::_thesis:_for_p_being_real_number_st_p_in_dom_(f_|_X)_holds_ f_|_X_is_continuous_in_p let p be real number ; ::_thesis: ( p in dom (f | X) implies f | X is_continuous_in p ) assume A3: p in dom (f | X) ; ::_thesis: f | X is_continuous_in p then A4: p in {x0} by A2; thus f | X is_continuous_in p ::_thesis: verum proof thus p in dom (f | X) by A3; :: according to NFCONT_3:def_1 ::_thesis: for s1 being Real_Sequence st rng s1 c= dom (f | X) & s1 is convergent & lim s1 = p holds ( (f | X) /* s1 is convergent & (f | X) /. p = lim ((f | X) /* s1) ) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom (f | X) & s1 is convergent & lim s1 = p implies ( (f | X) /* s1 is convergent & (f | X) /. p = lim ((f | X) /* s1) ) ) assume that A5: rng s1 c= dom (f | X) and s1 is convergent and lim s1 = p ; ::_thesis: ( (f | X) /* s1 is convergent & (f | X) /. p = lim ((f | X) /* s1) ) A6: (dom f) /\ {x0} c= {x0} by XBOOLE_1:17; rng s1 c= (dom f) /\ {x0} by A2, A5, RELAT_1:61; then A7: rng s1 c= {x0} by A6, XBOOLE_1:1; A8: now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_=_x0 let n be Element of NAT ; ::_thesis: s1 . n = x0 s1 . n in rng s1 by VALUED_0:28; hence s1 . n = x0 by A7, TARSKI:def_1; ::_thesis: verum end; A9: p = x0 by A4, TARSKI:def_1; A10: now__::_thesis:_for_g_being_Real_st_0_<_g_holds_ ex_n_being_Element_of_NAT_st_ for_m_being_Element_of_NAT_st_n_<=_m_holds_ ||.((((f_|_{x0})_/*_s1)_._m)_-_((f_|_{x0})_/._p)).||_<_g let g be Real; ::_thesis: ( 0 < g implies ex n being Element of NAT st for m being Element of NAT st n <= m holds ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g ) assume A11: 0 < g ; ::_thesis: ex n being Element of NAT st for m being Element of NAT st n <= m holds ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g take n = 0 ; ::_thesis: for m being Element of NAT st n <= m holds ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g let m be Element of NAT ; ::_thesis: ( n <= m implies ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g ) assume n <= m ; ::_thesis: ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| = ||.(((f | {x0}) /. (s1 . m)) - ((f | {x0}) /. x0)).|| by A2, A9, A5, FUNCT_2:109 .= ||.(((f | {x0}) /. x0) - ((f | {x0}) /. x0)).|| by A8 .= 0 by NORMSP_1:6 ; hence ||.((((f | {x0}) /* s1) . m) - ((f | {x0}) /. p)).|| < g by A11; ::_thesis: verum end; hence (f | X) /* s1 is convergent by A2, NORMSP_1:def_6; ::_thesis: (f | X) /. p = lim ((f | X) /* s1) hence lim ((f | X) /* s1) = (f | X) /. p by A2, A10, NORMSP_1:def_7; ::_thesis: verum end; end; hence for b1 being PartFunc of REAL, the carrier of S st b1 = f | X holds b1 is continuous by Def2; ::_thesis: verum end; hence for b1 being PartFunc of REAL, the carrier of S st b1 = f | X holds b1 is continuous ; ::_thesis: verum end; end; registration let S be RealNormSpace; let f1, f2 be continuous PartFunc of REAL, the carrier of S; clusterf1 + f2 -> continuous for PartFunc of REAL, the carrier of S; coherence for b1 being PartFunc of REAL, the carrier of S st b1 = f1 + f2 holds b1 is continuous proof set X = dom (f1 + f2); dom (f1 + f2) c= (dom f1) /\ (dom f2) by VFUNCT_1:def_1; then A1: ( dom (f1 + f2) c= dom f1 & dom (f1 + f2) c= dom f2 ) by XBOOLE_1:18; A2: ( f1 | (dom (f1 + f2)) is continuous & f2 | (dom (f1 + f2)) is continuous ) ; now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_dom_(f1_+_f2)_&_s1_is_convergent_&_lim_s1_in_dom_(f1_+_f2)_holds_ (_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._(lim_s1)_=_lim_((f1_+_f2)_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 in dom (f1 + f2) implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) ) assume that A3: rng s1 c= dom (f1 + f2) and A4: s1 is convergent and A5: lim s1 in dom (f1 + f2) ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) A6: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A1, A2, A3, A4, A5, Th16; then A7: (f1 /* s1) + (f2 /* s1) is convergent by NORMSP_1:19; A8: rng s1 c= (dom f1) /\ (dom f2) by A3, VFUNCT_1:def_1; ( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A1, A2, A3, A4, A5, Th16; then (f1 + f2) /. (lim s1) = (lim (f1 /* s1)) + (lim (f2 /* s1)) by A5, VFUNCT_1:def_1 .= lim ((f1 /* s1) + (f2 /* s1)) by A6, NORMSP_1:25 .= lim ((f1 + f2) /* s1) by A8, Th2 ; hence ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) by A8, A7, Th2; ::_thesis: verum end; then (f1 + f2) | (dom (f1 + f2)) is continuous by Th16; hence for b1 being PartFunc of REAL, the carrier of S st b1 = f1 + f2 holds b1 is continuous ; ::_thesis: verum end; clusterf1 - f2 -> continuous for PartFunc of REAL, the carrier of S; coherence for b1 being PartFunc of REAL, the carrier of S st b1 = f1 - f2 holds b1 is continuous proof set X = dom (f1 - f2); dom (f1 - f2) c= (dom f1) /\ (dom f2) by VFUNCT_1:def_2; then A9: ( dom (f1 - f2) c= dom f1 & dom (f1 - f2) c= dom f2 ) by XBOOLE_1:18; A10: ( f1 | (dom (f1 - f2)) is continuous & f2 | (dom (f1 - f2)) is continuous ) ; now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_dom_(f1_-_f2)_&_s1_is_convergent_&_lim_s1_in_dom_(f1_-_f2)_holds_ (_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._(lim_s1)_=_lim_((f1_-_f2)_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 in dom (f1 - f2) implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) ) assume that A11: rng s1 c= dom (f1 - f2) and A12: s1 is convergent and A13: lim s1 in dom (f1 - f2) ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) A14: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A9, A10, A11, A12, A13, Th16; then A15: (f1 /* s1) - (f2 /* s1) is convergent by NORMSP_1:20; A16: rng s1 c= (dom f1) /\ (dom f2) by A11, VFUNCT_1:def_2; ( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A9, A10, A11, A12, A13, Th16; then (f1 - f2) /. (lim s1) = (lim (f1 /* s1)) - (lim (f2 /* s1)) by A13, VFUNCT_1:def_2 .= lim ((f1 /* s1) - (f2 /* s1)) by A14, NORMSP_1:26 .= lim ((f1 - f2) /* s1) by A16, Th2 ; hence ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) by A16, A15, Th2; ::_thesis: verum end; then (f1 - f2) | (dom (f1 - f2)) is continuous by Th16; hence for b1 being PartFunc of REAL, the carrier of S st b1 = f1 - f2 holds b1 is continuous ; ::_thesis: verum end; end; theorem Th19: :: NFCONT_3:19 for S being RealNormSpace for X being set for f1, f2 being PartFunc of REAL, the carrier of S st X c= (dom f1) /\ (dom f2) & f1 | X is continuous & f2 | X is continuous holds ( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous ) proof let S be RealNormSpace; ::_thesis: for X being set for f1, f2 being PartFunc of REAL, the carrier of S st X c= (dom f1) /\ (dom f2) & f1 | X is continuous & f2 | X is continuous holds ( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous ) let X be set ; ::_thesis: for f1, f2 being PartFunc of REAL, the carrier of S st X c= (dom f1) /\ (dom f2) & f1 | X is continuous & f2 | X is continuous holds ( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous ) let f1, f2 be PartFunc of REAL, the carrier of S; ::_thesis: ( X c= (dom f1) /\ (dom f2) & f1 | X is continuous & f2 | X is continuous implies ( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous ) ) assume that A1: X c= (dom f1) /\ (dom f2) and A2: ( f1 | X is continuous & f2 | X is continuous ) ; ::_thesis: ( (f1 + f2) | X is continuous & (f1 - f2) | X is continuous ) A3: ( X c= dom f1 & X c= dom f2 ) by A1, XBOOLE_1:18; A4: ( X c= dom (f1 + f2) & X c= dom (f1 - f2) ) by A1, VFUNCT_1:def_1, VFUNCT_1:def_2; now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_ (_(f1_+_f2)_/*_s1_is_convergent_&_(f1_+_f2)_/._(lim_s1)_=_lim_((f1_+_f2)_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) ) assume that A5: ( rng s1 c= X & s1 is convergent ) and A6: lim s1 in X ; ::_thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) A7: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A3, A2, A5, A6, Th16; then A8: (f1 /* s1) + (f2 /* s1) is convergent by NORMSP_1:19; A9: rng s1 c= (dom f1) /\ (dom f2) by A1, A5, XBOOLE_1:1; ( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A3, A2, A5, A6, Th16; then (f1 + f2) /. (lim s1) = (lim (f1 /* s1)) + (lim (f2 /* s1)) by A4, A6, VFUNCT_1:def_1 .= lim ((f1 /* s1) + (f2 /* s1)) by A7, NORMSP_1:25 .= lim ((f1 + f2) /* s1) by A9, Th2 ; hence ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. (lim s1) = lim ((f1 + f2) /* s1) ) by A9, A8, Th2; ::_thesis: verum end; hence (f1 + f2) | X is continuous by A4, Th16; ::_thesis: (f1 - f2) | X is continuous now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_ (_(f1_-_f2)_/*_s1_is_convergent_&_(f1_-_f2)_/._(lim_s1)_=_lim_((f1_-_f2)_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) ) assume that A10: ( rng s1 c= X & s1 is convergent ) and A11: lim s1 in X ; ::_thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) A12: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A3, A2, A10, A11, Th16; then A13: (f1 /* s1) - (f2 /* s1) is convergent by NORMSP_1:20; A14: rng s1 c= (dom f1) /\ (dom f2) by A1, A10, XBOOLE_1:1; ( f1 /. (lim s1) = lim (f1 /* s1) & f2 /. (lim s1) = lim (f2 /* s1) ) by A3, A2, A10, A11, Th16; then (f1 - f2) /. (lim s1) = (lim (f1 /* s1)) - (lim (f2 /* s1)) by A4, A11, VFUNCT_1:def_2 .= lim ((f1 /* s1) - (f2 /* s1)) by A12, NORMSP_1:26 .= lim ((f1 - f2) /* s1) by A14, Th2 ; hence ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. (lim s1) = lim ((f1 - f2) /* s1) ) by A14, A13, Th2; ::_thesis: verum end; hence (f1 - f2) | X is continuous by A4, Th16; ::_thesis: verum end; theorem :: NFCONT_3:20 for S being RealNormSpace for X, X1 being set for f1, f2 being PartFunc of REAL, the carrier of S st X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous holds ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) proof let S be RealNormSpace; ::_thesis: for X, X1 being set for f1, f2 being PartFunc of REAL, the carrier of S st X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous holds ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) let X, X1 be set ; ::_thesis: for f1, f2 being PartFunc of REAL, the carrier of S st X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous holds ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) let f1, f2 be PartFunc of REAL, the carrier of S; ::_thesis: ( X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous implies ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) ) assume ( X c= dom f1 & X1 c= dom f2 ) ; ::_thesis: ( not f1 | X is continuous or not f2 | X1 is continuous or ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) ) then A1: X /\ X1 c= (dom f1) /\ (dom f2) by XBOOLE_1:27; assume ( f1 | X is continuous & f2 | X1 is continuous ) ; ::_thesis: ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) then ( f1 | (X /\ X1) is continuous & f2 | (X /\ X1) is continuous ) by Th18, XBOOLE_1:17; hence ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) by A1, Th19; ::_thesis: verum end; registration let S be RealNormSpace; let f be continuous PartFunc of REAL, the carrier of S; let r be Real; clusterr (#) f -> continuous for PartFunc of REAL, the carrier of S; coherence for b1 being PartFunc of REAL, the carrier of S st b1 = r (#) f holds b1 is continuous proof set X = dom f; A1: dom f = dom (r (#) f) by VFUNCT_1:def_4; then A2: (r (#) f) | (dom f) = r (#) f by RELAT_1:69; now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_dom_f_&_s1_is_convergent_&_lim_s1_in_dom_f_holds_ (_(r_(#)_f)_/*_s1_is_convergent_&_(r_(#)_f)_/._(lim_s1)_=_lim_((r_(#)_f)_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= dom f & s1 is convergent & lim s1 in dom f implies ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) ) assume that A3: rng s1 c= dom f and A4: s1 is convergent and A5: lim s1 in dom f ; ::_thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) f | (dom f) is continuous ; then A6: ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) by A3, A4, A5, Th16; then A7: r * (f /* s1) is convergent by NORMSP_1:22; (r (#) f) /. (lim s1) = r * (lim (f /* s1)) by A1, A5, A6, VFUNCT_1:def_4 .= lim (r * (f /* s1)) by A6, NORMSP_1:28 .= lim ((r (#) f) /* s1) by A3, Th4 ; hence ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) by A3, A7, Th4; ::_thesis: verum end; hence for b1 being PartFunc of REAL, the carrier of S st b1 = r (#) f holds b1 is continuous by A1, A2, Th16; ::_thesis: verum end; end; theorem Th21: :: NFCONT_3:21 for X being set for r being Real for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds (r (#) f) | X is continuous proof let X be set ; ::_thesis: for r being Real for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds (r (#) f) | X is continuous let r be Real; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds (r (#) f) | X is continuous let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds (r (#) f) | X is continuous let f be PartFunc of REAL, the carrier of S; ::_thesis: ( X c= dom f & f | X is continuous implies (r (#) f) | X is continuous ) assume that A1: X c= dom f and A2: f | X is continuous ; ::_thesis: (r (#) f) | X is continuous A3: X c= dom (r (#) f) by A1, VFUNCT_1:def_4; now__::_thesis:_for_s1_being_Real_Sequence_st_rng_s1_c=_X_&_s1_is_convergent_&_lim_s1_in_X_holds_ (_(r_(#)_f)_/*_s1_is_convergent_&_(r_(#)_f)_/._(lim_s1)_=_lim_((r_(#)_f)_/*_s1)_) let s1 be Real_Sequence; ::_thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) ) assume that A4: ( rng s1 c= X & s1 is convergent ) and A5: lim s1 in X ; ::_thesis: ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) A6: f /* s1 is convergent by A1, A2, A4, A5, Th16; then A7: r * (f /* s1) is convergent by NORMSP_1:22; f /. (lim s1) = lim (f /* s1) by A1, A2, A4, A5, Th16; then (r (#) f) /. (lim s1) = r * (lim (f /* s1)) by A3, A5, VFUNCT_1:def_4 .= lim (r * (f /* s1)) by A6, NORMSP_1:28 .= lim ((r (#) f) /* s1) by A1, A4, Th4, XBOOLE_1:1 ; hence ( (r (#) f) /* s1 is convergent & (r (#) f) /. (lim s1) = lim ((r (#) f) /* s1) ) by A1, A4, A7, Th4, XBOOLE_1:1; ::_thesis: verum end; hence (r (#) f) | X is continuous by A3, Th16; ::_thesis: verum end; theorem :: NFCONT_3:22 for X being set for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds ( ||.f.|| | X is continuous & (- f) | X is continuous ) proof let X be set ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds ( ||.f.|| | X is continuous & (- f) | X is continuous ) let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st X c= dom f & f | X is continuous holds ( ||.f.|| | X is continuous & (- f) | X is continuous ) let f be PartFunc of REAL, the carrier of S; ::_thesis: ( X c= dom f & f | X is continuous implies ( ||.f.|| | X is continuous & (- f) | X is continuous ) ) assume that A1: X c= dom f and A2: f | X is continuous ; ::_thesis: ( ||.f.|| | X is continuous & (- f) | X is continuous ) thus ||.f.|| | X is continuous ::_thesis: (- f) | X is continuous proof let r be real number ; :: according to FCONT_1:def_2 ::_thesis: ( not r in dom (||.f.|| | X) or ||.f.|| | X is_continuous_in r ) assume A3: r in dom (||.f.|| | X) ; ::_thesis: ||.f.|| | X is_continuous_in r then A4: ( r in dom ||.f.|| & r in X ) by RELAT_1:57; then r in dom f by NORMSP_0:def_3; then A5: r in dom (f | X) by A4, RELAT_1:57; then A6: f | X is_continuous_in r by A2, Def2; thus ||.f.|| | X is_continuous_in r ::_thesis: verum proof let s1 be Real_Sequence; :: according to FCONT_1:def_1 ::_thesis: ( not rng s1 c= dom (||.f.|| | X) or not s1 is convergent or not lim s1 = r or ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) . r = lim ((||.f.|| | X) /* s1) ) ) assume that A7: rng s1 c= dom (||.f.|| | X) and A8: ( s1 is convergent & lim s1 = r ) ; ::_thesis: ( (||.f.|| | X) /* s1 is convergent & (||.f.|| | X) . r = lim ((||.f.|| | X) /* s1) ) rng s1 c= (dom ||.f.||) /\ X by A7, RELAT_1:61; then rng s1 c= (dom f) /\ X by NORMSP_0:def_3; then A9: rng s1 c= dom (f | X) by RELAT_1:61; now__::_thesis:_for_n_being_Element_of_NAT_holds_||.((f_|_X)_/*_s1).||_._n_=_((||.f.||_|_X)_/*_s1)_._n let n be Element of NAT ; ::_thesis: ||.((f | X) /* s1).|| . n = ((||.f.|| | X) /* s1) . n A10: s1 . n in rng s1 by VALUED_0:28; then s1 . n in dom (f | X) by A9; then s1 . n in (dom f) /\ X by RELAT_1:61; then A11: ( s1 . n in X & s1 . n in dom f ) by XBOOLE_0:def_4; then A12: s1 . n in dom ||.f.|| by NORMSP_0:def_3; thus ||.((f | X) /* s1).|| . n = ||.(((f | X) /* s1) . n).|| by NORMSP_0:def_4 .= ||.((f | X) /. (s1 . n)).|| by A9, FUNCT_2:109 .= ||.(f /. (s1 . n)).|| by A9, A10, PARTFUN2:15 .= ||.f.|| . (s1 . n) by A12, NORMSP_0:def_3 .= (||.f.|| | X) . (s1 . n) by A11, FUNCT_1:49 .= ((||.f.|| | X) /* s1) . n by A7, FUNCT_2:108 ; ::_thesis: verum end; then A13: ||.((f | X) /* s1).|| = (||.f.|| | X) /* s1 by FUNCT_2:63; r in REAL by XREAL_0:def_1; then A14: ||.((f | X) /. r).|| = ||.(f /. r).|| by A5, PARTFUN2:15 .= ||.f.|| . r by A4, NORMSP_0:def_3 .= (||.f.|| | X) . r by A3, FUNCT_1:47 ; A15: (f | X) /* s1 is convergent by A6, A8, A9, Def1; hence (||.f.|| | X) /* s1 is convergent by A13, NORMSP_1:23; ::_thesis: (||.f.|| | X) . r = lim ((||.f.|| | X) /* s1) (f | X) /. r = lim ((f | X) /* s1) by A6, A8, A9, Def1; hence (||.f.|| | X) . r = lim ((||.f.|| | X) /* s1) by A15, A13, A14, LOPBAN_1:20; ::_thesis: verum end; end; ((- 1) (#) f) | X is continuous by A1, A2, Th21; hence (- f) | X is continuous by VFUNCT_1:23; ::_thesis: verum end; theorem :: NFCONT_3:23 for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f is total & ( for x1, x2 being real number holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being real number st f is_continuous_in x0 holds f | REAL is continuous proof let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st f is total & ( for x1, x2 being real number holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being real number st f is_continuous_in x0 holds f | REAL is continuous let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f is total & ( for x1, x2 being real number holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ) & ex x0 being real number st f is_continuous_in x0 implies f | REAL is continuous ) assume that A1: f is total and A2: for x1, x2 being real number holds f /. (x1 + x2) = (f /. x1) + (f /. x2) ; ::_thesis: ( for x0 being real number holds not f is_continuous_in x0 or f | REAL is continuous ) A3: dom f = REAL by A1, PARTFUN1:def_2; given x0 being real number such that A4: f is_continuous_in x0 ; ::_thesis: f | REAL is continuous reconsider z0 = 0 as real number ; (f /. z0) + (f /. z0) = f /. (z0 + z0) by A2; then (f /. z0) + ((f /. z0) - (f /. z0)) = (f /. z0) - (f /. z0) by RLVECT_1:28; then (f /. z0) + (0. S) = (f /. z0) - (f /. z0) by RLVECT_1:15; then A5: (f /. z0) + (0. S) = 0. S by RLVECT_1:15; A6: now__::_thesis:_for_x1_being_real_number_holds_-_(f_/._x1)_=_f_/._(-_x1) let x1 be real number ; ::_thesis: - (f /. x1) = f /. (- x1) 0. S = f /. (x1 + (- x1)) by A5, RLVECT_1:4; then 0. S = (f /. x1) + (f /. (- x1)) by A2; hence - (f /. x1) = f /. (- x1) by RLVECT_1:def_10; ::_thesis: verum end; A7: now__::_thesis:_for_x1,_x2_being_real_number_holds_f_/._(x1_-_x2)_=_(f_/._x1)_-_(f_/._x2) let x1, x2 be real number ; ::_thesis: f /. (x1 - x2) = (f /. x1) - (f /. x2) f /. (x1 - x2) = f /. (x1 + (- x2)) ; then f /. (x1 - x2) = (f /. x1) + (f /. (- x2)) by A2; hence f /. (x1 - x2) = (f /. x1) - (f /. x2) by A6; ::_thesis: verum end; now__::_thesis:_for_x1_being_real_number_ for_r_being_Real_st_x1_in_REAL_&_r_>_0_holds_ ex_s_being_real_number_st_ (_s_>_0_&_(_for_x2_being_real_number_st_x2_in_REAL_&_abs_(x2_-_x1)_<_s_holds_ ||.((f_/._x2)_-_(f_/._x1)).||_<_r_)_) let x1 be real number ; ::_thesis: for r being Real st x1 in REAL & r > 0 holds ex s being real number st ( s > 0 & ( for x2 being real number st x2 in REAL & abs (x2 - x1) < s holds ||.((f /. x2) - (f /. x1)).|| < r ) ) let r be Real; ::_thesis: ( x1 in REAL & r > 0 implies ex s being real number st ( s > 0 & ( for x2 being real number st x2 in REAL & abs (x2 - x1) < s holds ||.((f /. x2) - (f /. x1)).|| < r ) ) ) assume that x1 in REAL and A8: r > 0 ; ::_thesis: ex s being real number st ( s > 0 & ( for x2 being real number st x2 in REAL & abs (x2 - x1) < s holds ||.((f /. x2) - (f /. x1)).|| < r ) ) set y = x1 - x0; consider s being real number such that A9: 0 < s and A10: for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r by A4, A8, Th8; take s = s; ::_thesis: ( s > 0 & ( for x2 being real number st x2 in REAL & abs (x2 - x1) < s holds ||.((f /. x2) - (f /. x1)).|| < r ) ) thus s > 0 by A9; ::_thesis: for x2 being real number st x2 in REAL & abs (x2 - x1) < s holds ||.((f /. x2) - (f /. x1)).|| < r let x2 be real number ; ::_thesis: ( x2 in REAL & abs (x2 - x1) < s implies ||.((f /. x2) - (f /. x1)).|| < r ) assume that x2 in REAL and A11: abs (x2 - x1) < s ; ::_thesis: ||.((f /. x2) - (f /. x1)).|| < r A12: ( x2 - (x1 - x0) is Real & abs ((x2 - (x1 - x0)) - x0) = abs (x2 - x1) ) by XREAL_0:def_1; (x1 - x0) + x0 = x1 ; then ||.((f /. x2) - (f /. x1)).|| = ||.((f /. x2) - ((f /. (x1 - x0)) + (f /. x0))).|| by A2 .= ||.(((f /. x2) - (f /. (x1 - x0))) - (f /. x0)).|| by RLVECT_1:27 .= ||.((f /. (x2 - (x1 - x0))) - (f /. x0)).|| by A7 ; hence ||.((f /. x2) - (f /. x1)).|| < r by A3, A10, A11, A12; ::_thesis: verum end; hence f | REAL is continuous by A3, Th17; ::_thesis: verum end; theorem Th24: :: NFCONT_3:24 for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st dom f is compact & f | (dom f) is continuous holds rng f is compact proof let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st dom f is compact & f | (dom f) is continuous holds rng f is compact let f be PartFunc of REAL, the carrier of S; ::_thesis: ( dom f is compact & f | (dom f) is continuous implies rng f is compact ) assume that A1: dom f is compact and A2: f | (dom f) is continuous ; ::_thesis: rng f is compact now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_rng_f_holds_ ex_q2_being_Element_of_bool_[:NAT,_the_carrier_of_S:]_st_ (_q2_is_subsequence_of_s1_&_q2_is_convergent_&_lim_q2_in_rng_f_) let s1 be sequence of S; ::_thesis: ( rng s1 c= rng f implies ex q2 being Element of bool [:NAT, the carrier of S:] st ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) ) assume A3: rng s1 c= rng f ; ::_thesis: ex q2 being Element of bool [:NAT, the carrier of S:] st ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) defpred S1[ set , set ] means ( \$2 in dom f & f /. \$2 = s1 . \$1 ); A4: for n being Element of NAT ex p being Real st S1[n,p] proof let n be Element of NAT ; ::_thesis: ex p being Real st S1[n,p] dom s1 = NAT by FUNCT_2:def_1; then s1 . n in rng s1 by FUNCT_1:3; then consider p being Real such that A5: ( p in dom f & s1 . n = f . p ) by A3, PARTFUN1:3; take p ; ::_thesis: S1[n,p] thus S1[n,p] by A5, PARTFUN1:def_6; ::_thesis: verum end; consider q1 being Real_Sequence such that A6: for n being Element of NAT holds S1[n,q1 . n] from FUNCT_2:sch_3(A4); now__::_thesis:_for_x_being_set_st_x_in_rng_q1_holds_ x_in_dom_f let x be set ; ::_thesis: ( x in rng q1 implies x in dom f ) assume x in rng q1 ; ::_thesis: x in dom f then ex n being Element of NAT st x = q1 . n by FUNCT_2:113; hence x in dom f by A6; ::_thesis: verum end; then A7: rng q1 c= dom f by TARSKI:def_3; then consider s2 being Real_Sequence such that A8: s2 is subsequence of q1 and A9: s2 is convergent and A10: lim s2 in dom f by A1, RCOMP_1:def_3; take q2 = f /* s2; ::_thesis: ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) rng s2 c= rng q1 by A8, VALUED_0:21; then A11: rng s2 c= dom f by A7, XBOOLE_1:1; now__::_thesis:_for_n_being_Element_of_NAT_holds_(f_/*_q1)_._n_=_s1_._n let n be Element of NAT ; ::_thesis: (f /* q1) . n = s1 . n f /. (q1 . n) = s1 . n by A6; hence (f /* q1) . n = s1 . n by A7, FUNCT_2:109; ::_thesis: verum end; then A12: f /* q1 = s1 by FUNCT_2:63; lim s2 in dom (f | (dom f)) by A10; then f | (dom f) is_continuous_in lim s2 by A2, Def2; then A13: f is_continuous_in lim s2 ; then f /. (lim s2) = lim (f /* s2) by A9, A11, Def1; hence ( q2 is subsequence of s1 & q2 is convergent & lim q2 in rng f ) by A7, A12, A8, A9, A10, A13, A11, Def1, PARTFUN2:2, VALUED_0:22; ::_thesis: verum end; hence rng f is compact by NFCONT_1:def_2; ::_thesis: verum end; theorem :: NFCONT_3:25 for S being RealNormSpace for f being PartFunc of REAL, the carrier of S for Y being Subset of REAL st Y c= dom f & Y is compact & f | Y is continuous holds f .: Y is compact proof let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S for Y being Subset of REAL st Y c= dom f & Y is compact & f | Y is continuous holds f .: Y is compact let f be PartFunc of REAL, the carrier of S; ::_thesis: for Y being Subset of REAL st Y c= dom f & Y is compact & f | Y is continuous holds f .: Y is compact let Y be Subset of REAL; ::_thesis: ( Y c= dom f & Y is compact & f | Y is continuous implies f .: Y is compact ) assume that A1: Y c= dom f and A2: Y is compact and A3: f | Y is continuous ; ::_thesis: f .: Y is compact A4: (f | Y) | Y is continuous by A3; dom (f | Y) = (dom f) /\ Y by RELAT_1:61 .= Y by A1, XBOOLE_1:28 ; then rng (f | Y) is compact by A2, A4, Th24; hence f .: Y is compact by RELAT_1:115; ::_thesis: verum end; begin definition let S be RealNormSpace; let f be PartFunc of REAL, the carrier of S; attrf is Lipschitzian means :Def3: :: NFCONT_3:def 3 ex r being real number st ( 0 < r & ( for x1, x2 being real number st x1 in dom f & x2 in dom f holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ); end; :: deftheorem Def3 defines Lipschitzian NFCONT_3:def_3_:_ for S being RealNormSpace for f being PartFunc of REAL, the carrier of S holds ( f is Lipschitzian iff ex r being real number st ( 0 < r & ( for x1, x2 being real number st x1 in dom f & x2 in dom f holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ) ); theorem Th26: :: NFCONT_3:26 for X being set for S being RealNormSpace for f being PartFunc of REAL, the carrier of S holds ( f | X is Lipschitzian iff ex r being real number st ( 0 < r & ( for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ) ) proof let X be set ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S holds ( f | X is Lipschitzian iff ex r being real number st ( 0 < r & ( for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ) ) let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S holds ( f | X is Lipschitzian iff ex r being real number st ( 0 < r & ( for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ) ) let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f | X is Lipschitzian iff ex r being real number st ( 0 < r & ( for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ) ) thus ( f | X is Lipschitzian implies ex r being real number st ( 0 < r & ( for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ) ) ::_thesis: ( ex r being real number st ( 0 < r & ( for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ) implies f | X is Lipschitzian ) proof given r being real number such that A1: 0 < r and A2: for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.(((f | X) /. x1) - ((f | X) /. x2)).|| <= r * (abs (x1 - x2)) ; :: according to NFCONT_3:def_3 ::_thesis: ex r being real number st ( 0 < r & ( for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ) take r ; ::_thesis: ( 0 < r & ( for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) ) thus 0 < r by A1; ::_thesis: for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) A3: ( x1 in REAL & x2 in REAL ) by XREAL_0:def_1; assume A4: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) then ( (f | X) /. x1 = f /. x1 & (f | X) /. x2 = f /. x2 ) by PARTFUN2:15, A3; hence ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) by A2, A4; ::_thesis: verum end; given r being real number such that A5: 0 < r and A6: for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ; ::_thesis: f | X is Lipschitzian take r ; :: according to NFCONT_3:def_3 ::_thesis: ( 0 < r & ( for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.(((f | X) /. x1) - ((f | X) /. x2)).|| <= r * (abs (x1 - x2)) ) ) thus 0 < r by A5; ::_thesis: for x1, x2 being real number st x1 in dom (f | X) & x2 in dom (f | X) holds ||.(((f | X) /. x1) - ((f | X) /. x2)).|| <= r * (abs (x1 - x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies ||.(((f | X) /. x1) - ((f | X) /. x2)).|| <= r * (abs (x1 - x2)) ) A7: ( x1 in REAL & x2 in REAL ) by XREAL_0:def_1; assume A8: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; ::_thesis: ||.(((f | X) /. x1) - ((f | X) /. x2)).|| <= r * (abs (x1 - x2)) then ( (f | X) /. x1 = f /. x1 & (f | X) /. x2 = f /. x2 ) by PARTFUN2:15, A7; hence ||.(((f | X) /. x1) - ((f | X) /. x2)).|| <= r * (abs (x1 - x2)) by A6, A8; ::_thesis: verum end; registration let S be RealNormSpace; cluster empty Function-like -> Lipschitzian for Element of bool [:REAL, the carrier of S:]; coherence for b1 being PartFunc of REAL, the carrier of S st b1 is empty holds b1 is Lipschitzian proof let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f is empty implies f is Lipschitzian ) assume A1: f is empty ; ::_thesis: f is Lipschitzian take 1 ; :: according to NFCONT_3:def_3 ::_thesis: ( 0 < 1 & ( for x1, x2 being real number st x1 in dom f & x2 in dom f holds ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) ) ) thus ( 0 < 1 & ( for x1, x2 being real number st x1 in dom f & x2 in dom f holds ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) ) ) by A1; ::_thesis: verum end; end; registration let S be RealNormSpace; cluster empty Relation-like REAL -defined the carrier of S -valued Function-like for Element of bool [:REAL, the carrier of S:]; existence ex b1 being PartFunc of REAL, the carrier of S st b1 is empty proof take the empty PartFunc of REAL, the carrier of S ; ::_thesis: the empty PartFunc of REAL, the carrier of S is empty thus the empty PartFunc of REAL, the carrier of S is empty ; ::_thesis: verum end; end; registration let S be RealNormSpace; let f be Lipschitzian PartFunc of REAL, the carrier of S; let X be set ; clusterf | X -> Lipschitzian for PartFunc of REAL, the carrier of S; coherence for b1 being PartFunc of REAL, the carrier of S st b1 = f | X holds b1 is Lipschitzian proof consider r being real number such that A1: 0 < r and A2: for x1, x2 being real number st x1 in dom f & x2 in dom f holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) by Def3; now__::_thesis:_for_x1,_x2_being_real_number_st_x1_in_dom_(f_|_X)_&_x2_in_dom_(f_|_X)_holds_ ||.((f_/._x1)_-_(f_/._x2)).||_<=_r_*_(abs_(x1_-_x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) ) assume ( x1 in dom (f | X) & x2 in dom (f | X) ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) then ( x1 in dom f & x2 in dom f ) by RELAT_1:57; hence ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) by A2; ::_thesis: verum end; hence for b1 being PartFunc of REAL, the carrier of S st b1 = f | X holds b1 is Lipschitzian by A1, Th26; ::_thesis: verum end; end; theorem :: NFCONT_3:27 for X, X1 being set for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian & X1 c= X holds f | X1 is Lipschitzian proof let X, X1 be set ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian & X1 c= X holds f | X1 is Lipschitzian let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian & X1 c= X holds f | X1 is Lipschitzian let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f | X is Lipschitzian & X1 c= X implies f | X1 is Lipschitzian ) assume that A1: f | X is Lipschitzian and A2: X1 c= X ; ::_thesis: f | X1 is Lipschitzian f | X1 = (f | X) | X1 by A2, RELAT_1:74; hence f | X1 is Lipschitzian by A1; ::_thesis: verum end; registration let S be RealNormSpace; let f1, f2 be Lipschitzian PartFunc of REAL, the carrier of S; clusterf1 + f2 -> Lipschitzian for PartFunc of REAL, the carrier of S; coherence for b1 being PartFunc of REAL, the carrier of S st b1 = f1 + f2 holds b1 is Lipschitzian proof set X = dom f1; set X1 = dom f2; consider s being real number such that A1: 0 < s and A2: for x1, x2 being real number st x1 in dom (f1 | ((dom f1) /\ (dom f2))) & x2 in dom (f1 | ((dom f1) /\ (dom f2))) holds ||.((f1 /. x1) - (f1 /. x2)).|| <= s * (abs (x1 - x2)) by Th26; consider g being real number such that A3: 0 < g and A4: for x1, x2 being real number st x1 in dom (f2 | ((dom f1) /\ (dom f2))) & x2 in dom (f2 | ((dom f1) /\ (dom f2))) holds ||.((f2 /. x1) - (f2 /. x2)).|| <= g * (abs (x1 - x2)) by Th26; now__::_thesis:_ex_p_being_set_st_ (_0_<_p_&_(_for_x1,_x2_being_real_number_st_x1_in_dom_(f1_+_f2)_&_x2_in_dom_(f1_+_f2)_holds_ ||.(((f1_+_f2)_/._x1)_-_((f1_+_f2)_/._x2)).||_<=_p_*_(abs_(x1_-_x2))_)_) take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being real number st x1 in dom (f1 + f2) & x2 in dom (f1 + f2) holds ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * (abs (x1 - x2)) ) ) thus 0 < p by A1, A3; ::_thesis: for x1, x2 being real number st x1 in dom (f1 + f2) & x2 in dom (f1 + f2) holds ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * (abs (x1 - x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom (f1 + f2) & x2 in dom (f1 + f2) implies ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * (abs (x1 - x2)) ) assume A5: ( x1 in dom (f1 + f2) & x2 in dom (f1 + f2) ) ; ::_thesis: ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * (abs (x1 - x2)) ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| = ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 + f2) /. x2)).|| by A5, VFUNCT_1:def_1 .= ||.(((f1 /. x1) + (f2 /. x1)) - ((f1 /. x2) + (f2 /. x2))).|| by A5, VFUNCT_1:def_1 .= ||.((f1 /. x1) + ((f2 /. x1) - ((f1 /. x2) + (f2 /. x2)))).|| by RLVECT_1:28 .= ||.((f1 /. x1) + (((f2 /. x1) - (f1 /. x2)) - (f2 /. x2))).|| by RLVECT_1:27 .= ||.((f1 /. x1) + (((- (f1 /. x2)) + (f2 /. x1)) - (f2 /. x2))).|| .= ||.((f1 /. x1) + ((- (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:28 .= ||.(((f1 /. x1) - (f1 /. x2)) + ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:def_3 ; then A6: ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by NORMSP_1:def_1; dom (f2 | ((dom f1) /\ (dom f2))) = (dom f2) /\ ((dom f1) /\ (dom f2)) by RELAT_1:61 .= ((dom f2) /\ (dom f2)) /\ (dom f1) by XBOOLE_1:16 .= dom (f1 + f2) by VFUNCT_1:def_1 ; then A7: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * (abs (x1 - x2)) by A4, A5; dom (f1 | ((dom f1) /\ (dom f2))) = (dom f1) /\ ((dom f1) /\ (dom f2)) by RELAT_1:61 .= ((dom f1) /\ (dom f1)) /\ (dom f2) by XBOOLE_1:16 .= dom (f1 + f2) by VFUNCT_1:def_1 ; then ||.((f1 /. x1) - (f1 /. x2)).|| <= s * (abs (x1 - x2)) by A2, A5; then ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * (abs (x1 - x2))) + (g * (abs (x1 - x2))) by A7, XREAL_1:7; hence ||.(((f1 + f2) /. x1) - ((f1 + f2) /. x2)).|| <= p * (abs (x1 - x2)) by A6, XXREAL_0:2; ::_thesis: verum end; hence for b1 being PartFunc of REAL, the carrier of S st b1 = f1 + f2 holds b1 is Lipschitzian by Def3; ::_thesis: verum end; clusterf1 - f2 -> Lipschitzian for PartFunc of REAL, the carrier of S; coherence for b1 being PartFunc of REAL, the carrier of S st b1 = f1 - f2 holds b1 is Lipschitzian proof set X = dom f1; set X1 = dom f2; consider s being real number such that A8: 0 < s and A9: for x1, x2 being real number st x1 in dom (f1 | ((dom f1) /\ (dom f2))) & x2 in dom (f1 | ((dom f1) /\ (dom f2))) holds ||.((f1 /. x1) - (f1 /. x2)).|| <= s * (abs (x1 - x2)) by Th26; consider g being real number such that A10: 0 < g and A11: for x1, x2 being real number st x1 in dom (f2 | ((dom f1) /\ (dom f2))) & x2 in dom (f2 | ((dom f1) /\ (dom f2))) holds ||.((f2 /. x1) - (f2 /. x2)).|| <= g * (abs (x1 - x2)) by Th26; now__::_thesis:_ex_p_being_set_st_ (_0_<_p_&_(_for_x1,_x2_being_real_number_st_x1_in_dom_(f1_-_f2)_&_x2_in_dom_(f1_-_f2)_holds_ ||.(((f1_-_f2)_/._x1)_-_((f1_-_f2)_/._x2)).||_<=_p_*_(abs_(x1_-_x2))_)_) take p = s + g; ::_thesis: ( 0 < p & ( for x1, x2 being real number st x1 in dom (f1 - f2) & x2 in dom (f1 - f2) holds ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2)) ) ) thus 0 < p by A8, A10; ::_thesis: for x1, x2 being real number st x1 in dom (f1 - f2) & x2 in dom (f1 - f2) holds ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom (f1 - f2) & x2 in dom (f1 - f2) implies ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2)) ) assume A12: ( x1 in dom (f1 - f2) & x2 in dom (f1 - f2) ) ; ::_thesis: ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2)) ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| = ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 - f2) /. x2)).|| by A12, VFUNCT_1:def_2 .= ||.(((f1 /. x1) - (f2 /. x1)) - ((f1 /. x2) - (f2 /. x2))).|| by A12, VFUNCT_1:def_2 .= ||.((f1 /. x1) - ((f2 /. x1) + ((f1 /. x2) - (f2 /. x2)))).|| by RLVECT_1:27 .= ||.((f1 /. x1) - (((f1 /. x2) + (f2 /. x1)) - (f2 /. x2))).|| by RLVECT_1:28 .= ||.((f1 /. x1) - ((f1 /. x2) + ((f2 /. x1) - (f2 /. x2)))).|| by RLVECT_1:28 .= ||.(((f1 /. x1) - (f1 /. x2)) - ((f2 /. x1) - (f2 /. x2))).|| by RLVECT_1:27 ; then A13: ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| by NORMSP_1:3; dom (f2 | ((dom f1) /\ (dom f2))) = (dom f2) /\ ((dom f1) /\ (dom f2)) by RELAT_1:61 .= ((dom f2) /\ (dom f2)) /\ (dom f1) by XBOOLE_1:16 .= dom (f1 - f2) by VFUNCT_1:def_2 ; then A14: ||.((f2 /. x1) - (f2 /. x2)).|| <= g * (abs (x1 - x2)) by A11, A12; dom (f1 | ((dom f1) /\ (dom f2))) = (dom f1) /\ ((dom f1) /\ (dom f2)) by RELAT_1:61 .= ((dom f1) /\ (dom f1)) /\ (dom f2) by XBOOLE_1:16 .= dom (f1 - f2) by VFUNCT_1:def_2 ; then ||.((f1 /. x1) - (f1 /. x2)).|| <= s * (abs (x1 - x2)) by A9, A12; then ||.((f1 /. x1) - (f1 /. x2)).|| + ||.((f2 /. x1) - (f2 /. x2)).|| <= (s * (abs (x1 - x2))) + (g * (abs (x1 - x2))) by A14, XREAL_1:7; hence ||.(((f1 - f2) /. x1) - ((f1 - f2) /. x2)).|| <= p * (abs (x1 - x2)) by A13, XXREAL_0:2; ::_thesis: verum end; hence for b1 being PartFunc of REAL, the carrier of S st b1 = f1 - f2 holds b1 is Lipschitzian by Def3; ::_thesis: verum end; end; theorem :: NFCONT_3:28 for X, X1 being set for S being RealNormSpace for f1, f2 being PartFunc of REAL, the carrier of S st f1 | X is Lipschitzian & f2 | X1 is Lipschitzian holds (f1 + f2) | (X /\ X1) is Lipschitzian proof let X, X1 be set ; ::_thesis: for S being RealNormSpace for f1, f2 being PartFunc of REAL, the carrier of S st f1 | X is Lipschitzian & f2 | X1 is Lipschitzian holds (f1 + f2) | (X /\ X1) is Lipschitzian let S be RealNormSpace; ::_thesis: for f1, f2 being PartFunc of REAL, the carrier of S st f1 | X is Lipschitzian & f2 | X1 is Lipschitzian holds (f1 + f2) | (X /\ X1) is Lipschitzian let f1, f2 be PartFunc of REAL, the carrier of S; ::_thesis: ( f1 | X is Lipschitzian & f2 | X1 is Lipschitzian implies (f1 + f2) | (X /\ X1) is Lipschitzian ) A1: ( f1 | (X /\ X1) = (f1 | X) | (X /\ X1) & f2 | (X /\ X1) = (f2 | X1) | (X /\ X1) ) by RELAT_1:74, XBOOLE_1:17; A2: (f1 + f2) | (X /\ X1) = (f1 | (X /\ X1)) + (f2 | (X /\ X1)) by VFUNCT_1:27; assume ( f1 | X is Lipschitzian & f2 | X1 is Lipschitzian ) ; ::_thesis: (f1 + f2) | (X /\ X1) is Lipschitzian hence (f1 + f2) | (X /\ X1) is Lipschitzian by A1, A2; ::_thesis: verum end; theorem :: NFCONT_3:29 for X, X1 being set for S being RealNormSpace for f1, f2 being PartFunc of REAL, the carrier of S st f1 | X is Lipschitzian & f2 | X1 is Lipschitzian holds (f1 - f2) | (X /\ X1) is Lipschitzian proof let X, X1 be set ; ::_thesis: for S being RealNormSpace for f1, f2 being PartFunc of REAL, the carrier of S st f1 | X is Lipschitzian & f2 | X1 is Lipschitzian holds (f1 - f2) | (X /\ X1) is Lipschitzian let S be RealNormSpace; ::_thesis: for f1, f2 being PartFunc of REAL, the carrier of S st f1 | X is Lipschitzian & f2 | X1 is Lipschitzian holds (f1 - f2) | (X /\ X1) is Lipschitzian let f1, f2 be PartFunc of REAL, the carrier of S; ::_thesis: ( f1 | X is Lipschitzian & f2 | X1 is Lipschitzian implies (f1 - f2) | (X /\ X1) is Lipschitzian ) A1: ( f1 | (X /\ X1) = (f1 | X) | (X /\ X1) & f2 | (X /\ X1) = (f2 | X1) | (X /\ X1) ) by RELAT_1:74, XBOOLE_1:17; A2: (f1 - f2) | (X /\ X1) = (f1 | (X /\ X1)) - (f2 | (X /\ X1)) by VFUNCT_1:30; assume ( f1 | X is Lipschitzian & f2 | X1 is Lipschitzian ) ; ::_thesis: (f1 - f2) | (X /\ X1) is Lipschitzian hence (f1 - f2) | (X /\ X1) is Lipschitzian by A1, A2; ::_thesis: verum end; registration let S be RealNormSpace; let f be Lipschitzian PartFunc of REAL, the carrier of S; let p be Real; clusterp (#) f -> Lipschitzian for PartFunc of REAL, the carrier of S; coherence for b1 being PartFunc of REAL, the carrier of S st b1 = p (#) f holds b1 is Lipschitzian proof consider s being real number such that A1: 0 < s and A2: for x1, x2 being real number st x1 in dom f & x2 in dom f holds ||.((f /. x1) - (f /. x2)).|| <= s * (abs (x1 - x2)) by Def3; percases ( p = 0 or p <> 0 ) ; supposeA3: p = 0 ; ::_thesis: for b1 being PartFunc of REAL, the carrier of S st b1 = p (#) f holds b1 is Lipschitzian now__::_thesis:_ex_s_being_real_number_st_ (_0_<_s_&_(_for_x1,_x2_being_real_number_st_x1_in_dom_(p_(#)_f)_&_x2_in_dom_(p_(#)_f)_holds_ ||.(((p_(#)_f)_/._x1)_-_((p_(#)_f)_/._x2)).||_<=_s_*_(abs_(x1_-_x2))_)_) take s = s; ::_thesis: ( 0 < s & ( for x1, x2 being real number st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * (abs (x1 - x2)) ) ) thus 0 < s by A1; ::_thesis: for x1, x2 being real number st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * (abs (x1 - x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom (p (#) f) & x2 in dom (p (#) f) implies ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * (abs (x1 - x2)) ) assume A4: ( x1 in dom (p (#) f) & x2 in dom (p (#) f) ) ; ::_thesis: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * (abs (x1 - x2)) A5: 0 <= abs (x1 - x2) by COMPLEX1:46; ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| = ||.((p * (f /. x1)) - ((p (#) f) /. x2)).|| by A4, VFUNCT_1:def_4 .= ||.((p * (f /. x1)) - (p * (f /. x2))).|| by A4, VFUNCT_1:def_4 .= ||.((0. S) - (p * (f /. x2))).|| by A3, RLVECT_1:10 .= ||.((0. S) - (0. S)).|| by A3, RLVECT_1:10 .= 0 by NORMSP_1:6 ; hence ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= s * (abs (x1 - x2)) by A1, A5; ::_thesis: verum end; hence for b1 being PartFunc of REAL, the carrier of S st b1 = p (#) f holds b1 is Lipschitzian by Def3; ::_thesis: verum end; suppose p <> 0 ; ::_thesis: for b1 being PartFunc of REAL, the carrier of S st b1 = p (#) f holds b1 is Lipschitzian then 0 < abs p by COMPLEX1:47; then A6: 0 * s < (abs p) * s by A1, XREAL_1:68; now__::_thesis:_ex_g_being_Element_of_REAL_st_ (_0_<_g_&_(_for_x1,_x2_being_real_number_st_x1_in_dom_(p_(#)_f)_&_x2_in_dom_(p_(#)_f)_holds_ ||.(((p_(#)_f)_/._x1)_-_((p_(#)_f)_/._x2)).||_<=_g_*_(abs_(x1_-_x2))_)_) take g = (abs p) * s; ::_thesis: ( 0 < g & ( for x1, x2 being real number st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * (abs (x1 - x2)) ) ) A7: 0 <= abs p by COMPLEX1:46; thus 0 < g by A6; ::_thesis: for x1, x2 being real number st x1 in dom (p (#) f) & x2 in dom (p (#) f) holds ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * (abs (x1 - x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom (p (#) f) & x2 in dom (p (#) f) implies ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * (abs (x1 - x2)) ) assume A8: ( x1 in dom (p (#) f) & x2 in dom (p (#) f) ) ; ::_thesis: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * (abs (x1 - x2)) then A9: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| = ||.((p * (f /. x1)) - ((p (#) f) /. x2)).|| by VFUNCT_1:def_4 .= ||.((p * (f /. x1)) - (p * (f /. x2))).|| by A8, VFUNCT_1:def_4 .= ||.(p * ((f /. x1) - (f /. x2))).|| by RLVECT_1:34 .= (abs p) * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def_1 ; ( x1 in dom f & x2 in dom f ) by A8, VFUNCT_1:def_4; then (abs p) * ||.((f /. x1) - (f /. x2)).|| <= (abs p) * (s * (abs (x1 - x2))) by A2, A7, XREAL_1:64; hence ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| <= g * (abs (x1 - x2)) by A9; ::_thesis: verum end; hence for b1 being PartFunc of REAL, the carrier of S st b1 = p (#) f holds b1 is Lipschitzian by Def3; ::_thesis: verum end; end; end; end; theorem :: NFCONT_3:30 for X being set for p being Real for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian & X c= dom f holds (p (#) f) | X is Lipschitzian proof let X be set ; ::_thesis: for p being Real for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian & X c= dom f holds (p (#) f) | X is Lipschitzian let p be Real; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian & X c= dom f holds (p (#) f) | X is Lipschitzian let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian & X c= dom f holds (p (#) f) | X is Lipschitzian let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f | X is Lipschitzian & X c= dom f implies (p (#) f) | X is Lipschitzian ) (p (#) f) | X = p (#) (f | X) by VFUNCT_1:31; hence ( f | X is Lipschitzian & X c= dom f implies (p (#) f) | X is Lipschitzian ) ; ::_thesis: verum end; registration let S be RealNormSpace; let f be Lipschitzian PartFunc of REAL, the carrier of S; cluster||.f.|| -> Lipschitzian for PartFunc of REAL,REAL; coherence for b1 being PartFunc of REAL,REAL st b1 = ||.f.|| holds b1 is Lipschitzian proof consider s being real number such that A1: 0 < s and A2: for x1, x2 being real number st x1 in dom f & x2 in dom f holds ||.((f /. x1) - (f /. x2)).|| <= s * (abs (x1 - x2)) by Def3; now__::_thesis:_for_x1,_x2_being_real_number_st_x1_in_dom_||.f.||_&_x2_in_dom_||.f.||_holds_ abs_((||.f.||_._x1)_-_(||.f.||_._x2))_<=_s_*_(abs_(x1_-_x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom ||.f.|| & x2 in dom ||.f.|| implies abs ((||.f.|| . x1) - (||.f.|| . x2)) <= s * (abs (x1 - x2)) ) assume A3: ( x1 in dom ||.f.|| & x2 in dom ||.f.|| ) ; ::_thesis: abs ((||.f.|| . x1) - (||.f.|| . x2)) <= s * (abs (x1 - x2)) then ( x1 in dom f & x2 in dom f ) by NORMSP_0:def_3; then A4: ||.((f /. x1) - (f /. x2)).|| <= s * (abs (x1 - x2)) by A2; abs ((||.f.|| . x1) - (||.f.|| . x2)) = abs (||.(f /. x1).|| - (||.f.|| . x2)) by A3, NORMSP_0:def_3 .= abs (||.(f /. x1).|| - ||.(f /. x2).||) by A3, NORMSP_0:def_3 ; then abs ((||.f.|| . x1) - (||.f.|| . x2)) <= ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:9; hence abs ((||.f.|| . x1) - (||.f.|| . x2)) <= s * (abs (x1 - x2)) by A4, XXREAL_0:2; ::_thesis: verum end; hence for b1 being PartFunc of REAL,REAL st b1 = ||.f.|| holds b1 is Lipschitzian by A1, FCONT_1:def_3; ::_thesis: verum end; end; theorem :: NFCONT_3:31 for X being set for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian holds ( - (f | X) is Lipschitzian & (- f) | X is Lipschitzian & ||.f.|| | X is Lipschitzian ) proof let X be set ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian holds ( - (f | X) is Lipschitzian & (- f) | X is Lipschitzian & ||.f.|| | X is Lipschitzian ) let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st f | X is Lipschitzian holds ( - (f | X) is Lipschitzian & (- f) | X is Lipschitzian & ||.f.|| | X is Lipschitzian ) let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f | X is Lipschitzian implies ( - (f | X) is Lipschitzian & (- f) | X is Lipschitzian & ||.f.|| | X is Lipschitzian ) ) assume A1: f | X is Lipschitzian ; ::_thesis: ( - (f | X) is Lipschitzian & (- f) | X is Lipschitzian & ||.f.|| | X is Lipschitzian ) A2: - (f | X) = (- 1) (#) (f | X) by VFUNCT_1:23; hence - (f | X) is Lipschitzian by A1; ::_thesis: ( (- f) | X is Lipschitzian & ||.f.|| | X is Lipschitzian ) thus (- f) | X is Lipschitzian by A1, A2, VFUNCT_1:29; ::_thesis: ||.f.|| | X is Lipschitzian ||.f.|| | X = ||.(f | X).|| by VFUNCT_1:29; hence ||.f.|| | X is Lipschitzian by A1; ::_thesis: verum end; registration let S be RealNormSpace; cluster Function-like constant -> Lipschitzian for Element of bool [:REAL, the carrier of S:]; coherence for b1 being PartFunc of REAL, the carrier of S st b1 is constant holds b1 is Lipschitzian proof let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f is constant implies f is Lipschitzian ) assume A1: f is constant ; ::_thesis: f is Lipschitzian now__::_thesis:_for_x1,_x2_being_real_number_st_x1_in_dom_f_&_x2_in_dom_f_holds_ ||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_(abs_(x1_-_x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) ) assume A2: ( x1 in dom f & x2 in dom f ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) then f /. x1 = f . x1 by PARTFUN1:def_6 .= f . x2 by A1, A2, FUNCT_1:def_10 .= f /. x2 by A2, PARTFUN1:def_6 ; then ||.((f /. x1) - (f /. x2)).|| = 0 by NORMSP_1:6; hence ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) by COMPLEX1:46; ::_thesis: verum end; hence f is Lipschitzian by Def3; ::_thesis: verum end; end; registration let S be RealNormSpace; cluster Function-like Lipschitzian -> continuous for Element of bool [:REAL, the carrier of S:]; coherence for b1 being PartFunc of REAL, the carrier of S st b1 is Lipschitzian holds b1 is continuous proof let f be PartFunc of REAL, the carrier of S; ::_thesis: ( f is Lipschitzian implies f is continuous ) set X = dom f; assume f is Lipschitzian ; ::_thesis: f is continuous then consider r being real number such that A1: 0 < r and A2: for x1, x2 being real number st x1 in dom f & x2 in dom f holds ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) by Def3; now__::_thesis:_for_x0_being_real_number_st_x0_in_dom_f_holds_ f_is_continuous_in_x0 let x0 be real number ; ::_thesis: ( x0 in dom f implies f is_continuous_in x0 ) assume A3: x0 in dom f ; ::_thesis: f is_continuous_in x0 for r being Real st 0 < r holds ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < r ) ) proof let g be Real; ::_thesis: ( 0 < g implies ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < g ) ) ) assume A4: 0 < g ; ::_thesis: ex s being real number st ( 0 < s & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s holds ||.((f /. x1) - (f /. x0)).|| < g ) ) set s = g / r; take s9 = g / r; ::_thesis: ( 0 < s9 & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s9 holds ||.((f /. x1) - (f /. x0)).|| < g ) ) A5: now__::_thesis:_for_x1_being_real_number_st_x1_in_dom_f_&_abs_(x1_-_x0)_<_g_/_r_holds_ ||.((f_/._x1)_-_(f_/._x0)).||_<_g let x1 be real number ; ::_thesis: ( x1 in dom f & abs (x1 - x0) < g / r implies ||.((f /. x1) - (f /. x0)).|| < g ) assume that A6: x1 in dom f and A7: abs (x1 - x0) < g / r ; ::_thesis: ||.((f /. x1) - (f /. x0)).|| < g r * (abs (x1 - x0)) < (g / r) * r by A1, A7, XREAL_1:68; then A8: r * (abs (x1 - x0)) < g by A1, XCMPLX_1:87; ||.((f /. x1) - (f /. x0)).|| <= r * (abs (x1 - x0)) by A2, A3, A6; hence ||.((f /. x1) - (f /. x0)).|| < g by A8, XXREAL_0:2; ::_thesis: verum end; s9 = g * (r ") by XCMPLX_0:def_9; hence ( 0 < s9 & ( for x1 being real number st x1 in dom f & abs (x1 - x0) < s9 holds ||.((f /. x1) - (f /. x0)).|| < g ) ) by A1, A4, A5, XREAL_1:129; ::_thesis: verum end; hence f is_continuous_in x0 by A3, Th8; ::_thesis: verum end; hence f is continuous by Def2; ::_thesis: verum end; end; theorem :: NFCONT_3:32 for S being RealNormSpace for f being PartFunc of REAL, the carrier of S st ex r being Point of S st rng f = {r} holds f is continuous proof let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S st ex r being Point of S st rng f = {r} holds f is continuous let f be PartFunc of REAL, the carrier of S; ::_thesis: ( ex r being Point of S st rng f = {r} implies f is continuous ) given r being Point of S such that A1: rng f = {r} ; ::_thesis: f is continuous now__::_thesis:_for_x1,_x2_being_real_number_st_x1_in_dom_f_&_x2_in_dom_f_holds_ ||.((f_/._x1)_-_(f_/._x2)).||_<=_1_*_(abs_(x1_-_x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) ) assume A2: ( x1 in dom f & x2 in dom f ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) then f . x2 in rng f by FUNCT_1:def_3; then f /. x2 in rng f by A2, PARTFUN1:def_6; then A3: f /. x2 = r by A1, TARSKI:def_1; f . x1 in rng f by A2, FUNCT_1:def_3; then f /. x1 in rng f by A2, PARTFUN1:def_6; then f /. x1 = r by A1, TARSKI:def_1; then ||.((f /. x1) - (f /. x2)).|| = 0 by A3, NORMSP_1:6; hence ||.((f /. x1) - (f /. x2)).|| <= 1 * (abs (x1 - x2)) by COMPLEX1:46; ::_thesis: verum end; then f is Lipschitzian by Def3; hence f is continuous ; ::_thesis: verum end; theorem :: NFCONT_3:33 for X being set for S being RealNormSpace for f being PartFunc of REAL, the carrier of S for r, p being Point of S st ( for x0 being real number st x0 in X holds f /. x0 = (x0 * r) + p ) holds f | X is continuous proof let X be set ; ::_thesis: for S being RealNormSpace for f being PartFunc of REAL, the carrier of S for r, p being Point of S st ( for x0 being real number st x0 in X holds f /. x0 = (x0 * r) + p ) holds f | X is continuous let S be RealNormSpace; ::_thesis: for f being PartFunc of REAL, the carrier of S for r, p being Point of S st ( for x0 being real number st x0 in X holds f /. x0 = (x0 * r) + p ) holds f | X is continuous let f be PartFunc of REAL, the carrier of S; ::_thesis: for r, p being Point of S st ( for x0 being real number st x0 in X holds f /. x0 = (x0 * r) + p ) holds f | X is continuous let r, p be Point of S; ::_thesis: ( ( for x0 being real number st x0 in X holds f /. x0 = (x0 * r) + p ) implies f | X is continuous ) assume A1: for x0 being real number st x0 in X holds f /. x0 = (x0 * r) + p ; ::_thesis: f | X is continuous A2: now__::_thesis:_for_x1,_x2_being_real_number_st_x1_in_dom_(f_|_X)_&_x2_in_dom_(f_|_X)_holds_ ||.((f_/._x1)_-_(f_/._x2)).||_<=_(||.r.||_+_1)_*_(abs_(x1_-_x2)) let x1, x2 be real number ; ::_thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies ||.((f /. x1) - (f /. x2)).|| <= (||.r.|| + 1) * (abs (x1 - x2)) ) assume A3: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; ::_thesis: ||.((f /. x1) - (f /. x2)).|| <= (||.r.|| + 1) * (abs (x1 - x2)) then x2 in X ; then A4: f /. x2 = (x2 * r) + p by A1; A5: 0 <= abs (x1 - x2) by COMPLEX1:46; A6: x1 - x2 is Real by XREAL_0:def_1; x1 in X by A3; then f /. x1 = (x1 * r) + p by A1; then ||.((f /. x1) - (f /. x2)).|| = ||.((x1 * r) + (p - (p + (x2 * r)))).|| by A4, RLVECT_1:def_3 .= ||.((x1 * r) + ((p - p) - (x2 * r))).|| by RLVECT_1:27 .= ||.((x1 * r) + ((0. S) - (x2 * r))).|| by RLVECT_1:15 .= ||.((x1 * r) - (x2 * r)).|| by RLVECT_1:14 .= ||.((x1 - x2) * r).|| by RLVECT_1:35 .= (abs (x1 - x2)) * ||.r.|| by A6, NORMSP_1:def_1 ; then ||.((f /. x1) - (f /. x2)).|| + 0 <= (||.r.|| * (abs (x1 - x2))) + (1 * (abs (x1 - x2))) by A5, XREAL_1:7; hence ||.((f /. x1) - (f /. x2)).|| <= (||.r.|| + 1) * (abs (x1 - x2)) ; ::_thesis: verum end; 0 + 0 < ||.r.|| + 1 by NORMSP_1:4, XREAL_1:8; then f | X is Lipschitzian by A2, Th26; hence f | X is continuous ; ::_thesis: verum end;