:: PDIFF_9 semantic presentation begin theorem LM01CPre2: :: PDIFF_9:1 for S, T being RealNormSpace for f being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) for r being Real st 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds ||.(f . x).|| <= r * ||.x.|| ) holds ||.f.|| <= r proof let S, T be RealNormSpace; ::_thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators (S,T)) for r being Real st 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds ||.(f . x).|| <= r * ||.x.|| ) holds ||.f.|| <= r let f be Point of (R_NormSpace_of_BoundedLinearOperators (S,T)); ::_thesis: for r being Real st 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds ||.(f . x).|| <= r * ||.x.|| ) holds ||.f.|| <= r let r be Real; ::_thesis: ( 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds ||.(f . x).|| <= r * ||.x.|| ) implies ||.f.|| <= r ) assume AS: ( 0 <= r & ( for x being Point of S st ||.x.|| <= 1 holds ||.(f . x).|| <= r * ||.x.|| ) ) ; ::_thesis: ||.f.|| <= r P1: now__::_thesis:_for_x_being_Point_of_S_st_||.x.||_<=_1_holds_ ||.(f_._x).||_<=_r let x be Point of S; ::_thesis: ( ||.x.|| <= 1 implies ||.(f . x).|| <= r ) assume ||.x.|| <= 1 ; ::_thesis: ||.(f . x).|| <= r then ( ||.(f . x).|| <= r * ||.x.|| & r * ||.x.|| <= r * 1 ) by AS, XREAL_1:64; hence ||.(f . x).|| <= r by XXREAL_0:2; ::_thesis: verum end; reconsider g = f as Lipschitzian LinearOperator of S,T by LOPBAN_1:def_9; set PreNormS = PreNorms (modetrans (f,S,T)); Q1: for y being ext-real set st y in PreNorms (modetrans (f,S,T)) holds y <= r proof let y be ext-real set ; ::_thesis: ( y in PreNorms (modetrans (f,S,T)) implies y <= r ) assume y in PreNorms (modetrans (f,S,T)) ; ::_thesis: y <= r then consider x being VECTOR of S such that Q2: ( y = ||.((modetrans (f,S,T)) . x).|| & ||.x.|| <= 1 ) ; y = ||.(g . x).|| by Q2, LOPBAN_1:29; hence y <= r by P1, Q2; ::_thesis: verum end; set UBPreNormS = upper_bound (PreNorms (modetrans (f,S,T))); set dif = (upper_bound (PreNorms (modetrans (f,S,T)))) - r; now__::_thesis:_not_upper_bound_(PreNorms_(modetrans_(f,S,T)))_>_r assume upper_bound (PreNorms (modetrans (f,S,T))) > r ; ::_thesis: contradiction then D2: (upper_bound (PreNorms (modetrans (f,S,T)))) - r > 0 by XREAL_1:50; r is UpperBound of PreNorms (modetrans (f,S,T)) by Q1, XXREAL_2:def_1; then PreNorms (modetrans (f,S,T)) is bounded_above by XXREAL_2:def_10; then ex w being real set st ( w in PreNorms (modetrans (f,S,T)) & (upper_bound (PreNorms (modetrans (f,S,T)))) - ((upper_bound (PreNorms (modetrans (f,S,T)))) - r) < w ) by D2, SEQ_4:def_1; hence contradiction by Q1; ::_thesis: verum end; then upper_bound (PreNorms g) <= r by LOPBAN_1:def_11; hence ||.f.|| <= r by LOPBAN_1:30; ::_thesis: verum end; theorem NFCONT125: :: PDIFF_9:2 for Z being set for S being RealNormSpace for f being PartFunc of S,REAL holds ( f is_continuous_on Z iff ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) proof let Z be set ; ::_thesis: for S being RealNormSpace for f being PartFunc of S,REAL holds ( f is_continuous_on Z iff ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) let S be RealNormSpace; ::_thesis: for f being PartFunc of S,REAL holds ( f is_continuous_on Z iff ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) let f be PartFunc of S,REAL; ::_thesis: ( f is_continuous_on Z iff ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) thus ( f is_continuous_on Z implies ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) ) ::_thesis: ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) implies f is_continuous_on Z ) proof assume A1: f is_continuous_on Z ; ::_thesis: ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) then A2: Z c= dom f by NFCONT_1:def_8; now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_Z_&_s1_is_convergent_&_lim_s1_in_Z_holds_ (_f_/*_s1_is_convergent_&_f_/._(lim_s1)_=_lim_(f_/*_s1)_) let s1 be sequence of S; ::_thesis: ( rng s1 c= Z & s1 is convergent & lim s1 in Z implies ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) assume A3: ( rng s1 c= Z & s1 is convergent & lim s1 in Z ) ; ::_thesis: ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) then A6: f | Z is_continuous_in lim s1 by A1, NFCONT_1:def_8; dom (f | Z) = (dom f) /\ Z by PARTFUN2:15; then A7: dom (f | Z) = Z by A2, XBOOLE_1:28; now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_Z)_/*_s1)_._n_=_(f_/*_s1)_._n let n be Element of NAT ; ::_thesis: ((f | Z) /* s1) . n = (f /* s1) . n dom s1 = NAT by FUNCT_2:def_1; then A8: s1 . n in rng s1 by FUNCT_1:3; thus ((f | Z) /* s1) . n = (f | Z) /. (s1 . n) by A3, A7, FUNCT_2:109 .= f /. (s1 . n) by A3, A7, A8, PARTFUN2:15 .= (f /* s1) . n by A2, A3, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum end; then A9: (f | Z) /* s1 = f /* s1 by FUNCT_2:63; f /. (lim s1) = (f | Z) /. (lim s1) by A3, A7, PARTFUN2:15; hence ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) by A3, A7, A6, A9, NFCONT_1:def_6; ::_thesis: verum end; hence ( Z c= dom f & ( for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ) ) by A1, NFCONT_1:def_8; ::_thesis: verum end; assume that A10: Z c= dom f and A11: for s1 being sequence of S st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( f /* s1 is convergent & f /. (lim s1) = lim (f /* s1) ) ; ::_thesis: f is_continuous_on Z dom (f | Z) = (dom f) /\ Z by PARTFUN2:15; then A12: dom (f | Z) = Z by A10, XBOOLE_1:28; now__::_thesis:_for_x1_being_Point_of_S_st_x1_in_Z_holds_ f_|_Z_is_continuous_in_x1 let x1 be Point of S; ::_thesis: ( x1 in Z implies f | Z is_continuous_in x1 ) assume A13: x1 in Z ; ::_thesis: f | Z is_continuous_in x1 now__::_thesis:_for_s1_being_sequence_of_S_st_rng_s1_c=_dom_(f_|_Z)_&_s1_is_convergent_&_lim_s1_=_x1_holds_ (_(f_|_Z)_/*_s1_is_convergent_&_(f_|_Z)_/._x1_=_lim_((f_|_Z)_/*_s1)_) let s1 be sequence of S; ::_thesis: ( rng s1 c= dom (f | Z) & s1 is convergent & lim s1 = x1 implies ( (f | Z) /* s1 is convergent & (f | Z) /. x1 = lim ((f | Z) /* s1) ) ) assume A14: ( rng s1 c= dom (f | Z) & s1 is convergent & lim s1 = x1 ) ; ::_thesis: ( (f | Z) /* s1 is convergent & (f | Z) /. x1 = lim ((f | Z) /* s1) ) now__::_thesis:_for_n_being_Element_of_NAT_holds_((f_|_Z)_/*_s1)_._n_=_(f_/*_s1)_._n let n be Element of NAT ; ::_thesis: ((f | Z) /* s1) . n = (f /* s1) . n dom s1 = NAT by FUNCT_2:def_1; then A17: s1 . n in rng s1 by FUNCT_1:3; thus ((f | Z) /* s1) . n = (f | Z) /. (s1 . n) by A14, FUNCT_2:109 .= f /. (s1 . n) by A14, A17, PARTFUN2:15 .= (f /* s1) . n by A10, A12, A14, FUNCT_2:109, XBOOLE_1:1 ; ::_thesis: verum end; then A18: (f | Z) /* s1 = f /* s1 by FUNCT_2:63; (f | Z) /. (lim s1) = f /. (lim s1) by A13, A12, A14, PARTFUN2:15; hence ( (f | Z) /* s1 is convergent & (f | Z) /. x1 = lim ((f | Z) /* s1) ) by A11, A13, A12, A14, A18; ::_thesis: verum end; hence f | Z is_continuous_in x1 by A13, A12, NFCONT_1:def_6; ::_thesis: verum end; hence f is_continuous_on Z by A10, NFCONT_1:def_8; ::_thesis: verum end; theorem LMXTh0: :: PDIFF_9:3 for i being Element of NAT for f being PartFunc of (REAL i),REAL holds dom (<>* f) = dom f proof let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL holds dom (<>* f) = dom f let f be PartFunc of (REAL i),REAL; ::_thesis: dom (<>* f) = dom f rng f c= dom ((proj (1,1)) ") by PDIFF_1:2; hence dom (<>* f) = dom f by RELAT_1:27; ::_thesis: verum end; theorem :: PDIFF_9:4 for i being Element of NAT for Z being set for f being PartFunc of (REAL i),REAL st Z c= dom f holds dom ((<>* f) | Z) = Z proof let i be Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL i),REAL st Z c= dom f holds dom ((<>* f) | Z) = Z let Z be set ; ::_thesis: for f being PartFunc of (REAL i),REAL st Z c= dom f holds dom ((<>* f) | Z) = Z let f be PartFunc of (REAL i),REAL; ::_thesis: ( Z c= dom f implies dom ((<>* f) | Z) = Z ) assume Z c= dom f ; ::_thesis: dom ((<>* f) | Z) = Z then Z c= dom (<>* f) by LMXTh0; hence dom ((<>* f) | Z) = Z by RELAT_1:62; ::_thesis: verum end; theorem LMXTh1: :: PDIFF_9:5 for i being Element of NAT for Z being set for f being PartFunc of (REAL i),REAL holds <>* (f | Z) = (<>* f) | Z proof let i be Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL i),REAL holds <>* (f | Z) = (<>* f) | Z let Z be set ; ::_thesis: for f being PartFunc of (REAL i),REAL holds <>* (f | Z) = (<>* f) | Z let f be PartFunc of (REAL i),REAL; ::_thesis: <>* (f | Z) = (<>* f) | Z set W = (proj (1,1)) " ; rng (f | Z) c= dom ((proj (1,1)) ") by PDIFF_1:2; then dom (((proj (1,1)) ") * (f | Z)) = dom (f | Z) by RELAT_1:27 .= (dom f) /\ Z by RELAT_1:61 ; then P3: dom (((proj (1,1)) ") * (f | Z)) = (dom (<>* f)) /\ Z by LMXTh0; now__::_thesis:_for_x_being_set_st_x_in_dom_((<>*_f)_|_Z)_holds_ (<>*_(f_|_Z))_._x_=_((<>*_f)_|_Z)_._x let x be set ; ::_thesis: ( x in dom ((<>* f) | Z) implies (<>* (f | Z)) . x = ((<>* f) | Z) . x ) assume A1: x in dom ((<>* f) | Z) ; ::_thesis: (<>* (f | Z)) . x = ((<>* f) | Z) . x then x in (dom (<>* f)) /\ Z by RELAT_1:61; then x in (dom f) /\ Z by LMXTh0; then A2: ( x in Z & x in dom f ) by XBOOLE_0:def_4; dom (((proj (1,1)) ") * (f | Z)) = dom ((<>* f) | Z) by P3, RELAT_1:61; then (<>* (f | Z)) . x = ((proj (1,1)) ") . ((f | Z) . x) by A1, FUNCT_1:12 .= ((proj (1,1)) ") . (f . x) by A2, FUNCT_1:49 .= (((proj (1,1)) ") * f) . x by A2, FUNCT_1:13 ; hence (<>* (f | Z)) . x = ((<>* f) | Z) . x by A1, FUNCT_1:47; ::_thesis: verum end; hence <>* (f | Z) = (<>* f) | Z by P3, FUNCT_1:2, RELAT_1:61; ::_thesis: verum end; theorem XTh30: :: PDIFF_9:6 for i being Element of NAT for f being PartFunc of (REAL i),REAL for x being Element of REAL i st x in dom f holds ( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> ) proof let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL for x being Element of REAL i st x in dom f holds ( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> ) let f be PartFunc of (REAL i),REAL; ::_thesis: for x being Element of REAL i st x in dom f holds ( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> ) let x be Element of REAL i; ::_thesis: ( x in dom f implies ( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> ) ) set I = (proj (1,1)) " ; assume A1: x in dom f ; ::_thesis: ( (<>* f) . x = <*(f . x)*> & (<>* f) /. x = <*(f /. x)*> ) then (<>* f) . x = ((proj (1,1)) ") . (f . x) by FUNCT_1:13; hence A2: (<>* f) . x = <*(f . x)*> by PDIFF_1:1; ::_thesis: (<>* f) /. x = <*(f /. x)*> x in dom (<>* f) by A1, LMXTh0; then (<>* f) /. x = (<>* f) . x by PARTFUN1:def_6; hence (<>* f) /. x = <*(f /. x)*> by A1, A2, PARTFUN1:def_6; ::_thesis: verum end; theorem LMXTh10: :: PDIFF_9:7 for i being Element of NAT for f, g being PartFunc of (REAL i),REAL holds ( <>* (f + g) = (<>* f) + (<>* g) & <>* (f - g) = (<>* f) - (<>* g) ) proof let i be Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL i),REAL holds ( <>* (f + g) = (<>* f) + (<>* g) & <>* (f - g) = (<>* f) - (<>* g) ) let f, g be PartFunc of (REAL i),REAL; ::_thesis: ( <>* (f + g) = (<>* f) + (<>* g) & <>* (f - g) = (<>* f) - (<>* g) ) P1: ( dom (<>* (f + g)) = dom (f + g) & dom (<>* (f - g)) = dom (f - g) & dom (<>* f) = dom f & dom (<>* g) = dom g ) by LMXTh0; then ( dom (<>* (f + g)) = (dom (<>* f)) /\ (dom (<>* g)) & dom (<>* (f - g)) = (dom (<>* f)) /\ (dom (<>* g)) ) by VALUED_1:12, VALUED_1:def_1; then P4: ( dom (<>* (f + g)) = dom ((<>* f) + (<>* g)) & dom (<>* (f - g)) = dom ((<>* f) - (<>* g)) ) by VALUED_2:def_45, VALUED_2:def_46; now__::_thesis:_for_x_being_set_st_x_in_dom_(<>*_(f_+_g))_holds_ (<>*_(f_+_g))_._x_=_((<>*_f)_+_(<>*_g))_._x let x be set ; ::_thesis: ( x in dom (<>* (f + g)) implies (<>* (f + g)) . x = ((<>* f) + (<>* g)) . x ) assume A0: x in dom (<>* (f + g)) ; ::_thesis: (<>* (f + g)) . x = ((<>* f) + (<>* g)) . x then x in (dom f) /\ (dom g) by P1, VALUED_1:def_1; then ( x in dom f & x in dom g ) by XBOOLE_0:def_4; then A4: ( <*(f . x)*> = (<>* f) . x & <*(g . x)*> = (<>* g) . x ) by XTh30; (<>* (f + g)) . x = <*((f + g) . x)*> by XTh30, A0, P1 .= <*((f . x) + (g . x))*> by A0, P1, VALUED_1:def_1 .= ((<>* f) . x) + ((<>* g) . x) by A4, RVSUM_1:13 ; hence (<>* (f + g)) . x = ((<>* f) + (<>* g)) . x by P4, A0, VALUED_2:def_45; ::_thesis: verum end; hence <>* (f + g) = (<>* f) + (<>* g) by P4, FUNCT_1:2; ::_thesis: <>* (f - g) = (<>* f) - (<>* g) now__::_thesis:_for_x_being_set_st_x_in_dom_(<>*_(f_-_g))_holds_ (<>*_(f_-_g))_._x_=_((<>*_f)_-_(<>*_g))_._x let x be set ; ::_thesis: ( x in dom (<>* (f - g)) implies (<>* (f - g)) . x = ((<>* f) - (<>* g)) . x ) assume A0: x in dom (<>* (f - g)) ; ::_thesis: (<>* (f - g)) . x = ((<>* f) - (<>* g)) . x then x in (dom f) /\ (dom g) by P1, VALUED_1:12; then ( x in dom f & x in dom g ) by XBOOLE_0:def_4; then A4: ( <*(f . x)*> = (<>* f) . x & <*(g . x)*> = (<>* g) . x ) by XTh30; thus (<>* (f - g)) . x = <*((f - g) . x)*> by XTh30, A0, P1 .= <*((f . x) - (g . x))*> by A0, P1, VALUED_1:13 .= ((<>* f) . x) - ((<>* g) . x) by A4, RVSUM_1:29 .= ((<>* f) - (<>* g)) . x by P4, A0, VALUED_2:def_46 ; ::_thesis: verum end; hence <>* (f - g) = (<>* f) - (<>* g) by P4, FUNCT_1:2; ::_thesis: verum end; theorem LMXTh11: :: PDIFF_9:8 for i being Element of NAT for f being PartFunc of (REAL i),REAL for r being real number holds <>* (r (#) f) = r (#) (<>* f) proof let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL for r being real number holds <>* (r (#) f) = r (#) (<>* f) let f be PartFunc of (REAL i),REAL; ::_thesis: for r being real number holds <>* (r (#) f) = r (#) (<>* f) let r be real number ; ::_thesis: <>* (r (#) f) = r (#) (<>* f) P1: dom (<>* (r (#) f)) = dom (r (#) f) by LMXTh0; then P2: dom (<>* (r (#) f)) = dom f by VALUED_1:def_5; then P4: dom (<>* (r (#) f)) = dom (<>* f) by LMXTh0 .= dom (r (#) (<>* f)) by VALUED_2:def_39 ; now__::_thesis:_for_x_being_set_st_x_in_dom_(<>*_(r_(#)_f))_holds_ (<>*_(r_(#)_f))_._x_=_(r_(#)_(<>*_f))_._x let x be set ; ::_thesis: ( x in dom (<>* (r (#) f)) implies (<>* (r (#) f)) . x = (r (#) (<>* f)) . x ) assume A0: x in dom (<>* (r (#) f)) ; ::_thesis: (<>* (r (#) f)) . x = (r (#) (<>* f)) . x then (<>* (r (#) f)) . x = <*((r (#) f) . x)*> by P1, XTh30 .= <*(r * (f . x))*> by A0, P1, VALUED_1:def_5 .= r (#) <*(f . x)*> by RVSUM_1:47 .= r (#) ((<>* f) . x) by A0, P2, XTh30 ; hence (<>* (r (#) f)) . x = (r (#) (<>* f)) . x by A0, P4, VALUED_2:def_39; ::_thesis: verum end; hence <>* (r (#) f) = r (#) (<>* f) by P4, FUNCT_1:2; ::_thesis: verum end; XTh30D: for x being Real for y being Element of REAL 1 st <*x*> = y holds |.x.| = |.y.| proof let x be Real; ::_thesis: for y being Element of REAL 1 st <*x*> = y holds |.x.| = |.y.| let y be Element of REAL 1; ::_thesis: ( <*x*> = y implies |.x.| = |.y.| ) assume A1: <*x*> = y ; ::_thesis: |.x.| = |.y.| reconsider I = (proj (1,1)) " as Function of REAL,(REAL 1) by PDIFF_1:2; reconsider y0 = y as Point of (REAL-NS 1) by REAL_NS1:def_4; I . x = y by A1, PDIFF_1:1; then |.x.| = ||.y0.|| by PDIFF_1:3; hence |.x.| = |.y.| by REAL_NS1:1; ::_thesis: verum end; theorem LMXTh13: :: PDIFF_9:9 for i being Element of NAT for f being PartFunc of (REAL i),REAL for g being PartFunc of (REAL i),(REAL 1) st <>* f = g holds |.f.| = |.g.| proof let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL for g being PartFunc of (REAL i),(REAL 1) st <>* f = g holds |.f.| = |.g.| let f be PartFunc of (REAL i),REAL; ::_thesis: for g being PartFunc of (REAL i),(REAL 1) st <>* f = g holds |.f.| = |.g.| let g be PartFunc of (REAL i),(REAL 1); ::_thesis: ( <>* f = g implies |.f.| = |.g.| ) assume AS: <>* f = g ; ::_thesis: |.f.| = |.g.| A1: dom |.g.| = dom g by NFCONT_4:def_2 .= dom f by AS, LMXTh0 ; then A2: dom |.g.| = dom |.f.| by VALUED_1:def_11; now__::_thesis:_for_x_being_Element_of_REAL_i_st_x_in_dom_|.g.|_holds_ |.g.|_._x_=_|.f.|_._x let x be Element of REAL i; ::_thesis: ( x in dom |.g.| implies |.g.| . x = |.f.| . x ) assume A3: x in dom |.g.| ; ::_thesis: |.g.| . x = |.f.| . x then A6: g /. x = <*(f /. x)*> by AS, A1, XTh30; thus |.g.| . x = |.g.| /. x by A3, PARTFUN1:def_6 .= |.(g /. x).| by A3, NFCONT_4:def_2 .= |.(f /. x).| by A6, XTh30D .= |.(f . x).| by A1, A3, PARTFUN1:def_6 .= |.f.| . x by VALUED_1:18 ; ::_thesis: verum end; hence |.f.| = |.g.| by A2, PARTFUN1:5; ::_thesis: verum end; theorem OPEN: :: PDIFF_9:10 for m being non empty Element of NAT for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y holds ( X is open iff Y is open ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y holds ( X is open iff Y is open ) let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st X = Y holds ( X is open iff Y is open ) let Y be Subset of (REAL-NS m); ::_thesis: ( X = Y implies ( X is open iff Y is open ) ) assume A1: X = Y ; ::_thesis: ( X is open iff Y is open ) hereby ::_thesis: ( Y is open implies X is open ) assume X is open ; ::_thesis: Y is open then ex X0 being Subset of (REAL-NS m) st ( X0 = X & X0 is open ) by PDIFF_7:def_3; hence Y is open by A1; ::_thesis: verum end; thus ( Y is open implies X is open ) by A1, PDIFF_7:def_3; ::_thesis: verum end; theorem PDIFF75: :: PDIFF_9:11 for i, j being Element of NAT for q being Element of REAL st 1 <= i & i <= j holds |.((reproj (i,(0* j))) . q).| = |.q.| proof let i, j be Element of NAT ; ::_thesis: for q being Element of REAL st 1 <= i & i <= j holds |.((reproj (i,(0* j))) . q).| = |.q.| let q be Element of REAL ; ::_thesis: ( 1 <= i & i <= j implies |.((reproj (i,(0* j))) . q).| = |.q.| ) assume A1: ( 1 <= i & i <= j ) ; ::_thesis: |.((reproj (i,(0* j))) . q).| = |.q.| set y = 0* j; A3: (reproj (i,(0* j))) . q = Replace ((0* j),i,q) by PDIFF_1:def_5; 0* j in j -tuples_on REAL ; then ex s being Element of REAL * st ( s = 0* j & len s = j ) ; then A4: (reproj (i,(0* j))) . q = (((0* j) | (i -' 1)) ^ <*q*>) ^ ((0* j) /^ i) by A1, A3, FINSEQ_7:def_1; sqrt (Sum (sqr ((0* j) | (i -' 1)))) = |.(0* (i -' 1)).| by A1, PDIFF_7:2; then sqrt (Sum (sqr ((0* j) | (i -' 1)))) = 0 by EUCLID:7; then A5: Sum (sqr ((0* j) | (i -' 1))) = 0 by RVSUM_1:86, SQUARE_1:24; sqrt (Sum (sqr ((0* j) /^ i))) = |.(0* (j -' i)).| by PDIFF_7:3; then A6: sqrt (Sum (sqr ((0* j) /^ i))) = 0 by EUCLID:7; sqr ((((0* j) | (i -' 1)) ^ <*q*>) ^ ((0* j) /^ i)) = (sqr (((0* j) | (i -' 1)) ^ <*q*>)) ^ (sqr ((0* j) /^ i)) by TOPREAL7:10 .= ((sqr ((0* j) | (i -' 1))) ^ (sqr <*q*>)) ^ (sqr ((0* j) /^ i)) by TOPREAL7:10 .= ((sqr ((0* j) | (i -' 1))) ^ <*(q ^2)*>) ^ (sqr ((0* j) /^ i)) by RVSUM_1:55 ; then Sum (sqr ((((0* j) | (i -' 1)) ^ <*q*>) ^ ((0* j) /^ i))) = (Sum ((sqr ((0* j) | (i -' 1))) ^ <*(q ^2)*>)) + (Sum (sqr ((0* j) /^ i))) by RVSUM_1:75 .= ((Sum (sqr ((0* j) | (i -' 1)))) + (q ^2)) + (Sum (sqr ((0* j) /^ i))) by RVSUM_1:74 .= q ^2 by A5, A6, RVSUM_1:86, SQUARE_1:24 ; hence |.((reproj (i,(0* j))) . q).| = |.q.| by A4, COMPLEX1:72; ::_thesis: verum end; Lm5: for m being non empty Element of NAT for i being Element of NAT for x being Element of REAL m for Z being Subset of (REAL m) st Z is open & x in Z & 1 <= i & i <= m holds ex N being Neighbourhood of (proj (i,m)) . x st for z being Element of REAL st z in N holds (reproj (i,x)) . z in Z proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for x being Element of REAL m for Z being Subset of (REAL m) st Z is open & x in Z & 1 <= i & i <= m holds ex N being Neighbourhood of (proj (i,m)) . x st for z being Element of REAL st z in N holds (reproj (i,x)) . z in Z let i be Element of NAT ; ::_thesis: for x being Element of REAL m for Z being Subset of (REAL m) st Z is open & x in Z & 1 <= i & i <= m holds ex N being Neighbourhood of (proj (i,m)) . x st for z being Element of REAL st z in N holds (reproj (i,x)) . z in Z let x be Element of REAL m; ::_thesis: for Z being Subset of (REAL m) st Z is open & x in Z & 1 <= i & i <= m holds ex N being Neighbourhood of (proj (i,m)) . x st for z being Element of REAL st z in N holds (reproj (i,x)) . z in Z let Z be Subset of (REAL m); ::_thesis: ( Z is open & x in Z & 1 <= i & i <= m implies ex N being Neighbourhood of (proj (i,m)) . x st for z being Element of REAL st z in N holds (reproj (i,x)) . z in Z ) assume that A1: ( Z is open & x in Z ) and A3: ( 1 <= i & i <= m ) ; ::_thesis: ex N being Neighbourhood of (proj (i,m)) . x st for z being Element of REAL st z in N holds (reproj (i,x)) . z in Z consider r being Real such that A4: ( 0 < r & { y where y is Element of REAL m : |.(y - x).| < r } c= Z ) by A1, PDIFF_7:31; set N = ].(((proj (i,m)) . x) - r),(((proj (i,m)) . x) + r).[; reconsider N = ].(((proj (i,m)) . x) - r),(((proj (i,m)) . x) + r).[ as Neighbourhood of (proj (i,m)) . x by A4, RCOMP_1:def_6; take N ; ::_thesis: for z being Element of REAL st z in N holds (reproj (i,x)) . z in Z let z be Element of REAL ; ::_thesis: ( z in N implies (reproj (i,x)) . z in Z ) assume z in N ; ::_thesis: (reproj (i,x)) . z in Z then A5: |.(z - ((proj (i,m)) . x)).| < r by RCOMP_1:1; |.(((reproj (i,x)) . z) - x).| = |.((reproj (i,(0* m))) . (z - ((proj (i,m)) . x))).| by PDIFF_7:6 .= |.(z - ((proj (i,m)) . x)).| by A3, PDIFF75 ; then (reproj (i,x)) . z in { y where y is Element of REAL m : |.(y - x).| < r } by A5; hence (reproj (i,x)) . z in Z by A4; ::_thesis: verum end; theorem LMMMTh6: :: PDIFF_9:12 for j, i being Element of NAT for x being Element of REAL j holds x = (reproj (i,x)) . ((proj (i,j)) . x) proof let j, i be Element of NAT ; ::_thesis: for x being Element of REAL j holds x = (reproj (i,x)) . ((proj (i,j)) . x) let x be Element of REAL j; ::_thesis: x = (reproj (i,x)) . ((proj (i,j)) . x) set q = (reproj (i,x)) . ((proj (i,j)) . x); A1: ( dom ((reproj (i,x)) . ((proj (i,j)) . x)) = Seg j & dom x = Seg j ) by FINSEQ_1:89; A3: len x = j by A1, FINSEQ_1:def_3; for k being Nat st k in dom x holds x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k proof let k be Nat; ::_thesis: ( k in dom x implies x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k ) assume A5: k in dom x ; ::_thesis: x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k then A6: ( 1 <= k & k <= j ) by A1, FINSEQ_1:1; (reproj (i,x)) . ((proj (i,j)) . x) = Replace (x,i,((proj (i,j)) . x)) by PDIFF_1:def_5; then A9: ((reproj (i,x)) . ((proj (i,j)) . x)) . k = (Replace (x,i,((proj (i,j)) . x))) /. k by A5, A1, PARTFUN1:def_6; percases ( k = i or k <> i ) ; supposeA10: k = i ; ::_thesis: x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k then ((reproj (i,x)) . ((proj (i,j)) . x)) . k = (proj (i,j)) . x by A3, A6, A9, FINSEQ_7:8; hence x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k by A10, PDIFF_1:def_1; ::_thesis: verum end; suppose k <> i ; ::_thesis: x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k then ((reproj (i,x)) . ((proj (i,j)) . x)) . k = x /. k by A3, A6, A9, FINSEQ_7:10; hence x . k = ((reproj (i,x)) . ((proj (i,j)) . x)) . k by A5, PARTFUN1:def_6; ::_thesis: verum end; end; end; hence x = (reproj (i,x)) . ((proj (i,j)) . x) by A1, FINSEQ_1:13; ::_thesis: verum end; begin theorem MPDIFF633: :: PDIFF_9:13 for m, n being non empty Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X holds X is open proof let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X holds X is open let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X holds X is open let f be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_differentiable_on X implies X is open ) assume f is_differentiable_on X ; ::_thesis: X is open then ex X0 being Subset of (REAL-NS m) st ( X = X0 & X0 is open ) by PDIFF_6:33; hence X is open by PDIFF_7:def_3; ::_thesis: verum end; theorem MPDIFF632: :: PDIFF_9:14 for m, n being non empty Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),(REAL n) st X is open holds ( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_differentiable_in x ) ) ) proof let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),(REAL n) st X is open holds ( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_differentiable_in x ) ) ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),(REAL n) st X is open holds ( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_differentiable_in x ) ) ) let f be PartFunc of (REAL m),(REAL n); ::_thesis: ( X is open implies ( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_differentiable_in x ) ) ) ) assume X is open ; ::_thesis: ( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_differentiable_in x ) ) ) then ex X0 being Subset of (REAL-NS m) st ( X = X0 & X0 is open ) by PDIFF_7:def_3; hence ( f is_differentiable_on X iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_differentiable_in x ) ) ) by PDIFF_6:32; ::_thesis: verum end; definition let m, n be non empty Element of NAT ; let Z be set ; let f be PartFunc of (REAL m),(REAL n); assume A1: Z c= dom f ; funcf `| Z -> PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) means :Def1: :: PDIFF_9:def 1 ( dom it = Z & ( for x being Element of REAL m st x in Z holds it /. x = diff (f,x) ) ); existence ex b1 being PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) st ( dom b1 = Z & ( for x being Element of REAL m st x in Z holds b1 /. x = diff (f,x) ) ) proof defpred S1[ Element of REAL m, set ] means ( \$1 in Z & \$2 = diff (f,\$1) ); consider F being PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) such that A2: ( ( for x being Element of REAL m holds ( x in dom F iff ex z being Element of Funcs ((REAL m),(REAL n)) st S1[x,z] ) ) & ( for x being Element of REAL m st x in dom F holds S1[x,F . x] ) ) from SEQ_1:sch_2(); take F ; ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds F /. x = diff (f,x) ) ) A3: Z is Subset of (REAL m) by A1, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_Z_holds_ x_in_dom_F let x be set ; ::_thesis: ( x in Z implies x in dom F ) assume AS1: x in Z ; ::_thesis: x in dom F then reconsider z = x as Element of REAL m by A3; reconsider y = diff (f,z) as Element of Funcs ((REAL m),(REAL n)) by FUNCT_2:8; S1[z,y] by AS1; hence x in dom F by A2; ::_thesis: verum end; then A4: Z c= dom F by TARSKI:def_3; for y being set st y in dom F holds y in Z by A2; then dom F c= Z by TARSKI:def_3; hence dom F = Z by A4, XBOOLE_0:def_10; ::_thesis: for x being Element of REAL m st x in Z holds F /. x = diff (f,x) let x be Element of REAL m; ::_thesis: ( x in Z implies F /. x = diff (f,x) ) assume A5: x in Z ; ::_thesis: F /. x = diff (f,x) then F . x = diff (f,x) by A2, A4; hence F /. x = diff (f,x) by A5, A4, PARTFUN1:def_6; ::_thesis: verum end; uniqueness for b1, b2 being PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) st dom b1 = Z & ( for x being Element of REAL m st x in Z holds b1 /. x = diff (f,x) ) & dom b2 = Z & ( for x being Element of REAL m st x in Z holds b2 /. x = diff (f,x) ) holds b1 = b2 proof let F, G be PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))); ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds F /. x = diff (f,x) ) & dom G = Z & ( for x being Element of REAL m st x in Z holds G /. x = diff (f,x) ) implies F = G ) assume that A6: dom F = Z and A7: for x being Element of REAL m st x in Z holds F /. x = diff (f,x) and A8: dom G = Z and A9: for x being Element of REAL m st x in Z holds G /. x = diff (f,x) ; ::_thesis: F = G now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_dom_F_holds_ F_/._x_=_G_/._x let x be Element of REAL m; ::_thesis: ( x in dom F implies F /. x = G /. x ) assume A10: x in dom F ; ::_thesis: F /. x = G /. x then F /. x = diff (f,x) by A6, A7; hence F /. x = G /. x by A6, A9, A10; ::_thesis: verum end; hence F = G by A6, A8, PARTFUN2:1; ::_thesis: verum end; end; :: deftheorem Def1 defines `| PDIFF_9:def_1_:_ for m, n being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),(REAL n) st Z c= dom f holds for b5 being PartFunc of (REAL m),(Funcs ((REAL m),(REAL n))) holds ( b5 = f `| Z iff ( dom b5 = Z & ( for x being Element of REAL m st x in Z holds b5 /. x = diff (f,x) ) ) ); theorem :: PDIFF_9:15 for m, n being non empty Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) ) proof let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) ) let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) ) let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_differentiable_on X & g is_differentiable_on X implies ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) ) ) assume AS: ( f is_differentiable_on X & g is_differentiable_on X ) ; ::_thesis: ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) ) then A1: X is open by MPDIFF633; then A2: ( X c= dom f & X c= dom g ) by AS, MPDIFF632; dom (f + g) = (dom f) /\ (dom g) by VALUED_2:def_45; then A3: X c= dom (f + g) by A2, XBOOLE_1:19; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ f_+_g_is_differentiable_in_x let x be Element of REAL m; ::_thesis: ( x in X implies f + g is_differentiable_in x ) assume x in X ; ::_thesis: f + g is_differentiable_in x then ( f is_differentiable_in x & g is_differentiable_in x ) by AS, A1, MPDIFF632; hence f + g is_differentiable_in x by PDIFF_6:20; ::_thesis: verum end; hence f + g is_differentiable_on X by A3, A1, MPDIFF632; ::_thesis: for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) let x be Element of REAL m; ::_thesis: ( x in X implies ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) ) assume A5: x in X ; ::_thesis: ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) then ( f is_differentiable_in x & g is_differentiable_in x ) by AS, A1, MPDIFF632; then diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) by PDIFF_6:20; hence ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) by A3, A5, Def1; ::_thesis: verum end; theorem :: PDIFF_9:16 for m, n being non empty Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) ) proof let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) ) let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_differentiable_on X & g is_differentiable_on X holds ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) ) let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_differentiable_on X & g is_differentiable_on X implies ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) ) ) assume AS1: ( f is_differentiable_on X & g is_differentiable_on X ) ; ::_thesis: ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) ) then AS11: X is open by MPDIFF633; then P1: ( X c= dom f & X c= dom g ) by AS1, MPDIFF632; dom (f - g) = (dom f) /\ (dom g) by VALUED_2:def_46; then P3: X c= dom (f - g) by P1, XBOOLE_1:19; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ f_-_g_is_differentiable_in_x let x be Element of REAL m; ::_thesis: ( x in X implies f - g is_differentiable_in x ) assume x in X ; ::_thesis: f - g is_differentiable_in x then ( f is_differentiable_in x & g is_differentiable_in x ) by AS1, AS11, MPDIFF632; hence f - g is_differentiable_in x by PDIFF_6:21; ::_thesis: verum end; hence f - g is_differentiable_on X by P3, AS11, MPDIFF632; ::_thesis: for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) let x be Element of REAL m; ::_thesis: ( x in X implies ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) ) assume P7: x in X ; ::_thesis: ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) then ( f is_differentiable_in x & g is_differentiable_in x ) by AS1, AS11, MPDIFF632; then diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) by PDIFF_6:21; hence ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) by P3, P7, Def1; ::_thesis: verum end; theorem :: PDIFF_9:17 for m, n being non empty Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),(REAL n) for r being Real st f is_differentiable_on X holds ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) ) proof let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),(REAL n) for r being Real st f is_differentiable_on X holds ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),(REAL n) for r being Real st f is_differentiable_on X holds ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) ) let f be PartFunc of (REAL m),(REAL n); ::_thesis: for r being Real st f is_differentiable_on X holds ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) ) let r be Real; ::_thesis: ( f is_differentiable_on X implies ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) ) ) assume AS1: f is_differentiable_on X ; ::_thesis: ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) ) then AS11: X is open by MPDIFF633; then X c= dom f by AS1, MPDIFF632; then P3: X c= dom (r (#) f) by VALUED_2:def_39; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ r_(#)_f_is_differentiable_in_x let x be Element of REAL m; ::_thesis: ( x in X implies r (#) f is_differentiable_in x ) assume x in X ; ::_thesis: r (#) f is_differentiable_in x then f is_differentiable_in x by AS1, AS11, MPDIFF632; hence r (#) f is_differentiable_in x by PDIFF_6:22; ::_thesis: verum end; hence r (#) f is_differentiable_on X by P3, AS11, MPDIFF632; ::_thesis: for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) (diff (f,x)) let x be Element of REAL m; ::_thesis: ( x in X implies ((r (#) f) `| X) /. x = r (#) (diff (f,x)) ) assume P7: x in X ; ::_thesis: ((r (#) f) `| X) /. x = r (#) (diff (f,x)) then f is_differentiable_in x by AS1, AS11, MPDIFF632; then diff ((r (#) f),x) = r (#) (diff (f,x)) by PDIFF_6:22; hence ((r (#) f) `| X) /. x = r (#) (diff (f,x)) by P3, P7, Def1; ::_thesis: verum end; theorem LM01A: :: PDIFF_9:18 for j being Element of NAT for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) ex p being Point of (REAL-NS j) st ( p = f . <*1*> & ( for r being Real for x being Point of (REAL-NS 1) st x = <*r*> holds f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) ) proof let j be Element of NAT ; ::_thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) ex p being Point of (REAL-NS j) st ( p = f . <*1*> & ( for r being Real for x being Point of (REAL-NS 1) st x = <*r*> holds f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) ) let f be Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))); ::_thesis: ex p being Point of (REAL-NS j) st ( p = f . <*1*> & ( for r being Real for x being Point of (REAL-NS 1) st x = <*r*> holds f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) ) reconsider One = <*1*> as Element of REAL 1 by FINSEQ_2:98; reconsider L = f as Lipschitzian LinearOperator of (REAL-NS 1),(REAL-NS j) by LOPBAN_1:def_9; the carrier of (REAL-NS 1) = REAL 1 by REAL_NS1:def_4; then dom L = REAL 1 by FUNCT_2:def_1; then reconsider p = f . <*1*> as Point of (REAL-NS j) by FINSEQ_2:98, PARTFUN1:4; reconsider OneNS = One as VECTOR of (REAL-NS 1) by REAL_NS1:def_4; A1: now__::_thesis:_for_r_being_Real for_x_being_Point_of_(REAL-NS_1)_st_x_=_<*r*>_holds_ f_._x_=_r_*_p let r be Real; ::_thesis: for x being Point of (REAL-NS 1) st x = <*r*> holds f . x = r * p let x be Point of (REAL-NS 1); ::_thesis: ( x = <*r*> implies f . x = r * p ) assume x = <*r*> ; ::_thesis: f . x = r * p then P0: f . x = L . <*r*> ; <*r*> = <*(r * 1)*> .= r * <*1*> by RVSUM_1:47 .= r * OneNS by REAL_NS1:3 ; hence f . x = r * p by P0, LOPBAN_1:def_5; ::_thesis: verum end; now__::_thesis:_for_x_being_Point_of_(REAL-NS_1)_holds_||.(f_._x).||_=_||.p.||_*_||.x.|| let x be Point of (REAL-NS 1); ::_thesis: ||.(f . x).|| = ||.p.|| * ||.x.|| B0: the carrier of (REAL-NS 1) = REAL 1 by REAL_NS1:def_4; then reconsider x0 = x as FinSequence of REAL by FINSEQ_2:def_3; consider r being Element of REAL such that B2: x0 = <*r*> by B0, FINSEQ_2:97; thus ||.(f . x).|| = ||.(r * p).|| by A1, B2 .= (abs r) * ||.p.|| by NORMSP_1:def_1 .= ||.p.|| * ||.x.|| by B2, PDIFF_8:2 ; ::_thesis: verum end; hence ex p being Point of (REAL-NS j) st ( p = f . <*1*> & ( for r being Real for x being Point of (REAL-NS 1) st x = <*r*> holds f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) ) by A1; ::_thesis: verum end; theorem LM01C: :: PDIFF_9:19 for j being Element of NAT for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) ex p being Point of (REAL-NS j) st ( p = f . <*1*> & ||.p.|| = ||.f.|| ) proof let j be Element of NAT ; ::_thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) ex p being Point of (REAL-NS j) st ( p = f . <*1*> & ||.p.|| = ||.f.|| ) let f be Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))); ::_thesis: ex p being Point of (REAL-NS j) st ( p = f . <*1*> & ||.p.|| = ||.f.|| ) reconsider g = f as Lipschitzian LinearOperator of (REAL-NS 1),(REAL-NS j) by LOPBAN_1:def_9; consider p being Point of (REAL-NS j) such that P1: ( p = f . <*1*> & ( for r being Real for x being Point of (REAL-NS 1) st x = <*r*> holds f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) ) by LM01A; <*1*> in REAL 1 by FINSEQ_2:98; then reconsider One = <*1*> as Point of (REAL-NS 1) by REAL_NS1:def_4; ||.(g . One).|| <= ||.f.|| * ||.One.|| by LOPBAN_1:32; then ||.(g . One).|| <= ||.f.|| * (abs 1) by PDIFF_8:2; then P2: ||.(f . One).|| <= ||.f.|| * 1 by ABSVALUE:def_1; for x being Point of (REAL-NS 1) st ||.x.|| <= 1 holds ||.(f . x).|| <= ||.p.|| * ||.x.|| by P1; then ||.f.|| <= ||.p.|| by LM01CPre2; hence ex p being Point of (REAL-NS j) st ( p = f . <*1*> & ||.p.|| = ||.f.|| ) by P1, P2, XXREAL_0:1; ::_thesis: verum end; theorem LM01: :: PDIFF_9:20 for j being Element of NAT for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.f.|| * ||.x.|| proof let j be Element of NAT ; ::_thesis: for f being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))) for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.f.|| * ||.x.|| let f be Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS j))); ::_thesis: for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.f.|| * ||.x.|| let x be Point of (REAL-NS 1); ::_thesis: ||.(f . x).|| = ||.f.|| * ||.x.|| P1: ex p being Point of (REAL-NS j) st ( p = f . <*1*> & ( for r being Real for x being Point of (REAL-NS 1) st x = <*r*> holds f . x = r * p ) & ( for x being Point of (REAL-NS 1) holds ||.(f . x).|| = ||.p.|| * ||.x.|| ) ) by LM01A; ex q being Point of (REAL-NS j) st ( q = f . <*1*> & ||.f.|| = ||.q.|| ) by LM01C; hence ||.(f . x).|| = ||.f.|| * ||.x.|| by P1; ::_thesis: verum end; theorem LM02: :: PDIFF_9:21 for m, n being non empty Element of NAT for i being Element of NAT for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x being Element of REAL m for y being Point of (REAL-NS m) st x in X & x = y holds partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> proof let m, n be non empty Element of NAT ; ::_thesis: for i being Element of NAT for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x being Element of REAL m for y being Point of (REAL-NS m) st x in X & x = y holds partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x being Element of REAL m for y being Point of (REAL-NS m) st x in X & x = y holds partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x being Element of REAL m for y being Point of (REAL-NS m) st x in X & x = y holds partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x being Element of REAL m for y being Point of (REAL-NS m) st x in X & x = y holds partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x being Element of REAL m for y being Point of (REAL-NS m) st x in X & x = y holds partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> let Y be Subset of (REAL-NS m); ::_thesis: ( 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i implies for x being Element of REAL m for y being Point of (REAL-NS m) st x in X & x = y holds partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> ) assume AS0: ( 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i ) ; ::_thesis: for x being Element of REAL m for y being Point of (REAL-NS m) st x in X & x = y holds partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> let x be Element of REAL m; ::_thesis: for y being Point of (REAL-NS m) st x in X & x = y holds partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> let y be Point of (REAL-NS m); ::_thesis: ( x in X & x = y implies partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> ) assume AS: ( x in X & x = y ) ; ::_thesis: partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> then f is_partial_differentiable_in x,i by AS0, PDIFF_7:34; then ex g0 being PartFunc of (REAL-NS m),(REAL-NS n) ex y0 being Point of (REAL-NS m) st ( f = g0 & x = y0 & partdiff (f,x,i) = (partdiff (g0,y0,i)) . <*1*> ) by PDIFF_1:def_14; hence partdiff (f,x,i) = (partdiff (g,y,i)) . <*1*> by AS0, AS; ::_thesis: verum end; theorem LM03: :: PDIFF_9:22 for m, n being non empty Element of NAT for i being Element of NAT for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x0, x1 being Element of REAL m for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| proof let m, n be non empty Element of NAT ; ::_thesis: for i being Element of NAT for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x0, x1 being Element of REAL m for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x0, x1 being Element of REAL m for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x0, x1 being Element of REAL m for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x0, x1 being Element of REAL m for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i holds for x0, x1 being Element of REAL m for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| let Y be Subset of (REAL-NS m); ::_thesis: ( 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i implies for x0, x1 being Element of REAL m for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| ) assume AS0: ( 1 <= i & i <= m & X is open & g = f & X = Y & f is_partial_differentiable_on X,i ) ; ::_thesis: for x0, x1 being Element of REAL m for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| let x0, x1 be Element of REAL m; ::_thesis: for y0, y1 being Point of (REAL-NS m) st x0 = y0 & x1 = y1 & x0 in X & x1 in X holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| let y0, y1 be Point of (REAL-NS m); ::_thesis: ( x0 = y0 & x1 = y1 & x0 in X & x1 in X implies |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| ) assume AS1: ( x0 = y0 & x1 = y1 & x0 in X & x1 in X ) ; ::_thesis: |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| <*1*> is Element of REAL 1 by FINSEQ_2:98; then reconsider Pt1 = <*1*> as Point of (REAL-NS 1) by REAL_NS1:def_4; ( (f `partial| (X,i)) /. x1 = partdiff (f,x1,i) & (f `partial| (X,i)) /. x0 = partdiff (f,x0,i) ) by AS1, AS0, PDIFF_7:def_5; then ( (f `partial| (X,i)) /. x1 = (partdiff (g,y1,i)) . Pt1 & (f `partial| (X,i)) /. x0 = (partdiff (g,y0,i)) . Pt1 ) by LM02, AS0, AS1; then ((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0) = ((partdiff (g,y1,i)) . Pt1) - ((partdiff (g,y0,i)) . Pt1) by REAL_NS1:5; then A3: ((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0) = ((partdiff (g,y1,i)) - (partdiff (g,y0,i))) . Pt1 by LOPBAN_1:40; ||.Pt1.|| = abs 1 by PDIFF_8:2; then ||.Pt1.|| = 1 by ABSVALUE:def_1; then A4: ||.(((partdiff (g,y1,i)) - (partdiff (g,y0,i))) . Pt1).|| = ||.((partdiff (g,y1,i)) - (partdiff (g,y0,i))).|| * 1 by LM01; g is_partial_differentiable_on Y,i by AS0, PDIFF_7:33; then ( (g `partial| (Y,i)) /. y1 = partdiff (g,y1,i) & (g `partial| (Y,i)) /. y0 = partdiff (g,y0,i) ) by AS0, AS1, PDIFF_1:def_20; hence |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| by A3, A4, REAL_NS1:1; ::_thesis: verum end; theorem LM1Direct: :: PDIFF_9:23 for m, n being non empty Element of NAT for i being Element of NAT for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) proof let m, n be non empty Element of NAT ; ::_thesis: for i being Element of NAT for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st 1 <= i & i <= m & X is open & g = f & X = Y holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) let Y be Subset of (REAL-NS m); ::_thesis: ( 1 <= i & i <= m & X is open & g = f & X = Y implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) ) assume AS: ( 1 <= i & i <= m & X is open & g = f & X = Y ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X iff ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) hereby ::_thesis: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) assume A2: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) hence g is_partial_differentiable_on Y,i by AS, PDIFF_7:33; ::_thesis: g `partial| (Y,i) is_continuous_on Y then A3: dom (g `partial| (Y,i)) = Y by PDIFF_1:def_20; for y0 being Point of (REAL-NS m) for r being Real st y0 in Y & 0 < r holds ex s being Real st ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) ) proof let y0 be Point of (REAL-NS m); ::_thesis: for r being Real st y0 in Y & 0 < r holds ex s being Real st ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) ) let r be Real; ::_thesis: ( y0 in Y & 0 < r implies ex s being Real st ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) ) ) reconsider x0 = y0 as Element of REAL m by REAL_NS1:def_4; assume A4: ( y0 in Y & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) ) then consider s being Real such that A5: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) by AS, A2, PDIFF_7:38; take s ; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) ) thus 0 < s by A5; ::_thesis: for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r let y1 be Point of (REAL-NS m); ::_thesis: ( y1 in Y & ||.(y1 - y0).|| < s implies ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) reconsider x1 = y1 as Element of REAL m by REAL_NS1:def_4; assume A6: ( y1 in Y & ||.(y1 - y0).|| < s ) ; ::_thesis: ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r then A7: |.(x1 - x0).| < s by REAL_NS1:1, REAL_NS1:5; |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| = ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| by A4, A6, AS, A2, LM03; hence ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r by A7, A5, A6, AS; ::_thesis: verum end; hence g `partial| (Y,i) is_continuous_on Y by A3, NFCONT_1:19; ::_thesis: verum end; assume B1: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) then B2: f is_partial_differentiable_on X,i by AS, PDIFF_7:33; then B3: dom (f `partial| (X,i)) = X by PDIFF_7:def_5; for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) proof let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) ) reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4; assume B4: ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) then consider s being Real such that B5: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Y & ||.(y1 - y0).|| < s holds ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r ) ) by AS, B1, NFCONT_1:19; take s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) thus 0 < s by B5; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) reconsider y1 = x1 as Element of (REAL-NS m) by REAL_NS1:def_4; assume B6: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r |.(x1 - x0).| = ||.(y1 - y0).|| by REAL_NS1:1, REAL_NS1:5; then ||.(((g `partial| (Y,i)) /. y1) - ((g `partial| (Y,i)) /. y0)).|| < r by B5, B6, AS; hence |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r by B4, B6, AS, B2, LM03; ::_thesis: verum end; hence ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by B1, B3, AS, PDIFF_7:33, PDIFF_7:38; ::_thesis: verum end; theorem ThGdiff: :: PDIFF_9:24 for m, n being non empty Element of NAT for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ) proof let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ) let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ) let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ) let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ) let Y be Subset of (REAL-NS m); ::_thesis: ( X = Y & X is open & f = g implies ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ) ) assume AS0: ( X = Y & X is open & f = g ) ; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ) then A1: Y is open by OPEN; hereby ::_thesis: ( g is_differentiable_on Y & g `| Y is_continuous_on Y implies for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) assume AS1: for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_m_holds_ (_g_is_partial_differentiable_on_Y,i_&_g_`partial|_(Y,i)_is_continuous_on_Y_) let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) assume AS2: ( 1 <= i & i <= m ) ; ::_thesis: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) then ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS1; hence ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) by AS0, AS2, LM1Direct; ::_thesis: verum end; hence ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by A1, PDIFF_8:22; ::_thesis: verum end; assume AS3: ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) assume AS4: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) then ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) by A1, AS3, PDIFF_8:22; hence ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS0, AS4, LM1Direct; ::_thesis: verum end; theorem LM2: :: PDIFF_9:25 for m, n being non empty Element of NAT for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds ( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) proof let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds ( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds ( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds ( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st X is open & X c= dom f & g = f & X = Y holds ( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) let Y be Subset of (REAL-NS m); ::_thesis: ( X is open & X c= dom f & g = f & X = Y implies ( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) ) assume AS: ( X is open & X c= dom f & g = f & X = Y ) ; ::_thesis: ( g is_differentiable_on Y & g `| Y is_continuous_on Y iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) then O1: Y is open by OPEN; hereby ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) implies ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ) assume AS1: ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) ; ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) hence Z1: f is_differentiable_on X by AS, PDIFF_6:30; ::_thesis: for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) assume P2: ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) reconsider xx0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4; consider s being Real such that P3: ( 0 < s & ( for xx1 being Point of (REAL-NS m) st xx1 in Y & ||.(xx1 - xx0).|| < s holds ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r ) ) by AS, AS1, P2, NFCONT_1:19; take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) thus 0 < s by P3; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) assume P4: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| reconsider xx1 = x1 as Point of (REAL-NS m) by REAL_NS1:def_4; ||.(xx1 - xx0).|| < s by P4, REAL_NS1:1, REAL_NS1:5; then P5: ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r by P3, P4, AS; let v be Element of REAL m; ::_thesis: |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| reconsider vv = v as Point of (REAL-NS m) by REAL_NS1:def_4; f is_differentiable_in x0 by P2, AS, Z1, O1, PDIFF_6:32; then ex g being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st ( f = g & x0 = y & diff (f,x0) = diff (g,y) ) by PDIFF_1:def_8; then P8: ((g `| Y) /. xx0) . vv = (diff (f,x0)) . v by AS, P2, AS1, NDIFF_1:def_9; f is_differentiable_in x1 by P4, AS, Z1, O1, PDIFF_6:32; then ex g being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st ( f = g & x1 = y & diff (f,x1) = diff (g,y) ) by PDIFF_1:def_8; then P7: ((g `| Y) /. xx1) . vv = (diff (f,x1)) . v by AS, P4, AS1, NDIFF_1:def_9; reconsider g10 = ((g `| Y) /. xx1) - ((g `| Y) /. xx0) as Lipschitzian LinearOperator of (REAL-NS m),(REAL-NS n) by LOPBAN_1:def_9; (((g `| Y) /. xx1) . vv) - (((g `| Y) /. xx0) . vv) = g10 . vv by LOPBAN_1:40; then D2: ||.((((g `| Y) /. xx1) . vv) - (((g `| Y) /. xx0) . vv)).|| <= ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| * ||.vv.|| by LOPBAN_1:32; ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| * ||.vv.|| <= r * ||.vv.|| by P5, XREAL_1:64; then ||.((((g `| Y) /. xx1) . vv) - (((g `| Y) /. xx0) . vv)).|| <= r * ||.vv.|| by D2, XXREAL_0:2; then |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * ||.vv.|| by P7, P8, REAL_NS1:1, REAL_NS1:5; hence |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| by REAL_NS1:1; ::_thesis: verum end; assume P1: ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ; ::_thesis: ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) hence Z1: g is_differentiable_on Y by AS, PDIFF_6:30; ::_thesis: g `| Y is_continuous_on Y then Z2: dom (g `| Y) = Y by NDIFF_1:def_9; for x0 being Point of (REAL-NS m) for r being Real st x0 in Y & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - x0).|| < s holds ||.(((g `| Y) /. x1) - ((g `| Y) /. x0)).|| < r ) ) proof let xx0 be Point of (REAL-NS m); ::_thesis: for r being Real st xx0 in Y & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds ||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r ) ) let r0 be Real; ::_thesis: ( xx0 in Y & 0 < r0 implies ex s being Real st ( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds ||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r0 ) ) ) assume P2: ( xx0 in Y & 0 < r0 ) ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds ||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r0 ) ) set r = r0 / 2; P21: ( 0 < r0 / 2 & r0 / 2 < r0 ) by P2, XREAL_1:216; reconsider x0 = xx0 as Element of REAL m by REAL_NS1:def_4; consider s being Real such that P3: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= (r0 / 2) * |.v.| ) ) by P1, AS, P2; take s ; ::_thesis: ( 0 < s & ( for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds ||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r0 ) ) thus 0 < s by P3; ::_thesis: for x1 being Point of (REAL-NS m) st x1 in Y & ||.(x1 - xx0).|| < s holds ||.(((g `| Y) /. x1) - ((g `| Y) /. xx0)).|| < r0 let xx1 be Point of (REAL-NS m); ::_thesis: ( xx1 in Y & ||.(xx1 - xx0).|| < s implies ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r0 ) assume P4: ( xx1 in Y & ||.(xx1 - xx0).|| < s ) ; ::_thesis: ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r0 reconsider x1 = xx1 as Element of REAL m by REAL_NS1:def_4; P5: |.(x1 - x0).| < s by P4, REAL_NS1:1, REAL_NS1:5; now__::_thesis:_for_vv_being_Point_of_(REAL-NS_m)_st_||.vv.||_<=_1_holds_ ||.((((g_`|_Y)_/._xx1)_-_((g_`|_Y)_/._xx0))_._vv).||_<=_(r0_/_2)_*_||.vv.|| let vv be Point of (REAL-NS m); ::_thesis: ( ||.vv.|| <= 1 implies ||.((((g `| Y) /. xx1) - ((g `| Y) /. xx0)) . vv).|| <= (r0 / 2) * ||.vv.|| ) assume ||.vv.|| <= 1 ; ::_thesis: ||.((((g `| Y) /. xx1) - ((g `| Y) /. xx0)) . vv).|| <= (r0 / 2) * ||.vv.|| reconsider v = vv as Element of REAL m by REAL_NS1:def_4; f is_differentiable_in x0 by P2, AS, P1, O1, PDIFF_6:32; then ex g being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st ( f = g & x0 = y & diff (f,x0) = diff (g,y) ) by PDIFF_1:def_8; then P8: ((g `| Y) /. xx0) . vv = (diff (f,x0)) . v by AS, Z1, P2, NDIFF_1:def_9; f is_differentiable_in x1 by P4, AS, P1, O1, PDIFF_6:32; then ex g being PartFunc of (REAL-NS m),(REAL-NS n) ex y being Point of (REAL-NS m) st ( f = g & x1 = y & diff (f,x1) = diff (g,y) ) by PDIFF_1:def_8; then P7: ((g `| Y) /. xx1) . vv = (diff (f,x1)) . v by AS, Z1, P4, NDIFF_1:def_9; |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= (r0 / 2) * |.v.| by P5, P3, P4, AS; then |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= (r0 / 2) * ||.vv.|| by REAL_NS1:1; then ||.((((g `| Y) /. xx1) . vv) - (((g `| Y) /. xx0) . vv)).|| <= (r0 / 2) * ||.vv.|| by P7, P8, REAL_NS1:1, REAL_NS1:5; hence ||.((((g `| Y) /. xx1) - ((g `| Y) /. xx0)) . vv).|| <= (r0 / 2) * ||.vv.|| by LOPBAN_1:40; ::_thesis: verum end; then ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| <= r0 / 2 by P2, LM01CPre2; hence ||.(((g `| Y) /. xx1) - ((g `| Y) /. xx0)).|| < r0 by P21, XXREAL_0:2; ::_thesis: verum end; hence g `| Y is_continuous_on Y by Z2, NFCONT_1:19; ::_thesis: verum end; theorem CW01: :: PDIFF_9:26 for m, n being non empty Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),(REAL n) st X is open & X c= dom f holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) proof let m, n be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),(REAL n) st X is open & X c= dom f holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),(REAL n) st X is open & X c= dom f holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) let f be PartFunc of (REAL m),(REAL n); ::_thesis: ( X is open & X c= dom f implies ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) ) assume AS0: ( X is open & X c= dom f ) ; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) reconsider Y = X as Subset of (REAL-NS m) by REAL_NS1:def_4; ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4; then reconsider g = f as PartFunc of (REAL-NS m),(REAL-NS n) ; hereby ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) implies for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) assume for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) then ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by AS0, ThGdiff; hence ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) by AS0, LM2; ::_thesis: verum end; assume ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) then ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by AS0, LM2; hence for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS0, ThGdiff; ::_thesis: verum end; theorem :: PDIFF_9:27 for m, n being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) st f = g & f is_differentiable_on Z holds f `| Z = g `| Z proof let m, n be non empty Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) st f = g & f is_differentiable_on Z holds f `| Z = g `| Z let Z be set ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) st f = g & f is_differentiable_on Z holds f `| Z = g `| Z let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n) st f = g & f is_differentiable_on Z holds f `| Z = g `| Z let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: ( f = g & f is_differentiable_on Z implies f `| Z = g `| Z ) assume AS: ( f = g & f is_differentiable_on Z ) ; ::_thesis: f `| Z = g `| Z then P1: g is_differentiable_on Z by PDIFF_6:30; then P3: dom (g `| Z) = Z by NDIFF_1:def_9; AK: Z c= dom f by AS, PDIFF_6:def_4; then P2: dom (f `| Z) = Z by Def1; now__::_thesis:_for_x0_being_set_st_x0_in_dom_(f_`|_Z)_holds_ (f_`|_Z)_._x0_=_(g_`|_Z)_._x0 let x0 be set ; ::_thesis: ( x0 in dom (f `| Z) implies (f `| Z) . x0 = (g `| Z) . x0 ) assume P4: x0 in dom (f `| Z) ; ::_thesis: (f `| Z) . x0 = (g `| Z) . x0 then reconsider x = x0 as Element of REAL m ; reconsider Z1 = Z as Subset of (REAL-NS m) by P1, NDIFF_1:30; reconsider z = x as Point of (REAL-NS m) by REAL_NS1:def_4; Z1 is open by P1, NDIFF_1:32; then f is_differentiable_in x by AS, P2, P4, PDIFF_6:32; then P6: ex g0 being PartFunc of (REAL-NS m),(REAL-NS n) ex y0 being Point of (REAL-NS m) st ( f = g0 & x = y0 & diff (f,x) = diff (g0,y0) ) by PDIFF_1:def_8; thus (f `| Z) . x0 = (f `| Z) /. x by P4, PARTFUN1:def_6 .= diff (g,z) by AK, P2, P4, P6, AS, Def1 .= (g `| Z) /. x by P4, P2, P1, NDIFF_1:def_9 .= (g `| Z) . x0 by P4, P2, P3, PARTFUN1:def_6 ; ::_thesis: verum end; hence f `| Z = g `| Z by P2, P3, FUNCT_1:2; ::_thesis: verum end; theorem :: PDIFF_9:28 for m, n being non empty Element of NAT for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ) proof let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ) let f be PartFunc of (REAL m),(REAL n); ::_thesis: for g being PartFunc of (REAL-NS m),(REAL-NS n) for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ) let g be PartFunc of (REAL-NS m),(REAL-NS n); ::_thesis: for X being Subset of (REAL m) for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ) let X be Subset of (REAL m); ::_thesis: for Y being Subset of (REAL-NS m) st X = Y & X is open & f = g holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ) let Y be Subset of (REAL-NS m); ::_thesis: ( X = Y & X is open & f = g implies ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ) ) assume AS0: ( X = Y & X is open & f = g ) ; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ) then A1: Y is open by OPEN; hereby ::_thesis: ( f is_differentiable_on X & g `| Y is_continuous_on Y implies for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) assume AS1: for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( f is_differentiable_on X & g `| Y is_continuous_on Y ) now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_m_holds_ (_g_is_partial_differentiable_on_Y,i_&_g_`partial|_(Y,i)_is_continuous_on_Y_) let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) ) assume AS2: ( 1 <= i & i <= m ) ; ::_thesis: ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) then ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS1; hence ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) by AS0, AS2, LM1Direct; ::_thesis: verum end; then ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by A1, PDIFF_8:22; hence ( f is_differentiable_on X & g `| Y is_continuous_on Y ) by AS0, PDIFF_6:30; ::_thesis: verum end; assume ( f is_differentiable_on X & g `| Y is_continuous_on Y ) ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) then B2: ( g is_differentiable_on Y & g `| Y is_continuous_on Y ) by AS0, PDIFF_6:30; let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) assume AS4: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) then ( g is_partial_differentiable_on Y,i & g `partial| (Y,i) is_continuous_on Y ) by A1, B2, PDIFF_8:22; hence ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by AS0, AS4, LM1Direct; ::_thesis: verum end; theorem XTh350: :: PDIFF_9:29 for m, n being non empty Element of NAT for f, g being PartFunc of (REAL m),(REAL n) for x being Element of REAL m st f is_continuous_in x & g is_continuous_in x holds ( f + g is_continuous_in x & f - g is_continuous_in x ) proof let m, n be non empty Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) for x being Element of REAL m st f is_continuous_in x & g is_continuous_in x holds ( f + g is_continuous_in x & f - g is_continuous_in x ) let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: for x being Element of REAL m st f is_continuous_in x & g is_continuous_in x holds ( f + g is_continuous_in x & f - g is_continuous_in x ) let x be Element of REAL m; ::_thesis: ( f is_continuous_in x & g is_continuous_in x implies ( f + g is_continuous_in x & f - g is_continuous_in x ) ) assume A1: ( f is_continuous_in x & g is_continuous_in x ) ; ::_thesis: ( f + g is_continuous_in x & f - g is_continuous_in x ) reconsider y = x as Point of (REAL-NS m) by REAL_NS1:def_4; A20: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4; then reconsider f1 = f, g1 = g as PartFunc of (REAL-NS m),(REAL-NS n) ; ( f1 is_continuous_in y & g1 is_continuous_in y ) by A1, PDIFF_7:35; then A2: ( f1 + g1 is_continuous_in y & f1 - g1 is_continuous_in y ) by NFCONT_1:15; ( f + g = f1 + g1 & f - g = f1 - g1 ) by NFCONT_4:5, NFCONT_4:10, A20; hence ( f + g is_continuous_in x & f - g is_continuous_in x ) by A2, PDIFF_7:35; ::_thesis: verum end; theorem XTh351: :: PDIFF_9:30 for m, n being non empty Element of NAT for f being PartFunc of (REAL m),(REAL n) for x being Element of REAL m for r being Real st f is_continuous_in x holds r (#) f is_continuous_in x proof let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for x being Element of REAL m for r being Real st f is_continuous_in x holds r (#) f is_continuous_in x let f be PartFunc of (REAL m),(REAL n); ::_thesis: for x being Element of REAL m for r being Real st f is_continuous_in x holds r (#) f is_continuous_in x let x be Element of REAL m; ::_thesis: for r being Real st f is_continuous_in x holds r (#) f is_continuous_in x let r be Real; ::_thesis: ( f is_continuous_in x implies r (#) f is_continuous_in x ) assume A1: f is_continuous_in x ; ::_thesis: r (#) f is_continuous_in x reconsider y = x as Point of (REAL-NS m) by REAL_NS1:def_4; A20: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4; then reconsider f1 = f as PartFunc of (REAL-NS m),(REAL-NS n) ; f1 is_continuous_in y by A1, PDIFF_7:35; then A2: r (#) f1 is_continuous_in y by NFCONT_1:16; r (#) f = r (#) f1 by NFCONT_4:6, A20; hence r (#) f is_continuous_in x by A2, PDIFF_7:35; ::_thesis: verum end; theorem :: PDIFF_9:31 for m, n being non empty Element of NAT for f being PartFunc of (REAL m),(REAL n) for x being Element of REAL m st f is_continuous_in x holds - f is_continuous_in x proof let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for x being Element of REAL m st f is_continuous_in x holds - f is_continuous_in x let f be PartFunc of (REAL m),(REAL n); ::_thesis: for x being Element of REAL m st f is_continuous_in x holds - f is_continuous_in x let x be Element of REAL m; ::_thesis: ( f is_continuous_in x implies - f is_continuous_in x ) A1: - 1 is Real by XREAL_0:def_1; assume f is_continuous_in x ; ::_thesis: - f is_continuous_in x then (- 1) (#) f is_continuous_in x by A1, XTh351; hence - f is_continuous_in x by NFCONT_4:7; ::_thesis: verum end; theorem YTh354: :: PDIFF_9:32 for m, n being non empty Element of NAT for f being PartFunc of (REAL m),(REAL n) for x being Element of REAL m st f is_continuous_in x holds |.f.| is_continuous_in x proof let m, n be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),(REAL n) for x being Element of REAL m st f is_continuous_in x holds |.f.| is_continuous_in x let f be PartFunc of (REAL m),(REAL n); ::_thesis: for x being Element of REAL m st f is_continuous_in x holds |.f.| is_continuous_in x let x be Element of REAL m; ::_thesis: ( f is_continuous_in x implies |.f.| is_continuous_in x ) assume A1: f is_continuous_in x ; ::_thesis: |.f.| is_continuous_in x reconsider y = x as Point of (REAL-NS m) by REAL_NS1:def_4; A20: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4; then reconsider f1 = f as PartFunc of (REAL-NS m),(REAL-NS n) ; f1 is_continuous_in y by A1, PDIFF_7:35; then A2: ||.f1.|| is_continuous_in y by NFCONT_1:17; |.f.| = ||.f1.|| by NFCONT_4:9, A20; hence |.f.| is_continuous_in x by A2, NFCONT_4:21; ::_thesis: verum end; theorem XTh350X: :: PDIFF_9:33 for m, n being non empty Element of NAT for Z being set for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z & g is_continuous_on Z holds ( f + g is_continuous_on Z & f - g is_continuous_on Z ) proof let m, n be non empty Element of NAT ; ::_thesis: for Z being set for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z & g is_continuous_on Z holds ( f + g is_continuous_on Z & f - g is_continuous_on Z ) let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z & g is_continuous_on Z holds ( f + g is_continuous_on Z & f - g is_continuous_on Z ) let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_continuous_on Z & g is_continuous_on Z implies ( f + g is_continuous_on Z & f - g is_continuous_on Z ) ) assume A1: ( f is_continuous_on Z & g is_continuous_on Z ) ; ::_thesis: ( f + g is_continuous_on Z & f - g is_continuous_on Z ) A2: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4; then reconsider f1 = f, g1 = g as PartFunc of (REAL-NS m),(REAL-NS n) ; ( f1 is_continuous_on Z & g1 is_continuous_on Z ) by A1, PDIFF_7:37; then A3: ( f1 + g1 is_continuous_on Z & f1 - g1 is_continuous_on Z ) by NFCONT_1:25; ( f + g = f1 + g1 & f - g = f1 - g1 ) by NFCONT_4:5, NFCONT_4:10, A2; hence ( f + g is_continuous_on Z & f - g is_continuous_on Z ) by A3, PDIFF_7:37; ::_thesis: verum end; theorem XTh351X: :: PDIFF_9:34 for m, n being non empty Element of NAT for Z being set for r being Real for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds r (#) f is_continuous_on Z proof let m, n be non empty Element of NAT ; ::_thesis: for Z being set for r being Real for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds r (#) f is_continuous_on Z let Z be set ; ::_thesis: for r being Real for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds r (#) f is_continuous_on Z let r be Real; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds r (#) f is_continuous_on Z let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_continuous_on Z implies r (#) f is_continuous_on Z ) assume A1: f is_continuous_on Z ; ::_thesis: r (#) f is_continuous_on Z A2: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4; then reconsider f1 = f as PartFunc of (REAL-NS m),(REAL-NS n) ; f1 is_continuous_on Z by A1, PDIFF_7:37; then A3: r (#) f1 is_continuous_on Z by NFCONT_1:27; r (#) f1 = r (#) f by NFCONT_4:6, A2; hence r (#) f is_continuous_on Z by A3, PDIFF_7:37; ::_thesis: verum end; theorem :: PDIFF_9:35 for m, n being non empty Element of NAT for Z being set for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds - f is_continuous_on Z proof let m, n be non empty Element of NAT ; ::_thesis: for Z being set for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds - f is_continuous_on Z let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds - f is_continuous_on Z let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_continuous_on Z implies - f is_continuous_on Z ) assume A1: f is_continuous_on Z ; ::_thesis: - f is_continuous_on Z - 1 is Real by XREAL_0:def_1; then (- 1) (#) f is_continuous_on Z by A1, XTh351X; hence - f is_continuous_on Z by NFCONT_4:7; ::_thesis: verum end; theorem XDef60: :: PDIFF_9:36 for i being Element of NAT for f being PartFunc of (REAL i),REAL for x0 being Element of REAL i holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds ex s being Real st ( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds |.((f /. x) - (f /. x0)).| < r ) ) ) ) ) proof let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL for x0 being Element of REAL i holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds ex s being Real st ( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds |.((f /. x) - (f /. x0)).| < r ) ) ) ) ) let f be PartFunc of (REAL i),REAL; ::_thesis: for x0 being Element of REAL i holds ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds ex s being Real st ( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds |.((f /. x) - (f /. x0)).| < r ) ) ) ) ) let x0 be Element of REAL i; ::_thesis: ( f is_continuous_in x0 iff ( x0 in dom f & ( for r being Real st 0 < r holds ex s being Real st ( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds |.((f /. x) - (f /. x0)).| < r ) ) ) ) ) hereby ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds ex s being Real st ( 0 < s & ( for x being Element of REAL i st x in dom f & |.(x - x0).| < s holds |.((f /. x) - (f /. x0)).| < r ) ) ) implies f is_continuous_in x0 ) assume f is_continuous_in x0 ; ::_thesis: ( x0 in dom f & ( for r being Real st 0 < r holds ex s being Real st ( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds |.((f /. a) - (f /. x0)).| < r ) ) ) ) then consider y0 being Point of (REAL-NS i), g being PartFunc of (REAL-NS i),REAL such that P1: ( x0 = y0 & f = g & g is_continuous_in y0 ) by NFCONT_4:def_4; thus x0 in dom f by P1, NFCONT_1:8; ::_thesis: for r being Real st 0 < r holds ex s being Real st ( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds |.((f /. a) - (f /. x0)).| < r ) ) let r be Real; ::_thesis: ( 0 < r implies ex s being Real st ( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds |.((f /. a) - (f /. x0)).| < r ) ) ) assume 0 < r ; ::_thesis: ex s being Real st ( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds |.((f /. a) - (f /. x0)).| < r ) ) then consider s being Real such that P3: ( 0 < s & ( for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r ) ) by P1, NFCONT_1:8; take s = s; ::_thesis: ( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds |.((f /. a) - (f /. x0)).| < r ) ) thus 0 < s by P3; ::_thesis: for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds |.((f /. a) - (f /. x0)).| < r let a be Element of REAL i; ::_thesis: ( a in dom f & |.(a - x0).| < s implies |.((f /. a) - (f /. x0)).| < r ) assume P4: ( a in dom f & |.(a - x0).| < s ) ; ::_thesis: |.((f /. a) - (f /. x0)).| < r reconsider y1 = a as Point of (REAL-NS i) by REAL_NS1:def_4; ||.(y1 - y0).|| = |.(a - x0).| by P1, REAL_NS1:1, REAL_NS1:5; hence |.((f /. a) - (f /. x0)).| < r by P1, P3, P4; ::_thesis: verum end; assume P1: ( x0 in dom f & ( for r being Real st 0 < r holds ex s being Real st ( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds |.((f /. a) - (f /. x0)).| < r ) ) ) ) ; ::_thesis: f is_continuous_in x0 reconsider y0 = x0 as Point of (REAL-NS i) by REAL_NS1:def_4; reconsider g = f as PartFunc of (REAL-NS i),REAL by REAL_NS1:def_4; now__::_thesis:_for_r_being_Real_st_0_<_r_holds_ ex_s_being_Real_st_ (_0_<_s_&_(_for_y1_being_Point_of_(REAL-NS_i)_st_y1_in_dom_g_&_||.(y1_-_y0).||_<_s_holds_ |.((g_/._y1)_-_(g_/._y0)).|_<_r_)_) let r be Real; ::_thesis: ( 0 < r implies ex s being Real st ( 0 < s & ( for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r ) ) ) assume 0 < r ; ::_thesis: ex s being Real st ( 0 < s & ( for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r ) ) then consider s being Real such that P3: ( 0 < s & ( for a being Element of REAL i st a in dom f & |.(a - x0).| < s holds |.((f /. a) - (f /. x0)).| < r ) ) by P1; take s = s; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r ) ) thus 0 < s by P3; ::_thesis: for y1 being Point of (REAL-NS i) st y1 in dom g & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r hereby ::_thesis: verum let y1 be Point of (REAL-NS i); ::_thesis: ( y1 in dom g & ||.(y1 - y0).|| < s implies |.((g /. y1) - (g /. y0)).| < r ) assume P4: ( y1 in dom g & ||.(y1 - y0).|| < s ) ; ::_thesis: |.((g /. y1) - (g /. y0)).| < r reconsider a = y1 as Element of REAL i by REAL_NS1:def_4; ||.(y1 - y0).|| = |.(a - x0).| by REAL_NS1:1, REAL_NS1:5; hence |.((g /. y1) - (g /. y0)).| < r by P3, P4; ::_thesis: verum end; end; then g is_continuous_in y0 by P1, NFCONT_1:8; hence f is_continuous_in x0 by NFCONT_4:def_4; ::_thesis: verum end; theorem XTh35: :: PDIFF_9:37 for m being non empty Element of NAT for f being PartFunc of (REAL m),REAL for x0 being Element of REAL m holds ( f is_continuous_in x0 iff <>* f is_continuous_in x0 ) proof let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL for x0 being Element of REAL m holds ( f is_continuous_in x0 iff <>* f is_continuous_in x0 ) let f be PartFunc of (REAL m),REAL; ::_thesis: for x0 being Element of REAL m holds ( f is_continuous_in x0 iff <>* f is_continuous_in x0 ) let x0 be Element of REAL m; ::_thesis: ( f is_continuous_in x0 iff <>* f is_continuous_in x0 ) set g = <>* f; hereby ::_thesis: ( <>* f is_continuous_in x0 implies f is_continuous_in x0 ) assume P1: f is_continuous_in x0 ; ::_thesis: <>* f is_continuous_in x0 then P2: x0 in dom f by XDef60; then P3: x0 in dom (<>* f) by LMXTh0; now__::_thesis:_for_r_being_Real_st_0_<_r_holds_ ex_s_being_Real_st_ (_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_dom_(<>*_f)_&_|.(x1_-_x0).|_<_s_holds_ |.(((<>*_f)_/._x1)_-_((<>*_f)_/._x0)).|_<_r_)_) let r be Real; ::_thesis: ( 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) ) ) assume 0 < r ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) ) then consider s being Real such that P5: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) by P1, XDef60; take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) ) thus 0 < s by P5; ::_thesis: for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r hereby ::_thesis: verum let x1 be Element of REAL m; ::_thesis: ( x1 in dom (<>* f) & |.(x1 - x0).| < s implies |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) assume P6: ( x1 in dom (<>* f) & |.(x1 - x0).| < s ) ; ::_thesis: |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r then P8: x1 in dom f by LMXTh0; then P7: |.((f /. x1) - (f /. x0)).| < r by P5, P6; ( (<>* f) /. x1 = <*(f /. x1)*> & (<>* f) /. x0 = <*(f /. x0)*> ) by P2, P8, XTh30; then ((<>* f) /. x1) - ((<>* f) /. x0) = <*((f /. x1) - (f /. x0))*> by RVSUM_1:29; hence |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r by P7, XTh30D; ::_thesis: verum end; end; hence <>* f is_continuous_in x0 by P3, PDIFF_7:36; ::_thesis: verum end; assume A1: <>* f is_continuous_in x0 ; ::_thesis: f is_continuous_in x0 then x0 in dom (<>* f) by PDIFF_7:36; then P2: x0 in dom f by LMXTh0; now__::_thesis:_for_r_being_Real_st_0_<_r_holds_ ex_s_being_Real_st_ (_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_dom_f_&_|.(x1_-_x0).|_<_s_holds_ |.((f_/._x1)_-_(f_/._x0)).|_<_r_)_) let r be Real; ::_thesis: ( 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ) assume 0 < r ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) then consider s being Real such that P4: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom (<>* f) & |.(x1 - x0).| < s holds |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) ) by A1, PDIFF_7:36; take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) thus 0 < s by P4; ::_thesis: for x1 being Element of REAL m st x1 in dom f & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r hereby ::_thesis: verum let x1 be Element of REAL m; ::_thesis: ( x1 in dom f & |.(x1 - x0).| < s implies |.((f /. x1) - (f /. x0)).| < r ) assume P5: ( x1 in dom f & |.(x1 - x0).| < s ) ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r then x1 in dom (<>* f) by LMXTh0; then P6: |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r by P4, P5; ( (<>* f) /. x1 = <*(f /. x1)*> & (<>* f) /. x0 = <*(f /. x0)*> ) by P2, P5, XTh30; then ((<>* f) /. x1) - ((<>* f) /. x0) = <*((f /. x1) - (f /. x0))*> by RVSUM_1:29; hence |.((f /. x1) - (f /. x0)).| < r by P6, XTh30D; ::_thesis: verum end; end; hence f is_continuous_in x0 by P2, XDef60; ::_thesis: verum end; theorem :: PDIFF_9:38 for m being non empty Element of NAT for f, g being PartFunc of (REAL m),REAL for x0 being Element of REAL m st f is_continuous_in x0 & g is_continuous_in x0 holds ( f + g is_continuous_in x0 & f - g is_continuous_in x0 ) proof let m be non empty Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL for x0 being Element of REAL m st f is_continuous_in x0 & g is_continuous_in x0 holds ( f + g is_continuous_in x0 & f - g is_continuous_in x0 ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: for x0 being Element of REAL m st f is_continuous_in x0 & g is_continuous_in x0 holds ( f + g is_continuous_in x0 & f - g is_continuous_in x0 ) let x0 be Element of REAL m; ::_thesis: ( f is_continuous_in x0 & g is_continuous_in x0 implies ( f + g is_continuous_in x0 & f - g is_continuous_in x0 ) ) assume ( f is_continuous_in x0 & g is_continuous_in x0 ) ; ::_thesis: ( f + g is_continuous_in x0 & f - g is_continuous_in x0 ) then ( <>* f is_continuous_in x0 & <>* g is_continuous_in x0 ) by XTh35; then A2: ( (<>* f) + (<>* g) is_continuous_in x0 & (<>* f) - (<>* g) is_continuous_in x0 ) by XTh350; ( (<>* f) + (<>* g) = <>* (f + g) & (<>* f) - (<>* g) = <>* (f - g) ) by LMXTh10; hence ( f + g is_continuous_in x0 & f - g is_continuous_in x0 ) by A2, XTh35; ::_thesis: verum end; theorem :: PDIFF_9:39 for m being non empty Element of NAT for f being PartFunc of (REAL m),REAL for x0 being Element of REAL m for r being Real st f is_continuous_in x0 holds r (#) f is_continuous_in x0 proof let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL for x0 being Element of REAL m for r being Real st f is_continuous_in x0 holds r (#) f is_continuous_in x0 let f be PartFunc of (REAL m),REAL; ::_thesis: for x0 being Element of REAL m for r being Real st f is_continuous_in x0 holds r (#) f is_continuous_in x0 let x0 be Element of REAL m; ::_thesis: for r being Real st f is_continuous_in x0 holds r (#) f is_continuous_in x0 let r be Real; ::_thesis: ( f is_continuous_in x0 implies r (#) f is_continuous_in x0 ) assume f is_continuous_in x0 ; ::_thesis: r (#) f is_continuous_in x0 then <>* f is_continuous_in x0 by XTh35; then A2: r (#) (<>* f) is_continuous_in x0 by XTh351; r (#) (<>* f) = <>* (r (#) f) by LMXTh11; hence r (#) f is_continuous_in x0 by A2, XTh35; ::_thesis: verum end; theorem :: PDIFF_9:40 for m being non empty Element of NAT for f being PartFunc of (REAL m),REAL for x0 being Element of REAL m st f is_continuous_in x0 holds |.f.| is_continuous_in x0 proof let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL for x0 being Element of REAL m st f is_continuous_in x0 holds |.f.| is_continuous_in x0 let f be PartFunc of (REAL m),REAL; ::_thesis: for x0 being Element of REAL m st f is_continuous_in x0 holds |.f.| is_continuous_in x0 let x0 be Element of REAL m; ::_thesis: ( f is_continuous_in x0 implies |.f.| is_continuous_in x0 ) assume f is_continuous_in x0 ; ::_thesis: |.f.| is_continuous_in x0 then <>* f is_continuous_in x0 by XTh35; then |.(<>* f).| is_continuous_in x0 by YTh354; hence |.f.| is_continuous_in x0 by LMXTh13; ::_thesis: verum end; XTh360: for i being Element of NAT for f being PartFunc of (REAL i),REAL for g being PartFunc of (REAL-NS i),REAL for x being Element of REAL i for y being Point of (REAL-NS i) st f = g & x = y holds ( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds ( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ) proof let i be Element of NAT ; ::_thesis: for f being PartFunc of (REAL i),REAL for g being PartFunc of (REAL-NS i),REAL for x being Element of REAL i for y being Point of (REAL-NS i) st f = g & x = y holds ( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds ( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ) let f be PartFunc of (REAL i),REAL; ::_thesis: for g being PartFunc of (REAL-NS i),REAL for x being Element of REAL i for y being Point of (REAL-NS i) st f = g & x = y holds ( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds ( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ) let g be PartFunc of (REAL-NS i),REAL; ::_thesis: for x being Element of REAL i for y being Point of (REAL-NS i) st f = g & x = y holds ( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds ( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ) let x be Element of REAL i; ::_thesis: for y being Point of (REAL-NS i) st f = g & x = y holds ( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds ( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ) let y be Point of (REAL-NS i); ::_thesis: ( f = g & x = y implies ( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds ( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ) ) assume AS: ( f = g & x = y ) ; ::_thesis: ( f is_continuous_in x iff ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds ( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ) hereby ::_thesis: ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds ( g /* s is convergent & g /. y = lim (g /* s) ) ) implies f is_continuous_in x ) assume f is_continuous_in x ; ::_thesis: ( y in dom g & ( for s1 being sequence of (REAL-NS i) st rng s1 c= dom g & s1 is convergent & lim s1 = y holds ( g /* s1 is convergent & g /. y = lim (g /* s1) ) ) ) then g is_continuous_in y by AS, NFCONT_4:21; hence ( y in dom g & ( for s1 being sequence of (REAL-NS i) st rng s1 c= dom g & s1 is convergent & lim s1 = y holds ( g /* s1 is convergent & g /. y = lim (g /* s1) ) ) ) by NFCONT_1:def_6; ::_thesis: verum end; hereby ::_thesis: verum assume ( y in dom g & ( for s being sequence of (REAL-NS i) st rng s c= dom g & s is convergent & lim s = y holds ( g /* s is convergent & g /. y = lim (g /* s) ) ) ) ; ::_thesis: f is_continuous_in x then g is_continuous_in y by NFCONT_1:def_6; hence f is_continuous_in x by AS, NFCONT_4:21; ::_thesis: verum end; end; theorem :: PDIFF_9:41 for i being Element of NAT for f, g being PartFunc of (REAL i),REAL for x being Element of REAL i st f is_continuous_in x & g is_continuous_in x holds f (#) g is_continuous_in x proof let i be Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL i),REAL for x being Element of REAL i st f is_continuous_in x & g is_continuous_in x holds f (#) g is_continuous_in x let g1, g2 be PartFunc of (REAL i),REAL; ::_thesis: for x being Element of REAL i st g1 is_continuous_in x & g2 is_continuous_in x holds g1 (#) g2 is_continuous_in x let x be Element of REAL i; ::_thesis: ( g1 is_continuous_in x & g2 is_continuous_in x implies g1 (#) g2 is_continuous_in x ) assume A2: ( g1 is_continuous_in x & g2 is_continuous_in x ) ; ::_thesis: g1 (#) g2 is_continuous_in x reconsider y = x as Point of (REAL-NS i) by REAL_NS1:def_4; reconsider f1 = g1, f2 = g2 as PartFunc of (REAL-NS i),REAL by REAL_NS1:def_4; A3: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by VALUED_1:def_4; ( f1 is_continuous_in y & f2 is_continuous_in y ) by A2, NFCONT_4:21; then X1: ( y in dom f1 & y in dom f2 ) by NFCONT_1:def_6; then X2: y in dom (f1 (#) f2) by A3, XBOOLE_0:def_4; now__::_thesis:_for_s1_being_sequence_of_(REAL-NS_i)_st_rng_s1_c=_dom_(f1_(#)_f2)_&_s1_is_convergent_&_lim_s1_=_y_holds_ (_(f1_(#)_f2)_/*_s1_is_convergent_&_(f1_(#)_f2)_/._y_=_lim_((f1_(#)_f2)_/*_s1)_) let s1 be sequence of (REAL-NS i); ::_thesis: ( rng s1 c= dom (f1 (#) f2) & s1 is convergent & lim s1 = y implies ( (f1 (#) f2) /* s1 is convergent & (f1 (#) f2) /. y = lim ((f1 (#) f2) /* s1) ) ) assume that A22: rng s1 c= dom (f1 (#) f2) and A23: ( s1 is convergent & lim s1 = y ) ; ::_thesis: ( (f1 (#) f2) /* s1 is convergent & (f1 (#) f2) /. y = lim ((f1 (#) f2) /* s1) ) ( dom (f1 (#) f2) c= dom f1 & dom (f1 (#) f2) c= dom f2 ) by A3, XBOOLE_1:17; then A24: ( rng s1 c= dom f1 & rng s1 c= dom f2 ) by A22, XBOOLE_1:1; then A25: ( f1 /* s1 is convergent & f2 /* s1 is convergent ) by A2, A23, XTh360; then (f1 /* s1) (#) (f2 /* s1) is convergent ; hence (f1 (#) f2) /* s1 is convergent by A3, A22, RFUNCT_2:8; ::_thesis: (f1 (#) f2) /. y = lim ((f1 (#) f2) /* s1) ( f1 . y = f1 /. y & f2 . y = f2 /. y ) by X1, PARTFUN1:def_6; then A29: ( f1 . y = lim (f1 /* s1) & f2 . y = lim (f2 /* s1) ) by A2, A23, A24, XTh360; thus (f1 (#) f2) /. y = (f1 (#) f2) . y by X2, PARTFUN1:def_6 .= (f1 . y) * (f2 . y) by VALUED_1:5 .= lim ((f1 /* s1) (#) (f2 /* s1)) by A29, A25, SEQ_2:15 .= lim ((f1 (#) f2) /* s1) by A3, A22, RFUNCT_2:8 ; ::_thesis: verum end; hence g1 (#) g2 is_continuous_in x by XTh360, X2; ::_thesis: verum end; definition let m be non empty Element of NAT ; let Z be set ; let f be PartFunc of (REAL m),REAL; predf is_continuous_on Z means :XDef7: :: PDIFF_9:def 2 for x0 being Element of REAL m st x0 in Z holds f | Z is_continuous_in x0; end; :: deftheorem XDef7 defines is_continuous_on PDIFF_9:def_2_:_ for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL holds ( f is_continuous_on Z iff for x0 being Element of REAL m st x0 in Z holds f | Z is_continuous_in x0 ); theorem XTh360B: :: PDIFF_9:42 for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL-NS m),REAL st f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL-NS m),REAL st f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL-NS m),REAL st f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL-NS m),REAL st f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) let g be PartFunc of (REAL-NS m),REAL; ::_thesis: ( f = g implies ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) ) assume AS: f = g ; ::_thesis: ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) hereby ::_thesis: ( g is_continuous_on Z implies ( Z c= dom f & f is_continuous_on Z ) ) assume P2: Z c= dom f ; ::_thesis: ( f is_continuous_on Z implies g is_continuous_on Z ) assume P0: f is_continuous_on Z ; ::_thesis: g is_continuous_on Z now__::_thesis:_for_x0_being_Point_of_(REAL-NS_m)_st_x0_in_Z_holds_ g_|_Z_is_continuous_in_x0 let x0 be Point of (REAL-NS m); ::_thesis: ( x0 in Z implies g | Z is_continuous_in x0 ) assume P3: x0 in Z ; ::_thesis: g | Z is_continuous_in x0 reconsider y0 = x0 as Element of REAL m by REAL_NS1:def_4; f | Z is_continuous_in y0 by P3, P0, XDef7; hence g | Z is_continuous_in x0 by AS, NFCONT_4:21; ::_thesis: verum end; hence g is_continuous_on Z by AS, P2, NFCONT_1:def_8; ::_thesis: verum end; assume P1: g is_continuous_on Z ; ::_thesis: ( Z c= dom f & f is_continuous_on Z ) hence Z c= dom f by AS, NFCONT_1:def_8; ::_thesis: f is_continuous_on Z let x0 be Element of REAL m; :: according to PDIFF_9:def_2 ::_thesis: ( x0 in Z implies f | Z is_continuous_in x0 ) assume P3: x0 in Z ; ::_thesis: f | Z is_continuous_in x0 reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4; g | Z is_continuous_in y0 by P3, P1, NFCONT_1:def_8; hence f | Z is_continuous_in x0 by AS, NFCONT_4:21; ::_thesis: verum end; theorem XTh360C: :: PDIFF_9:43 for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL-NS m),REAL st f = g & Z c= dom f holds ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds ( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL-NS m),REAL st f = g & Z c= dom f holds ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds ( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL-NS m),REAL st f = g & Z c= dom f holds ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds ( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL-NS m),REAL st f = g & Z c= dom f holds ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds ( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) let g be PartFunc of (REAL-NS m),REAL; ::_thesis: ( f = g & Z c= dom f implies ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds ( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) ) assume AS: f = g ; ::_thesis: ( not Z c= dom f or ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds ( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) ) assume A0: Z c= dom f ; ::_thesis: ( f is_continuous_on Z iff for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds ( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) hereby ::_thesis: ( ( for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds ( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ) implies f is_continuous_on Z ) assume f is_continuous_on Z ; ::_thesis: for s1 being sequence of (REAL-NS m) st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( g /* s1 is convergent & g /. (lim s1) = lim (g /* s1) ) then g is_continuous_on Z by A0, XTh360B, AS; hence for s1 being sequence of (REAL-NS m) st rng s1 c= Z & s1 is convergent & lim s1 in Z holds ( g /* s1 is convergent & g /. (lim s1) = lim (g /* s1) ) by NFCONT125; ::_thesis: verum end; assume for s being sequence of (REAL-NS m) st rng s c= Z & s is convergent & lim s in Z holds ( g /* s is convergent & g /. (lim s) = lim (g /* s) ) ; ::_thesis: f is_continuous_on Z then g is_continuous_on Z by AS, A0, NFCONT125; hence f is_continuous_on Z by XTh360B, AS; ::_thesis: verum end; theorem XTh37: :: PDIFF_9:44 for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g implies ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) ) assume A1: <>* f = g ; ::_thesis: ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) then A0: <>* (f | Z) = g | Z by LMXTh1; hereby ::_thesis: ( g is_continuous_on Z implies ( Z c= dom f & f is_continuous_on Z ) ) assume AK: Z c= dom f ; ::_thesis: ( f is_continuous_on Z implies g is_continuous_on Z ) assume A2: f is_continuous_on Z ; ::_thesis: g is_continuous_on Z A3: Z c= dom g by LMXTh0, A1, AK; now__::_thesis:_for_x0_being_Element_of_REAL_m_st_x0_in_Z_holds_ g_|_Z_is_continuous_in_x0 let x0 be Element of REAL m; ::_thesis: ( x0 in Z implies g | Z is_continuous_in x0 ) assume x0 in Z ; ::_thesis: g | Z is_continuous_in x0 then f | Z is_continuous_in x0 by A2, XDef7; hence g | Z is_continuous_in x0 by A0, XTh35; ::_thesis: verum end; hence g is_continuous_on Z by A3, PDIFF_7:def_7; ::_thesis: verum end; assume A5: g is_continuous_on Z ; ::_thesis: ( Z c= dom f & f is_continuous_on Z ) then Z c= dom g by PDIFF_7:def_7; hence Z c= dom f by LMXTh0, A1; ::_thesis: f is_continuous_on Z let x0 be Element of REAL m; :: according to PDIFF_9:def_2 ::_thesis: ( x0 in Z implies f | Z is_continuous_in x0 ) assume x0 in Z ; ::_thesis: f | Z is_continuous_in x0 then g | Z is_continuous_in x0 by A5, PDIFF_7:def_7; hence f | Z is_continuous_in x0 by A0, XTh35; ::_thesis: verum end; theorem XTh38: :: PDIFF_9:45 for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL st Z c= dom f holds ( f is_continuous_on Z iff for x0 being Element of REAL m for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ) proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL st Z c= dom f holds ( f is_continuous_on Z iff for x0 being Element of REAL m for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ) let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL st Z c= dom f holds ( f is_continuous_on Z iff for x0 being Element of REAL m for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ) let f be PartFunc of (REAL m),REAL; ::_thesis: ( Z c= dom f implies ( f is_continuous_on Z iff for x0 being Element of REAL m for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ) ) set g = <>* f; assume A2: Z c= dom f ; ::_thesis: ( f is_continuous_on Z iff for x0 being Element of REAL m for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ) hereby ::_thesis: ( ( for x0 being Element of REAL m for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ) implies f is_continuous_on Z ) assume f is_continuous_on Z ; ::_thesis: for x0 being Element of REAL m for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) then A1: <>* f is_continuous_on Z by A2, XTh37; thus for x0 being Element of REAL m for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ::_thesis: verum proof let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) let r be Real; ::_thesis: ( x0 in Z & 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ) assume A3: ( x0 in Z & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) then consider s being Real such that A4: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) ) by A1, PDIFF_7:38; take s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) thus 0 < s by A4; ::_thesis: for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r hereby ::_thesis: verum let x1 be Element of REAL m; ::_thesis: ( x1 in Z & |.(x1 - x0).| < s implies |.((f /. x1) - (f /. x0)).| < r ) assume A5: ( x1 in Z & |.(x1 - x0).| < s ) ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r then A6: |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r by A4; ( (<>* f) /. x1 = <*(f /. x1)*> & (<>* f) /. x0 = <*(f /. x0)*> ) by A5, A2, A3, XTh30; then ((<>* f) /. x1) - ((<>* f) /. x0) = <*((f /. x1) - (f /. x0))*> by RVSUM_1:29; hence |.((f /. x1) - (f /. x0)).| < r by A6, XTh30D; ::_thesis: verum end; end; end; assume A7: for x0 being Element of REAL m for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ; ::_thesis: f is_continuous_on Z A70: Z c= dom (<>* f) by A2, LMXTh0; for y0 being Element of REAL m for r being Real st y0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - y0).| < s holds |.(((<>* f) /. y1) - ((<>* f) /. y0)).| < r ) ) proof let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds |.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r ) ) let r be Real; ::_thesis: ( x0 in Z & 0 < r implies ex s being Real st ( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds |.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r ) ) ) assume A8: ( x0 in Z & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds |.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r ) ) then consider s being Real such that A9: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) by A7; take s ; ::_thesis: ( 0 < s & ( for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds |.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r ) ) thus 0 < s by A9; ::_thesis: for y1 being Element of REAL m st y1 in Z & |.(y1 - x0).| < s holds |.(((<>* f) /. y1) - ((<>* f) /. x0)).| < r hereby ::_thesis: verum let x1 be Element of REAL m; ::_thesis: ( x1 in Z & |.(x1 - x0).| < s implies |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r ) assume A10: ( x1 in Z & |.(x1 - x0).| < s ) ; ::_thesis: |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r then A11: |.((f /. x1) - (f /. x0)).| < r by A9; ( (<>* f) /. x1 = <*(f /. x1)*> & (<>* f) /. x0 = <*(f /. x0)*> ) by A2, A10, A8, XTh30; then ((<>* f) /. x1) - ((<>* f) /. x0) = <*((f /. x1) - (f /. x0))*> by RVSUM_1:29; hence |.(((<>* f) /. x1) - ((<>* f) /. x0)).| < r by A11, XTh30D; ::_thesis: verum end; end; then <>* f is_continuous_on Z by A70, PDIFF_7:38; hence f is_continuous_on Z by XTh37; ::_thesis: verum end; theorem :: PDIFF_9:46 for m being non empty Element of NAT for Z being set for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds ( f + g is_continuous_on Z & f - g is_continuous_on Z ) proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds ( f + g is_continuous_on Z & f - g is_continuous_on Z ) let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds ( f + g is_continuous_on Z & f - g is_continuous_on Z ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g implies ( f + g is_continuous_on Z & f - g is_continuous_on Z ) ) assume ( f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g ) ; ::_thesis: ( f + g is_continuous_on Z & f - g is_continuous_on Z ) then ( <>* f is_continuous_on Z & <>* g is_continuous_on Z ) by XTh37; then P2: ( (<>* f) + (<>* g) is_continuous_on Z & (<>* f) - (<>* g) is_continuous_on Z ) by XTh350X; ( (<>* f) + (<>* g) = <>* (f + g) & (<>* f) - (<>* g) = <>* (f - g) ) by LMXTh10; hence ( f + g is_continuous_on Z & f - g is_continuous_on Z ) by P2, XTh37; ::_thesis: verum end; theorem :: PDIFF_9:47 for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL for r being Real st Z c= dom f & f is_continuous_on Z holds r (#) f is_continuous_on Z proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL for r being Real st Z c= dom f & f is_continuous_on Z holds r (#) f is_continuous_on Z let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL for r being Real st Z c= dom f & f is_continuous_on Z holds r (#) f is_continuous_on Z let f be PartFunc of (REAL m),REAL; ::_thesis: for r being Real st Z c= dom f & f is_continuous_on Z holds r (#) f is_continuous_on Z let r be Real; ::_thesis: ( Z c= dom f & f is_continuous_on Z implies r (#) f is_continuous_on Z ) assume ( Z c= dom f & f is_continuous_on Z ) ; ::_thesis: r (#) f is_continuous_on Z then <>* f is_continuous_on Z by XTh37; then P2: r (#) (<>* f) is_continuous_on Z by XTh351X; r (#) (<>* f) = <>* (r (#) f) by LMXTh11; hence r (#) f is_continuous_on Z by P2, XTh37; ::_thesis: verum end; theorem :: PDIFF_9:48 for m being non empty Element of NAT for Z being set for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds f (#) g is_continuous_on Z proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds f (#) g is_continuous_on Z let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g holds f (#) g is_continuous_on Z let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g implies f (#) g is_continuous_on Z ) assume A1: ( f is_continuous_on Z & g is_continuous_on Z ) ; ::_thesis: ( not Z c= dom f or not Z c= dom g or f (#) g is_continuous_on Z ) assume AK: ( Z c= dom f & Z c= dom g ) ; ::_thesis: f (#) g is_continuous_on Z reconsider f1 = f, g1 = g as PartFunc of (REAL-NS m),REAL by REAL_NS1:def_4; P2: Z c= (dom f1) /\ (dom g1) by AK, XBOOLE_1:19; AA: dom (f1 (#) g1) = (dom f1) /\ (dom g1) by VALUED_1:def_4; now__::_thesis:_for_s1_being_sequence_of_(REAL-NS_m)_st_rng_s1_c=_Z_&_s1_is_convergent_&_lim_s1_in_Z_holds_ (_(f1_(#)_g1)_/*_s1_is_convergent_&_(f1_(#)_g1)_/._(lim_s1)_=_lim_((f1_(#)_g1)_/*_s1)_) let s1 be sequence of (REAL-NS m); ::_thesis: ( rng s1 c= Z & s1 is convergent & lim s1 in Z implies ( (f1 (#) g1) /* s1 is convergent & (f1 (#) g1) /. (lim s1) = lim ((f1 (#) g1) /* s1) ) ) assume A23: ( rng s1 c= Z & s1 is convergent & lim s1 in Z ) ; ::_thesis: ( (f1 (#) g1) /* s1 is convergent & (f1 (#) g1) /. (lim s1) = lim ((f1 (#) g1) /* s1) ) then A25: ( f1 /* s1 is convergent & g1 /* s1 is convergent ) by AK, XTh360C, A1; then A28: (f1 /* s1) (#) (g1 /* s1) is convergent ; A26: rng s1 c= (dom f1) /\ (dom g1) by P2, A23, XBOOLE_1:1; hence (f1 (#) g1) /* s1 is convergent by A28, RFUNCT_2:8; ::_thesis: (f1 (#) g1) /. (lim s1) = lim ((f1 (#) g1) /* s1) set y = lim s1; ( f1 . (lim s1) = f1 /. (lim s1) & g1 . (lim s1) = g1 /. (lim s1) ) by A23, AK, PARTFUN1:def_6; then A29: ( f1 . (lim s1) = lim (f1 /* s1) & g1 . (lim s1) = lim (g1 /* s1) ) by A23, AK, XTh360C, A1; thus (f1 (#) g1) /. (lim s1) = (f1 (#) g1) . (lim s1) by A23, P2, AA, PARTFUN1:def_6 .= (f1 . (lim s1)) * (g1 . (lim s1)) by VALUED_1:5 .= lim ((f1 /* s1) (#) (g1 /* s1)) by A29, A25, SEQ_2:15 .= lim ((f1 (#) g1) /* s1) by A26, RFUNCT_2:8 ; ::_thesis: verum end; hence f (#) g is_continuous_on Z by P2, AA, XTh360C; ::_thesis: verum end; theorem PDIFF736X: :: PDIFF_9:49 for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL-NS m),REAL st f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL-NS m),REAL st f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL-NS m),REAL st f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL-NS m),REAL st f = g holds ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) let g be PartFunc of (REAL-NS m),REAL; ::_thesis: ( f = g implies ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) ) assume AS0: f = g ; ::_thesis: ( ( Z c= dom f & f is_continuous_on Z ) iff g is_continuous_on Z ) hereby ::_thesis: ( g is_continuous_on Z implies ( Z c= dom f & f is_continuous_on Z ) ) assume P0: Z c= dom f ; ::_thesis: ( f is_continuous_on Z implies g is_continuous_on Z ) assume Q0: f is_continuous_on Z ; ::_thesis: g is_continuous_on Z now__::_thesis:_for_y0_being_Point_of_(REAL-NS_m) for_r_being_Real_st_y0_in_Z_&_0_<_r_holds_ ex_s_being_Real_st_ (_0_<_s_&_(_for_y1_being_Point_of_(REAL-NS_m)_st_y1_in_Z_&_||.(y1_-_y0).||_<_s_holds_ |.((g_/._y1)_-_(g_/._y0)).|_<_r_)_) let y0 be Point of (REAL-NS m); ::_thesis: for r being Real st y0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r ) ) let r be Real; ::_thesis: ( y0 in Z & 0 < r implies ex s being Real st ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r ) ) ) reconsider x0 = y0 as Element of REAL m by REAL_NS1:def_4; assume ( y0 in Z & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r ) ) then consider s being Real such that A7: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) by P0, Q0, XTh38; take s = s; ::_thesis: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r ) ) thus 0 < s by A7; ::_thesis: for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r let y1 be Point of (REAL-NS m); ::_thesis: ( y1 in Z & ||.(y1 - y0).|| < s implies |.((g /. y1) - (g /. y0)).| < r ) assume A8: ( y1 in Z & ||.(y1 - y0).|| < s ) ; ::_thesis: |.((g /. y1) - (g /. y0)).| < r reconsider x1 = y1 as Element of REAL m by REAL_NS1:def_4; ||.(y1 - y0).|| = |.(x1 - x0).| by REAL_NS1:1, REAL_NS1:5; hence |.((g /. y1) - (g /. y0)).| < r by AS0, A8, A7; ::_thesis: verum end; hence g is_continuous_on Z by AS0, P0, NFCONT_1:20; ::_thesis: verum end; assume Q1: g is_continuous_on Z ; ::_thesis: ( Z c= dom f & f is_continuous_on Z ) then A60: Z c= dom f by AS0, NFCONT_1:20; now__::_thesis:_for_x0_being_Element_of_REAL_m for_r_being_Real_st_x0_in_Z_&_0_<_r_holds_ ex_s_being_Real_st_ (_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_Z_&_|.(x1_-_x0).|_<_s_holds_ |.((f_/._x1)_-_(f_/._x0)).|_<_r_)_) let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in Z & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) let r be Real; ::_thesis: ( x0 in Z & 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) ) reconsider y0 = x0 as Point of (REAL-NS m) by REAL_NS1:def_4; assume ( x0 in Z & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) then consider s being Real such that A7: ( 0 < s & ( for y1 being Point of (REAL-NS m) st y1 in Z & ||.(y1 - y0).|| < s holds |.((g /. y1) - (g /. y0)).| < r ) ) by Q1, NFCONT_1:20; take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r ) ) thus 0 < s by A7; ::_thesis: for x1 being Element of REAL m st x1 in Z & |.(x1 - x0).| < s holds |.((f /. x1) - (f /. x0)).| < r let x1 be Element of REAL m; ::_thesis: ( x1 in Z & |.(x1 - x0).| < s implies |.((f /. x1) - (f /. x0)).| < r ) assume A8: ( x1 in Z & |.(x1 - x0).| < s ) ; ::_thesis: |.((f /. x1) - (f /. x0)).| < r reconsider y1 = x1 as Point of (REAL-NS m) by REAL_NS1:def_4; ||.(y1 - y0).|| = |.(x1 - x0).| by REAL_NS1:1, REAL_NS1:5; hence |.((f /. x1) - (f /. x0)).| < r by AS0, A8, A7; ::_thesis: verum end; hence ( Z c= dom f & f is_continuous_on Z ) by A60, XTh38; ::_thesis: verum end; theorem :: PDIFF_9:50 for m, n being non empty Element of NAT for Z being set for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds |.f.| is_continuous_on Z proof let m, n be non empty Element of NAT ; ::_thesis: for Z being set for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds |.f.| is_continuous_on Z let Z be set ; ::_thesis: for f, g being PartFunc of (REAL m),(REAL n) st f is_continuous_on Z holds |.f.| is_continuous_on Z let f, g be PartFunc of (REAL m),(REAL n); ::_thesis: ( f is_continuous_on Z implies |.f.| is_continuous_on Z ) assume A1: f is_continuous_on Z ; ::_thesis: |.f.| is_continuous_on Z A2: ( the carrier of (REAL-NS m) = REAL m & the carrier of (REAL-NS n) = REAL n ) by REAL_NS1:def_4; then reconsider f1 = f as PartFunc of (REAL-NS m),(REAL-NS n) ; f1 is_continuous_on Z by A1, PDIFF_7:37; then A3: ||.f1.|| is_continuous_on Z by NFCONT_1:28; ||.f1.|| = |.f.| by NFCONT_4:9, A2; hence |.f.| is_continuous_on Z by A3, PDIFF736X; ::_thesis: verum end; theorem PDIFF620X: :: PDIFF_9:51 for m being non empty Element of NAT for f, g being PartFunc of (REAL m),REAL for x being Element of REAL m st f is_differentiable_in x & g is_differentiable_in x holds ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) proof let m be non empty Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL for x being Element of REAL m st f is_differentiable_in x & g is_differentiable_in x holds ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: for x being Element of REAL m st f is_differentiable_in x & g is_differentiable_in x holds ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) let x be Element of REAL m; ::_thesis: ( f is_differentiable_in x & g is_differentiable_in x implies ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) ) assume ( f is_differentiable_in x & g is_differentiable_in x ) ; ::_thesis: ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) then P2: ( <>* f is_differentiable_in x & <>* g is_differentiable_in x ) by PDIFF_7:def_1; then P4: ( (<>* f) + (<>* g) is_differentiable_in x & (<>* f) - (<>* g) is_differentiable_in x ) by PDIFF_6:20, PDIFF_6:21; (<>* f) + (<>* g) = <>* (f + g) by LMXTh10; hence f + g is_differentiable_in x by P4, PDIFF_7:def_1; ::_thesis: ( diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) & f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) thus diff ((f + g),x) = (proj (1,1)) * (diff (((<>* f) + (<>* g)),x)) by LMXTh10 .= (proj (1,1)) * ((diff ((<>* f),x)) + (diff ((<>* g),x))) by P2, PDIFF_6:20 .= (diff (f,x)) + (diff (g,x)) by INTEGR15:15 ; ::_thesis: ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) (<>* f) - (<>* g) = <>* (f - g) by LMXTh10; hence f - g is_differentiable_in x by P4, PDIFF_7:def_1; ::_thesis: diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) thus diff ((f - g),x) = (proj (1,1)) * (diff (((<>* f) - (<>* g)),x)) by LMXTh10 .= (proj (1,1)) * ((diff ((<>* f),x)) - (diff ((<>* g),x))) by P2, PDIFF_6:21 .= (diff (f,x)) - (diff (g,x)) by INTEGR15:15 ; ::_thesis: verum end; theorem PDIFF622X: :: PDIFF_9:52 for m being non empty Element of NAT for f being PartFunc of (REAL m),REAL for r being Real for x being Element of REAL m st f is_differentiable_in x holds ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) proof let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL for r being Real for x being Element of REAL m st f is_differentiable_in x holds ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) let f be PartFunc of (REAL m),REAL; ::_thesis: for r being Real for x being Element of REAL m st f is_differentiable_in x holds ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) let r be Real; ::_thesis: for x being Element of REAL m st f is_differentiable_in x holds ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) let x be Element of REAL m; ::_thesis: ( f is_differentiable_in x implies ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) ) assume f is_differentiable_in x ; ::_thesis: ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) then P1: <>* f is_differentiable_in x by PDIFF_7:def_1; then P2: r (#) (<>* f) is_differentiable_in x by PDIFF_6:22; r (#) (<>* f) = <>* (r (#) f) by LMXTh11; hence r (#) f is_differentiable_in x by P2, PDIFF_7:def_1; ::_thesis: diff ((r (#) f),x) = r (#) (diff (f,x)) thus diff ((r (#) f),x) = (proj (1,1)) * (diff ((r (#) (<>* f)),x)) by LMXTh11 .= (proj (1,1)) * (r (#) (diff ((<>* f),x))) by P1, PDIFF_6:22 .= r (#) (diff (f,x)) by INTEGR15:16 ; ::_thesis: verum end; definition let Z be set ; let m be non empty Element of NAT ; let f be PartFunc of (REAL m),REAL; predf is_differentiable_on Z means :XDef4: :: PDIFF_9:def 3 for x being Element of REAL m st x in Z holds f | Z is_differentiable_in x; end; :: deftheorem XDef4 defines is_differentiable_on PDIFF_9:def_3_:_ for Z being set for m being non empty Element of NAT for f being PartFunc of (REAL m),REAL holds ( f is_differentiable_on Z iff for x being Element of REAL m st x in Z holds f | Z is_differentiable_in x ); theorem YTh30: :: PDIFF_9:53 for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds ( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z ) proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds ( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z ) let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds ( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z ) let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g holds ( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z ) let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g implies ( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z ) ) assume A1: <>* f = g ; ::_thesis: ( ( Z c= dom f & f is_differentiable_on Z ) iff g is_differentiable_on Z ) AN: dom (<>* f) = dom f by LMXTh0; hereby ::_thesis: ( g is_differentiable_on Z implies ( Z c= dom f & f is_differentiable_on Z ) ) assume AK: Z c= dom f ; ::_thesis: ( f is_differentiable_on Z implies g is_differentiable_on Z ) assume A3: f is_differentiable_on Z ; ::_thesis: g is_differentiable_on Z A40: Z c= dom g by AK, LMXTh0, A1; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_Z_holds_ g_|_Z_is_differentiable_in_x let x be Element of REAL m; ::_thesis: ( x in Z implies g | Z is_differentiable_in x ) assume x in Z ; ::_thesis: g | Z is_differentiable_in x then f | Z is_differentiable_in x by A3, XDef4; then <>* (f | Z) is_differentiable_in x by PDIFF_7:def_1; hence g | Z is_differentiable_in x by A1, LMXTh1; ::_thesis: verum end; hence g is_differentiable_on Z by A40, PDIFF_6:def_4; ::_thesis: verum end; assume A5: g is_differentiable_on Z ; ::_thesis: ( Z c= dom f & f is_differentiable_on Z ) hence Z c= dom f by AN, A1, PDIFF_6:def_4; ::_thesis: f is_differentiable_on Z hereby :: according to PDIFF_9:def_3 ::_thesis: verum let x be Element of REAL m; ::_thesis: ( x in Z implies f | Z is_differentiable_in x ) assume x in Z ; ::_thesis: f | Z is_differentiable_in x then A60: g | Z is_differentiable_in x by A5, PDIFF_6:def_4; g | Z = <>* (f | Z) by A1, LMXTh1; hence f | Z is_differentiable_in x by A60, PDIFF_7:def_1; ::_thesis: verum end; end; theorem YTh32: :: PDIFF_9:54 for m being non empty Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL st X c= dom f & X is open holds ( f is_differentiable_on X iff for x being Element of REAL m st x in X holds f is_differentiable_in x ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL st X c= dom f & X is open holds ( f is_differentiable_on X iff for x being Element of REAL m st x in X holds f is_differentiable_in x ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL st X c= dom f & X is open holds ( f is_differentiable_on X iff for x being Element of REAL m st x in X holds f is_differentiable_in x ) let f be PartFunc of (REAL m),REAL; ::_thesis: ( X c= dom f & X is open implies ( f is_differentiable_on X iff for x being Element of REAL m st x in X holds f is_differentiable_in x ) ) set g = <>* f; assume AK: X c= dom f ; ::_thesis: ( not X is open or ( f is_differentiable_on X iff for x being Element of REAL m st x in X holds f is_differentiable_in x ) ) assume X is open ; ::_thesis: ( f is_differentiable_on X iff for x being Element of REAL m st x in X holds f is_differentiable_in x ) then ex Z0 being Subset of (REAL-NS m) st ( Z0 = X & Z0 is open ) by PDIFF_7:def_3; then A2: ( <>* f is_differentiable_on X iff ( X c= dom (<>* f) & ( for x being Element of REAL m st x in X holds <>* f is_differentiable_in x ) ) ) by PDIFF_6:32; hereby ::_thesis: ( ( for x being Element of REAL m st x in X holds f is_differentiable_in x ) implies f is_differentiable_on X ) assume A3: f is_differentiable_on X ; ::_thesis: for x being Element of REAL m st x in X holds f is_differentiable_in x let x be Element of REAL m; ::_thesis: ( x in X implies f is_differentiable_in x ) assume x in X ; ::_thesis: f is_differentiable_in x then <>* f is_differentiable_in x by AK, A2, A3, YTh30; hence f is_differentiable_in x by PDIFF_7:def_1; ::_thesis: verum end; assume A4: for x being Element of REAL m st x in X holds f is_differentiable_in x ; ::_thesis: f is_differentiable_on X now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ <>*_f_is_differentiable_in_x let x be Element of REAL m; ::_thesis: ( x in X implies <>* f is_differentiable_in x ) assume x in X ; ::_thesis: <>* f is_differentiable_in x then f is_differentiable_in x by A4; hence <>* f is_differentiable_in x by PDIFF_7:def_1; ::_thesis: verum end; hence f is_differentiable_on X by AK, A2, LMXTh0, YTh30; ::_thesis: verum end; theorem YTh33: :: PDIFF_9:55 for m being non empty Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL st X c= dom f & f is_differentiable_on X holds X is open proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL st X c= dom f & f is_differentiable_on X holds X is open let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL st X c= dom f & f is_differentiable_on X holds X is open let f be PartFunc of (REAL m),REAL; ::_thesis: ( X c= dom f & f is_differentiable_on X implies X is open ) reconsider g = <>* f as PartFunc of (REAL m),(REAL 1) ; assume ( X c= dom f & f is_differentiable_on X ) ; ::_thesis: X is open then g is_differentiable_on X by YTh30; then ex Z0 being Subset of (REAL-NS m) st ( X = Z0 & Z0 is open ) by PDIFF_6:33; hence X is open by PDIFF_7:def_3; ::_thesis: verum end; definition let m be non empty Element of NAT ; let Z be set ; let f be PartFunc of (REAL m),REAL; assume AK: Z c= dom f ; funcf `| Z -> PartFunc of (REAL m),(Funcs ((REAL m),REAL)) means :XDef1: :: PDIFF_9:def 4 ( dom it = Z & ( for x being Element of REAL m st x in Z holds it /. x = diff (f,x) ) ); existence ex b1 being PartFunc of (REAL m),(Funcs ((REAL m),REAL)) st ( dom b1 = Z & ( for x being Element of REAL m st x in Z holds b1 /. x = diff (f,x) ) ) proof defpred S1[ Element of REAL m, set ] means ( \$1 in Z & \$2 = diff (f,\$1) ); consider F being PartFunc of (REAL m),(Funcs ((REAL m),REAL)) such that A2: ( ( for x being Element of REAL m holds ( x in dom F iff ex z being Element of Funcs ((REAL m),REAL) st S1[x,z] ) ) & ( for x being Element of REAL m st x in dom F holds S1[x,F . x] ) ) from SEQ_1:sch_2(); take F ; ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds F /. x = diff (f,x) ) ) A3: Z is Subset of (REAL m) by AK, XBOOLE_1:1; now__::_thesis:_for_x_being_set_st_x_in_Z_holds_ x_in_dom_F let x be set ; ::_thesis: ( x in Z implies x in dom F ) assume AS1: x in Z ; ::_thesis: x in dom F then reconsider z = x as Element of REAL m by A3; reconsider y = diff (f,z) as Element of Funcs ((REAL m),REAL) by FUNCT_2:8; S1[z,y] by AS1; hence x in dom F by A2; ::_thesis: verum end; then A4: Z c= dom F by TARSKI:def_3; for y being set st y in dom F holds y in Z by A2; then dom F c= Z by TARSKI:def_3; hence dom F = Z by A4, XBOOLE_0:def_10; ::_thesis: for x being Element of REAL m st x in Z holds F /. x = diff (f,x) hereby ::_thesis: verum let x be Element of REAL m; ::_thesis: ( x in Z implies F /. x = diff (f,x) ) assume A5: x in Z ; ::_thesis: F /. x = diff (f,x) then F . x = diff (f,x) by A2, A4; hence F /. x = diff (f,x) by A5, A4, PARTFUN1:def_6; ::_thesis: verum end; end; uniqueness for b1, b2 being PartFunc of (REAL m),(Funcs ((REAL m),REAL)) st dom b1 = Z & ( for x being Element of REAL m st x in Z holds b1 /. x = diff (f,x) ) & dom b2 = Z & ( for x being Element of REAL m st x in Z holds b2 /. x = diff (f,x) ) holds b1 = b2 proof let F, G be PartFunc of (REAL m),(Funcs ((REAL m),REAL)); ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds F /. x = diff (f,x) ) & dom G = Z & ( for x being Element of REAL m st x in Z holds G /. x = diff (f,x) ) implies F = G ) assume that A6: ( dom F = Z & ( for x being Element of REAL m st x in Z holds F /. x = diff (f,x) ) ) and A8: ( dom G = Z & ( for x being Element of REAL m st x in Z holds G /. x = diff (f,x) ) ) ; ::_thesis: F = G now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_dom_F_holds_ F_/._x_=_G_/._x let x be Element of REAL m; ::_thesis: ( x in dom F implies F /. x = G /. x ) assume A10: x in dom F ; ::_thesis: F /. x = G /. x then F /. x = diff (f,x) by A6; hence F /. x = G /. x by A6, A8, A10; ::_thesis: verum end; hence F = G by A6, A8, PARTFUN2:1; ::_thesis: verum end; end; :: deftheorem XDef1 defines `| PDIFF_9:def_4_:_ for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL st Z c= dom f holds for b4 being PartFunc of (REAL m),(Funcs ((REAL m),REAL)) holds ( b4 = f `| Z iff ( dom b4 = Z & ( for x being Element of REAL m st x in Z holds b4 /. x = diff (f,x) ) ) ); theorem :: PDIFF_9:56 for m being non empty Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X c= dom f & f is_differentiable_on X holds ( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X c= dom f & f is_differentiable_on X holds ( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X c= dom f & f is_differentiable_on X holds ( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) ) let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X c= dom f & f is_differentiable_on X holds ( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) ) let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g & X c= dom f & f is_differentiable_on X implies ( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) ) ) assume AS: ( <>* f = g & X c= dom f & f is_differentiable_on X ) ; ::_thesis: ( g is_differentiable_on X & ( for x being Element of REAL m st x in X holds (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) ) hence g is_differentiable_on X by YTh30; ::_thesis: for x being Element of REAL m st x in X holds (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) AA: dom f = dom (<>* f) by LMXTh0; let x be Element of REAL m; ::_thesis: ( x in X implies (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) ) assume P4: x in X ; ::_thesis: (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) then (f `| X) /. x = diff (f,x) by AS, XDef1; hence (f `| X) /. x = (proj (1,1)) * ((g `| X) /. x) by AA, AS, P4, Def1; ::_thesis: verum end; theorem :: PDIFF_9:57 for m being non empty Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) ) let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X implies ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) ) ) assume AK: ( X c= dom f & X c= dom g ) ; ::_thesis: ( not f is_differentiable_on X or not g is_differentiable_on X or ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) ) ) assume AS1: ( f is_differentiable_on X & g is_differentiable_on X ) ; ::_thesis: ( f + g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) ) then P0: X is open by AK, YTh33; dom (f + g) = (dom f) /\ (dom g) by VALUED_1:def_1; then P3: X c= dom (f + g) by AK, XBOOLE_1:19; P5: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ (_f_+_g_is_differentiable_in_x_&_diff_((f_+_g),x)_=_(diff_(f,x))_+_(diff_(g,x))_) let x be Element of REAL m; ::_thesis: ( x in X implies ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) ) ) assume x in X ; ::_thesis: ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) ) then ( f is_differentiable_in x & g is_differentiable_in x ) by AK, AS1, P0, YTh32; hence ( f + g is_differentiable_in x & diff ((f + g),x) = (diff (f,x)) + (diff (g,x)) ) by PDIFF620X; ::_thesis: verum end; then for x being Element of REAL m st x in X holds f + g is_differentiable_in x ; hence f + g is_differentiable_on X by P3, P0, YTh32; ::_thesis: for x being Element of REAL m st x in X holds ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) let x be Element of REAL m; ::_thesis: ( x in X implies ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) ) assume P7: x in X ; ::_thesis: ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) then ((f + g) `| X) /. x = diff ((f + g),x) by P3, XDef1; then ((f + g) `| X) /. x = (diff (f,x)) + (diff (g,x)) by P7, P5; then ((f + g) `| X) /. x = ((f `| X) /. x) + (diff (g,x)) by AK, P7, XDef1; hence ((f + g) `| X) /. x = ((f `| X) /. x) + ((g `| X) /. x) by AK, P7, XDef1; ::_thesis: verum end; theorem :: PDIFF_9:58 for m being non empty Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X holds ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X c= dom f & X c= dom g & f is_differentiable_on X & g is_differentiable_on X implies ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) ) assume AK: ( X c= dom f & X c= dom g ) ; ::_thesis: ( not f is_differentiable_on X or not g is_differentiable_on X or ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) ) assume AS1: ( f is_differentiable_on X & g is_differentiable_on X ) ; ::_thesis: ( f - g is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) ) then P0: X is open by AK, YTh33; dom (f - g) = (dom f) /\ (dom g) by VALUED_1:12; then P3: X c= dom (f - g) by AK, XBOOLE_1:19; P5: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ (_f_-_g_is_differentiable_in_x_&_diff_((f_-_g),x)_=_(diff_(f,x))_-_(diff_(g,x))_) let x be Element of REAL m; ::_thesis: ( x in X implies ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) ) assume x in X ; ::_thesis: ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) then ( f is_differentiable_in x & g is_differentiable_in x ) by AK, AS1, P0, YTh32; hence ( f - g is_differentiable_in x & diff ((f - g),x) = (diff (f,x)) - (diff (g,x)) ) by PDIFF620X; ::_thesis: verum end; then for x being Element of REAL m st x in X holds f - g is_differentiable_in x ; hence f - g is_differentiable_on X by P3, P0, YTh32; ::_thesis: for x being Element of REAL m st x in X holds ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) let x be Element of REAL m; ::_thesis: ( x in X implies ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) ) assume P7: x in X ; ::_thesis: ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) then ((f - g) `| X) /. x = diff ((f - g),x) by P3, XDef1; then ((f - g) `| X) /. x = (diff (f,x)) - (diff (g,x)) by P7, P5; then ((f - g) `| X) /. x = ((f `| X) /. x) - (diff (g,x)) by AK, P7, XDef1; hence ((f - g) `| X) /. x = ((f `| X) /. x) - ((g `| X) /. x) by AK, P7, XDef1; ::_thesis: verum end; theorem :: PDIFF_9:59 for m being non empty Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for r being Real st X c= dom f & f is_differentiable_on X holds ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for r being Real st X c= dom f & f is_differentiable_on X holds ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL for r being Real st X c= dom f & f is_differentiable_on X holds ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) ) let f be PartFunc of (REAL m),REAL; ::_thesis: for r being Real st X c= dom f & f is_differentiable_on X holds ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) ) let r be Real; ::_thesis: ( X c= dom f & f is_differentiable_on X implies ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) ) ) assume AK: X c= dom f ; ::_thesis: ( not f is_differentiable_on X or ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) ) ) assume AS1: f is_differentiable_on X ; ::_thesis: ( r (#) f is_differentiable_on X & ( for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) ) then P0: X is open by AK, YTh33; P3: X c= dom (r (#) f) by AK, VALUED_1:def_5; P5: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ (_r_(#)_f_is_differentiable_in_x_&_diff_((r_(#)_f),x)_=_r_(#)_(diff_(f,x))_) let x be Element of REAL m; ::_thesis: ( x in X implies ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) ) assume x in X ; ::_thesis: ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) then f is_differentiable_in x by AS1, P0, AK, YTh32; hence ( r (#) f is_differentiable_in x & diff ((r (#) f),x) = r (#) (diff (f,x)) ) by PDIFF622X; ::_thesis: verum end; then for x being Element of REAL m st x in X holds r (#) f is_differentiable_in x ; hence r (#) f is_differentiable_on X by P3, P0, YTh32; ::_thesis: for x being Element of REAL m st x in X holds ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) let x be Element of REAL m; ::_thesis: ( x in X implies ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) ) assume P7: x in X ; ::_thesis: ((r (#) f) `| X) /. x = r (#) ((f `| X) /. x) then ((r (#) f) `| X) /. x = diff ((r (#) f),x) by P3, XDef1; hence ((r (#) f) `| X) /. x = r (#) (diff (f,x)) by P7, P5 .= r (#) ((f `| X) /. x) by AK, P7, XDef1 ; ::_thesis: verum end; definition let m be non empty Element of NAT ; let Z be set ; let i be Element of NAT ; let f be PartFunc of (REAL m),REAL; predf is_partial_differentiable_on Z,i means :CWDef19: :: PDIFF_9:def 5 ( Z c= dom f & ( for x being Element of REAL m st x in Z holds f | Z is_partial_differentiable_in x,i ) ); end; :: deftheorem CWDef19 defines is_partial_differentiable_on PDIFF_9:def_5_:_ for m being non empty Element of NAT for Z being set for i being Element of NAT for f being PartFunc of (REAL m),REAL holds ( f is_partial_differentiable_on Z,i iff ( Z c= dom f & ( for x being Element of REAL m st x in Z holds f | Z is_partial_differentiable_in x,i ) ) ); definition let m be non empty Element of NAT ; let Z be set ; let i be Element of NAT ; let f be PartFunc of (REAL m),REAL; assume A1: f is_partial_differentiable_on Z,i ; funcf `partial| (Z,i) -> PartFunc of (REAL m),REAL means :DefPDX: :: PDIFF_9:def 6 ( dom it = Z & ( for x being Element of REAL m st x in Z holds it /. x = partdiff (f,x,i) ) ); existence ex b1 being PartFunc of (REAL m),REAL st ( dom b1 = Z & ( for x being Element of REAL m st x in Z holds b1 /. x = partdiff (f,x,i) ) ) proof deffunc H1( Element of REAL m) -> Element of REAL = partdiff (f,\$1,i); defpred S1[ Element of REAL m] means \$1 in Z; consider F being PartFunc of (REAL m),REAL such that A2: ( ( for x being Element of REAL m holds ( x in dom F iff S1[x] ) ) & ( for x being Element of REAL m st x in dom F holds F . x = H1(x) ) ) from SEQ_1:sch_3(); take F ; ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds F /. x = partdiff (f,x,i) ) ) now__::_thesis:_for_y_being_set_st_y_in_Z_holds_ y_in_dom_F Z c= dom f by A1, CWDef19; then A3: Z is Subset of (REAL m) by XBOOLE_1:1; let y be set ; ::_thesis: ( y in Z implies y in dom F ) assume y in Z ; ::_thesis: y in dom F hence y in dom F by A2, A3; ::_thesis: verum end; then A4: Z c= dom F by TARSKI:def_3; for y being set st y in dom F holds y in Z by A2; then dom F c= Z by TARSKI:def_3; hence dom F = Z by A4, XBOOLE_0:def_10; ::_thesis: for x being Element of REAL m st x in Z holds F /. x = partdiff (f,x,i) hereby ::_thesis: verum let x be Element of REAL m; ::_thesis: ( x in Z implies F /. x = partdiff (f,x,i) ) assume x in Z ; ::_thesis: F /. x = partdiff (f,x,i) then A5: x in dom F by A2; then F . x = partdiff (f,x,i) by A2; hence F /. x = partdiff (f,x,i) by A5, PARTFUN1:def_6; ::_thesis: verum end; end; uniqueness for b1, b2 being PartFunc of (REAL m),REAL st dom b1 = Z & ( for x being Element of REAL m st x in Z holds b1 /. x = partdiff (f,x,i) ) & dom b2 = Z & ( for x being Element of REAL m st x in Z holds b2 /. x = partdiff (f,x,i) ) holds b1 = b2 proof let F, G be PartFunc of (REAL m),REAL; ::_thesis: ( dom F = Z & ( for x being Element of REAL m st x in Z holds F /. x = partdiff (f,x,i) ) & dom G = Z & ( for x being Element of REAL m st x in Z holds G /. x = partdiff (f,x,i) ) implies F = G ) assume that A6: ( dom F = Z & ( for x being Element of REAL m st x in Z holds F /. x = partdiff (f,x,i) ) ) and A8: ( dom G = Z & ( for x being Element of REAL m st x in Z holds G /. x = partdiff (f,x,i) ) ) ; ::_thesis: F = G now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_dom_F_holds_ F_/._x_=_G_/._x let x be Element of REAL m; ::_thesis: ( x in dom F implies F /. x = G /. x ) assume A10: x in dom F ; ::_thesis: F /. x = G /. x then F /. x = partdiff (f,x,i) by A6; hence F /. x = G /. x by A6, A8, A10; ::_thesis: verum end; hence F = G by A6, A8, PARTFUN2:1; ::_thesis: verum end; end; :: deftheorem DefPDX defines `partial| PDIFF_9:def_6_:_ for m being non empty Element of NAT for Z being set for i being Element of NAT for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_on Z,i holds for b5 being PartFunc of (REAL m),REAL holds ( b5 = f `partial| (Z,i) iff ( dom b5 = Z & ( for x being Element of REAL m st x in Z holds b5 /. x = partdiff (f,x,i) ) ) ); theorem PDIFF734: :: PDIFF_9:60 for m being non empty Element of NAT for i being Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i ) ) ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i ) ) ) let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i ) ) ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i ) ) ) let f be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i ) ) ) ) assume that A1: X is open and A2: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i iff ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i ) ) ) thus ( f is_partial_differentiable_on X,i implies ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i ) ) ) ::_thesis: ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i ) implies f is_partial_differentiable_on X,i ) proof assume A3: f is_partial_differentiable_on X,i ; ::_thesis: ( X c= dom f & ( for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i ) ) hence A4: X c= dom f by CWDef19; ::_thesis: for x being Element of REAL m st x in X holds f is_partial_differentiable_in x,i let nx0 be Element of REAL m; ::_thesis: ( nx0 in X implies f is_partial_differentiable_in nx0,i ) reconsider x0 = (proj (i,m)) . nx0 as Element of REAL ; assume A5: nx0 in X ; ::_thesis: f is_partial_differentiable_in nx0,i then f | X is_partial_differentiable_in nx0,i by A3, CWDef19; then (f | X) * (reproj (i,nx0)) is_differentiable_in x0 by PDIFF_1:def_11; then consider N0 being Neighbourhood of x0 such that A6: N0 c= dom ((f | X) * (reproj (i,nx0))) and A7: ex L being LinearFunc ex R being RestFunc st for x being Real st x in N0 holds (((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by FDIFF_1:def_4; consider L being LinearFunc, R being RestFunc such that A8: for x being Element of REAL st x in N0 holds (((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by A7; consider N1 being Neighbourhood of x0 such that A9: for x being Element of REAL st x in N1 holds (reproj (i,nx0)) . x in X by A1, A2, A5, Lm5; A10: now__::_thesis:_for_x_being_Element_of_REAL_st_x_in_N1_holds_ (reproj_(i,nx0))_._x_in_dom_(f_|_X) let x be Element of REAL ; ::_thesis: ( x in N1 implies (reproj (i,nx0)) . x in dom (f | X) ) assume x in N1 ; ::_thesis: (reproj (i,nx0)) . x in dom (f | X) then (reproj (i,nx0)) . x in X by A9; then (reproj (i,nx0)) . x in (dom f) /\ X by A4, XBOOLE_0:def_4; hence (reproj (i,nx0)) . x in dom (f | X) by RELAT_1:61; ::_thesis: verum end; consider N being Neighbourhood of x0 such that NXX: ( N c= N0 & N c= N1 ) by RCOMP_1:17; (f | X) * (reproj (i,nx0)) c= f * (reproj (i,nx0)) by RELAT_1:29, RELAT_1:59; then A11: dom ((f | X) * (reproj (i,nx0))) c= dom (f * (reproj (i,nx0))) by RELAT_1:11; N c= dom ((f | X) * (reproj (i,nx0))) by A6, NXX, XBOOLE_1:1; then A12: N c= dom (f * (reproj (i,nx0))) by A11, XBOOLE_1:1; now__::_thesis:_for_x_being_Element_of_REAL_st_x_in_N_holds_ ((f_*_(reproj_(i,nx0)))_._x)_-_((f_*_(reproj_(i,nx0)))_._x0)_=_(L_._(x_-_x0))_+_(R_._(x_-_x0)) let x be Element of REAL ; ::_thesis: ( x in N implies ((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) ) assume A13: x in N ; ::_thesis: ((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) then (reproj (i,nx0)) . x in dom (f | X) by A10, NXX; then A17: ( (reproj (i,nx0)) . x in dom f & (reproj (i,nx0)) . x in X ) by RELAT_1:57; (reproj (i,nx0)) . x0 in dom (f | X) by A10, RCOMP_1:16; then A19: ( (reproj (i,nx0)) . x0 in dom f & (reproj (i,nx0)) . x0 in X ) by RELAT_1:57; A15: dom (reproj (i,nx0)) = REAL by FUNCT_2:def_1; then A20: ((f | X) * (reproj (i,nx0))) . x = (f | X) . ((reproj (i,nx0)) . x) by FUNCT_1:13 .= f . ((reproj (i,nx0)) . x) by A17, FUNCT_1:49 .= (f * (reproj (i,nx0))) . x by A15, FUNCT_1:13 ; ((f | X) * (reproj (i,nx0))) . x0 = (f | X) . ((reproj (i,nx0)) . x0) by A15, FUNCT_1:13 .= f . ((reproj (i,nx0)) . x0) by A19, FUNCT_1:49 .= (f * (reproj (i,nx0))) . x0 by A15, FUNCT_1:13 ; hence ((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by A8, A13, NXX, A20; ::_thesis: verum end; then f * (reproj (i,nx0)) is_differentiable_in x0 by A12, FDIFF_1:def_4; hence f is_partial_differentiable_in nx0,i by PDIFF_1:def_11; ::_thesis: verum end; assume that A21: X c= dom f and A22: for nx being Element of REAL m st nx in X holds f is_partial_differentiable_in nx,i ; ::_thesis: f is_partial_differentiable_on X,i thus X c= dom f by A21; :: according to PDIFF_9:def_5 ::_thesis: for x being Element of REAL m st x in X holds f | X is_partial_differentiable_in x,i now__::_thesis:_for_nx0_being_Element_of_REAL_m_st_nx0_in_X_holds_ f_|_X_is_partial_differentiable_in_nx0,i let nx0 be Element of REAL m; ::_thesis: ( nx0 in X implies f | X is_partial_differentiable_in nx0,i ) assume A23: nx0 in X ; ::_thesis: f | X is_partial_differentiable_in nx0,i then A24: f is_partial_differentiable_in nx0,i by A22; reconsider x0 = (proj (i,m)) . nx0 as Element of REAL ; f * (reproj (i,nx0)) is_differentiable_in x0 by A24, PDIFF_1:def_11; then consider N0 being Neighbourhood of x0 such that N0 c= dom (f * (reproj (i,nx0))) and A25: ex L being LinearFunc ex R being RestFunc st for x being Element of REAL st x in N0 holds ((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by FDIFF_1:def_4; consider N1 being Neighbourhood of x0 such that A26: for x being Element of REAL st x in N1 holds (reproj (i,nx0)) . x in X by A1, A2, A23, Lm5; A27: now__::_thesis:_for_x_being_Element_of_REAL_st_x_in_N1_holds_ (reproj_(i,nx0))_._x_in_dom_(f_|_X) let x be Element of REAL ; ::_thesis: ( x in N1 implies (reproj (i,nx0)) . x in dom (f | X) ) assume x in N1 ; ::_thesis: (reproj (i,nx0)) . x in dom (f | X) then (reproj (i,nx0)) . x in X by A26; then (reproj (i,nx0)) . x in (dom f) /\ X by A21, XBOOLE_0:def_4; hence (reproj (i,nx0)) . x in dom (f | X) by RELAT_1:61; ::_thesis: verum end; A28: N1 c= dom ((f | X) * (reproj (i,nx0))) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in N1 or z in dom ((f | X) * (reproj (i,nx0))) ) assume A29: z in N1 ; ::_thesis: z in dom ((f | X) * (reproj (i,nx0))) then A30: z in REAL ; reconsider x = z as Element of REAL by A29; A31: (reproj (i,nx0)) . x in dom (f | X) by A29, A27; z in dom (reproj (i,nx0)) by A30, FUNCT_2:def_1; hence z in dom ((f | X) * (reproj (i,nx0))) by A31, FUNCT_1:11; ::_thesis: verum end; consider N being Neighbourhood of x0 such that NXX: ( N c= N0 & N c= N1 ) by RCOMP_1:17; A32: N c= dom ((f | X) * (reproj (i,nx0))) by NXX, A28, XBOOLE_1:1; consider L being LinearFunc, R being RestFunc such that A33: for x being Element of REAL st x in N0 holds ((f * (reproj (i,nx0))) . x) - ((f * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by A25; now__::_thesis:_for_x_being_Element_of_REAL_st_x_in_N_holds_ (((f_|_X)_*_(reproj_(i,nx0)))_._x)_-_(((f_|_X)_*_(reproj_(i,nx0)))_._x0)_=_(L_._(x_-_x0))_+_(R_._(x_-_x0)) let x be Element of REAL ; ::_thesis: ( x in N implies (((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) ) assume A34: x in N ; ::_thesis: (((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) A36: dom (reproj (i,nx0)) = REAL by FUNCT_2:def_1; (reproj (i,nx0)) . x in dom (f | X) by A27, A34, NXX; then A38: (reproj (i,nx0)) . x in (dom f) /\ X by RELAT_1:61; (reproj (i,nx0)) . x0 in dom (f | X) by A27, RCOMP_1:16; then A41: (reproj (i,nx0)) . x0 in (dom f) /\ X by RELAT_1:61; A43: ((f | X) * (reproj (i,nx0))) . x = (f | X) . ((reproj (i,nx0)) /. x) by A36, FUNCT_1:13 .= f . ((reproj (i,nx0)) . x) by A38, FUNCT_1:48 .= (f * (reproj (i,nx0))) . x by A36, FUNCT_1:13 ; ((f | X) * (reproj (i,nx0))) . x0 = (f | X) . ((reproj (i,nx0)) . x0) by A36, FUNCT_1:13 .= f . ((reproj (i,nx0)) . x0) by A41, FUNCT_1:48 .= (f * (reproj (i,nx0))) . x0 by A36, FUNCT_1:13 ; hence (((f | X) * (reproj (i,nx0))) . x) - (((f | X) * (reproj (i,nx0))) . x0) = (L . (x - x0)) + (R . (x - x0)) by A43, A34, A33, NXX; ::_thesis: verum end; then (f | X) * (reproj (i,nx0)) is_differentiable_in x0 by A32, FDIFF_1:def_4; hence f | X is_partial_differentiable_in nx0,i by PDIFF_1:def_11; ::_thesis: verum end; hence for x being Element of REAL m st x in X holds f | X is_partial_differentiable_in x,i ; ::_thesis: verum end; theorem CW020: :: PDIFF_9:61 for m being non empty Element of NAT for i being Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i ) let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i ) let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i ) let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g & X is open & 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i ) ) assume AS: ( <>* f = g & X is open & 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i iff g is_partial_differentiable_on X,i ) hereby ::_thesis: ( g is_partial_differentiable_on X,i implies f is_partial_differentiable_on X,i ) assume P2: f is_partial_differentiable_on X,i ; ::_thesis: g is_partial_differentiable_on X,i then X c= dom f by AS, PDIFF734; then P3: X c= dom g by LMXTh0, AS; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ g_is_partial_differentiable_in_x,i let x be Element of REAL m; ::_thesis: ( x in X implies g is_partial_differentiable_in x,i ) assume x in X ; ::_thesis: g is_partial_differentiable_in x,i then f is_partial_differentiable_in x,i by P2, AS, PDIFF734; hence g is_partial_differentiable_in x,i by AS, PDIFF_1:18; ::_thesis: verum end; hence g is_partial_differentiable_on X,i by P3, AS, PDIFF_7:34; ::_thesis: verum end; hereby ::_thesis: verum assume P3: g is_partial_differentiable_on X,i ; ::_thesis: f is_partial_differentiable_on X,i then X c= dom g by AS, PDIFF_7:34; then P4: X c= dom f by LMXTh0, AS; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ f_is_partial_differentiable_in_x,i let x be Element of REAL m; ::_thesis: ( x in X implies f is_partial_differentiable_in x,i ) assume x in X ; ::_thesis: f is_partial_differentiable_in x,i then g is_partial_differentiable_in x,i by P3, AS, PDIFF_7:34; hence f is_partial_differentiable_in x,i by AS, PDIFF_1:18; ::_thesis: verum end; hence f is_partial_differentiable_on X,i by AS, PDIFF734, P4; ::_thesis: verum end; end; theorem CW021: :: PDIFF_9:62 for m being non empty Element of NAT for i being Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X ) let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X ) let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) st <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X ) let g be PartFunc of (REAL m),(REAL 1); ::_thesis: ( <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i implies ( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X ) ) assume AS: ( <>* f = g & X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i ) ; ::_thesis: ( f `partial| (X,i) is_continuous_on X iff g `partial| (X,i) is_continuous_on X ) then P1: g is_partial_differentiable_on X,i by CW020; set ff = f `partial| (X,i); set gg = g `partial| (X,i); EQ1: for x, y being Element of REAL m st x in X & y in X holds |.(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y)).| = |.(((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y)).| proof let x, y be Element of REAL m; ::_thesis: ( x in X & y in X implies |.(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y)).| = |.(((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y)).| ) assume EQ2: ( x in X & y in X ) ; ::_thesis: |.(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y)).| = |.(((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y)).| then EQ3: ( (f `partial| (X,i)) /. x = partdiff (f,x,i) & (f `partial| (X,i)) /. y = partdiff (f,y,i) ) by AS, DefPDX; EQ5: ( (g `partial| (X,i)) /. x = partdiff (g,x,i) & (g `partial| (X,i)) /. y = partdiff (g,y,i) ) by P1, EQ2, PDIFF_7:def_5; ( g is_partial_differentiable_in x,i & g is_partial_differentiable_in y,i ) by P1, EQ2, AS, PDIFF_7:34; then ( partdiff (g,x,i) = <*(partdiff (f,x,i))*> & partdiff (g,y,i) = <*(partdiff (f,y,i))*> ) by AS, PDIFF_1:19; then ((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y) = <*(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y))*> by EQ3, EQ5, RVSUM_1:29; hence |.(((f `partial| (X,i)) /. x) - ((f `partial| (X,i)) /. y)).| = |.(((g `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. y)).| by XTh30D; ::_thesis: verum end; D1: dom (g `partial| (X,i)) = X by P1, PDIFF_7:def_5; D2: dom (f `partial| (X,i)) = X by DefPDX, AS; hereby ::_thesis: ( g `partial| (X,i) is_continuous_on X implies f `partial| (X,i) is_continuous_on X ) assume Q2: f `partial| (X,i) is_continuous_on X ; ::_thesis: g `partial| (X,i) is_continuous_on X now__::_thesis:_for_x0_being_Element_of_REAL_m for_r_being_Real_st_x0_in_X_&_0_<_r_holds_ ex_s_being_Real_st_ (_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_X_&_|.(x1_-_x0).|_<_s_holds_ |.(((g_`partial|_(X,i))_/._x1)_-_((g_`partial|_(X,i))_/._x0)).|_<_r_)_) let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) ) let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) ) ) assume Q40: ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) ) then consider s being Real such that Q41: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) by D2, Q2, XTh38; take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) ) thus 0 < s by Q41; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) assume Q42: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r then |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r by Q41; hence |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r by Q40, Q42, EQ1; ::_thesis: verum end; hence g `partial| (X,i) is_continuous_on X by D1, PDIFF_7:38; ::_thesis: verum end; hereby ::_thesis: verum assume Q2: g `partial| (X,i) is_continuous_on X ; ::_thesis: f `partial| (X,i) is_continuous_on X now__::_thesis:_for_x0_being_Element_of_REAL_m for_r_being_Real_st_x0_in_X_&_0_<_r_holds_ ex_s_being_Real_st_ (_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_X_&_|.(x1_-_x0).|_<_s_holds_ |.(((f_`partial|_(X,i))_/._x1)_-_((f_`partial|_(X,i))_/._x0)).|_<_r_)_) let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) ) assume Q40: ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) then consider s being Real such that Q41: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r ) ) by Q2, PDIFF_7:38; take s = s; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) ) thus 0 < s by Q41; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r ) assume Q42: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r then |.(((g `partial| (X,i)) /. x1) - ((g `partial| (X,i)) /. x0)).| < r by Q41; hence |.(((f `partial| (X,i)) /. x1) - ((f `partial| (X,i)) /. x0)).| < r by Q40, Q42, EQ1; ::_thesis: verum end; hence f `partial| (X,i) is_continuous_on X by XTh38, D2; ::_thesis: verum end; end; CW022: for m being non empty Element of NAT for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) for x1, x0, v being Element of REAL m st <>* f = g holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).| proof let m be non empty Element of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL for g being PartFunc of (REAL m),(REAL 1) for x1, x0, v being Element of REAL m st <>* f = g holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).| let f be PartFunc of (REAL m),REAL; ::_thesis: for g being PartFunc of (REAL m),(REAL 1) for x1, x0, v being Element of REAL m st <>* f = g holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).| let g be PartFunc of (REAL m),(REAL 1); ::_thesis: for x1, x0, v being Element of REAL m st <>* f = g holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).| let x1, x0, v be Element of REAL m; ::_thesis: ( <>* f = g implies |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).| ) assume AS: <>* f = g ; ::_thesis: |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).| set I = proj (1,1); reconsider w0 = (diff (g,x0)) . v, w1 = (diff (g,x1)) . v as Point of (REAL-NS 1) by REAL_NS1:def_4; ( dom (diff (g,x1)) = REAL m & dom (diff (g,x0)) = REAL m ) by FUNCT_2:def_1; then ( (diff (f,x0)) . v = (proj (1,1)) . w0 & (diff (f,x1)) . v = (proj (1,1)) . w1 ) by AS, FUNCT_1:13; then ((diff (f,x1)) . v) - ((diff (f,x0)) . v) = (proj (1,1)) . (w1 - w0) by PDIFF_1:4; then |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = ||.(w1 - w0).|| by PDIFF_1:4; hence |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| = |.(((diff (g,x1)) . v) - ((diff (g,x0)) . v)).| by REAL_NS1:1, REAL_NS1:5; ::_thesis: verum end; theorem :: PDIFF_9:63 for m being non empty Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL st X is open & X c= dom f holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL st X is open & X c= dom f holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) let X be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL st X is open & X c= dom f holds ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) let f be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & X c= dom f implies ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) ) set g = <>* f; assume AS1: ( X is open & X c= dom f ) ; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) iff ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ) then AS2: X c= dom (<>* f) by LMXTh0; hereby ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) implies for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) assume P1: for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ; ::_thesis: ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) P3: for i being Element of NAT st 1 <= i & i <= m holds ( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X ) proof let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X ) ) assume P20: ( 1 <= i & i <= m ) ; ::_thesis: ( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X ) then ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) by P1; hence ( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X ) by AS1, CW020, CW021, P20; ::_thesis: verum end; then <>* f is_differentiable_on X by CW01, AS1, AS2; hence f is_differentiable_on X by YTh30; ::_thesis: for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) thus for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ::_thesis: verum proof let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) assume ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) then consider s being Real such that Q2: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) ) by P3, CW01, AS1, AS2; take s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) thus 0 < s by Q2; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) assume Q3: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| let v be Element of REAL m; ::_thesis: |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| by Q3, Q2; hence |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| by CW022; ::_thesis: verum end; end; now__::_thesis:_(_f_is_differentiable_on_X_&_(_for_x0_being_Element_of_REAL_m for_r_being_Real_st_x0_in_X_&_0_<_r_holds_ ex_s_being_Real_st_ (_0_<_s_&_(_for_x1_being_Element_of_REAL_m_st_x1_in_X_&_|.(x1_-_x0).|_<_s_holds_ for_v_being_Element_of_REAL_m_holds_|.(((diff_(f,x1))_._v)_-_((diff_(f,x0))_._v)).|_<=_r_*_|.v.|_)_)_)_implies_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_m_holds_ (_f_is_partial_differentiable_on_X,i_&_f_`partial|_(X,i)_is_continuous_on_X_)_) assume P1: ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) ) ; ::_thesis: for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) then P2: <>* f is_differentiable_on X by AS1, YTh30; P3: for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) ) proof let x0 be Element of REAL m; ::_thesis: for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) ) let r be Real; ::_thesis: ( x0 in X & 0 < r implies ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) ) ) assume ( x0 in X & 0 < r ) ; ::_thesis: ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) ) then consider s being Real such that Q2: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) by P1; take s ; ::_thesis: ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) ) thus 0 < s by Q2; ::_thesis: for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| let x1 be Element of REAL m; ::_thesis: ( x1 in X & |.(x1 - x0).| < s implies for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| ) assume Q3: ( x1 in X & |.(x1 - x0).| < s ) ; ::_thesis: for v being Element of REAL m holds |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| let v be Element of REAL m; ::_thesis: |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| by Q3, Q2; hence |.(((diff ((<>* f),x1)) . v) - ((diff ((<>* f),x0)) . v)).| <= r * |.v.| by CW022; ::_thesis: verum end; thus for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ::_thesis: verum proof let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= m implies ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) assume P4: ( 1 <= i & i <= m ) ; ::_thesis: ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) then P5: ( <>* f is_partial_differentiable_on X,i & (<>* f) `partial| (X,i) is_continuous_on X ) by P3, CW01, AS2, AS1, P2; hence f is_partial_differentiable_on X,i by P4, CW020, AS1; ::_thesis: f `partial| (X,i) is_continuous_on X hence f `partial| (X,i) is_continuous_on X by P4, P5, CW021, AS1; ::_thesis: verum end; end; hence ( f is_differentiable_on X & ( for x0 being Element of REAL m for r being Real st x0 in X & 0 < r holds ex s being Real st ( 0 < s & ( for x1 being Element of REAL m st x1 in X & |.(x1 - x0).| < s holds for v being Element of REAL m holds |.(((diff (f,x1)) . v) - ((diff (f,x0)) . v)).| <= r * |.v.| ) ) ) implies for i being Element of NAT st 1 <= i & i <= m holds ( f is_partial_differentiable_on X,i & f `partial| (X,i) is_continuous_on X ) ) ; ::_thesis: verum end; LM1291: for i, k being Element of NAT for f, g being PartFunc of (REAL i),REAL for x being Element of REAL i holds (f * (reproj (k,x))) (#) (g * (reproj (k,x))) = (f (#) g) * (reproj (k,x)) proof let i, k be Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL i),REAL for x being Element of REAL i holds (f * (reproj (k,x))) (#) (g * (reproj (k,x))) = (f (#) g) * (reproj (k,x)) let f1, f2 be PartFunc of (REAL i),REAL; ::_thesis: for x being Element of REAL i holds (f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x))) = (f1 (#) f2) * (reproj (k,x)) let x be Element of REAL i; ::_thesis: (f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x))) = (f1 (#) f2) * (reproj (k,x)) A1: dom (reproj (k,x)) = REAL by FUNCT_2:def_1; A2: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by VALUED_1:def_4; for s being Element of REAL holds ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) ) proof let s be Element of REAL ; ::_thesis: ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) ) ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff (reproj (k,x)) . s in (dom f1) /\ (dom f2) ) by A2, A1, FUNCT_1:11; then ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff ( (reproj (k,x)) . s in dom f1 & (reproj (k,x)) . s in dom f2 ) ) by XBOOLE_0:def_4; then ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff ( s in dom (f1 * (reproj (k,x))) & s in dom (f2 * (reproj (k,x))) ) ) by A1, FUNCT_1:11; then ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in (dom (f1 * (reproj (k,x)))) /\ (dom (f2 * (reproj (k,x)))) ) by XBOOLE_0:def_4; hence ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) ) by VALUED_1:def_4; ::_thesis: verum end; then for s being set holds ( s in dom ((f1 (#) f2) * (reproj (k,x))) iff s in dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) ) ; then A3: dom ((f1 (#) f2) * (reproj (k,x))) = dom ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) by TARSKI:1; for z being Element of REAL st z in dom ((f1 (#) f2) * (reproj (k,x))) holds ((f1 (#) f2) * (reproj (k,x))) . z = ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) . z proof let z be Element of REAL ; ::_thesis: ( z in dom ((f1 (#) f2) * (reproj (k,x))) implies ((f1 (#) f2) * (reproj (k,x))) . z = ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) . z ) assume A5: z in dom ((f1 (#) f2) * (reproj (k,x))) ; ::_thesis: ((f1 (#) f2) * (reproj (k,x))) . z = ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) . z then (reproj (k,x)) . z in (dom f1) /\ (dom f2) by A2, FUNCT_1:11; then ( (reproj (k,x)) . z in dom f1 & (reproj (k,x)) . z in dom f2 ) by XBOOLE_0:def_4; then ( z in dom (f1 * (reproj (k,x))) & z in dom (f2 * (reproj (k,x))) ) by A1, FUNCT_1:11; then A13: ( f1 . ((reproj (k,x)) . z) = (f1 * (reproj (k,x))) . z & f2 . ((reproj (k,x)) . z) = (f2 * (reproj (k,x))) . z ) by FUNCT_1:12; thus ((f1 (#) f2) * (reproj (k,x))) . z = (f1 (#) f2) . ((reproj (k,x)) . z) by A5, FUNCT_1:12 .= (f1 . ((reproj (k,x)) . z)) * (f2 . ((reproj (k,x)) . z)) by VALUED_1:5 .= ((f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x)))) . z by A3, A5, A13, VALUED_1:def_4 ; ::_thesis: verum end; hence (f1 * (reproj (k,x))) (#) (f2 * (reproj (k,x))) = (f1 (#) f2) * (reproj (k,x)) by A3, PARTFUN1:5; ::_thesis: verum end; theorem MPDIFF129: :: PDIFF_9:64 for m being non empty Element of NAT for i being Element of NAT for f, g being PartFunc of (REAL m),REAL for x being Element of REAL m st f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i holds ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for f, g being PartFunc of (REAL m),REAL for x being Element of REAL m st f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i holds ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) let i be Element of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL for x being Element of REAL m st f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i holds ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: for x being Element of REAL m st f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i holds ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) let x be Element of REAL m; ::_thesis: ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i implies ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) assume ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i ) ; ::_thesis: ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) then P1: ( f * (reproj (i,x)) is_differentiable_in (proj (i,m)) . x & g * (reproj (i,x)) is_differentiable_in (proj (i,m)) . x ) by PDIFF_1:def_11; set y = (proj (i,m)) . x; dom (reproj (i,x)) = REAL by FUNCT_2:def_1; then P7: ( (f * (reproj (i,x))) . ((proj (i,m)) . x) = f . ((reproj (i,x)) . ((proj (i,m)) . x)) & (g * (reproj (i,x))) . ((proj (i,m)) . x) = g . ((reproj (i,x)) . ((proj (i,m)) . x)) ) by FUNCT_1:13; then P6: (f * (reproj (i,x))) . ((proj (i,m)) . x) = f . x by LMMMTh6; (f * (reproj (i,x))) (#) (g * (reproj (i,x))) = (f (#) g) * (reproj (i,x)) by LM1291; then (f (#) g) * (reproj (i,x)) is_differentiable_in (proj (i,m)) . x by P1, FDIFF_1:16; hence f (#) g is_partial_differentiable_in x,i by PDIFF_1:def_11; ::_thesis: partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) thus partdiff ((f (#) g),x,i) = diff (((f * (reproj (i,x))) (#) (g * (reproj (i,x)))),((proj (i,m)) . x)) by LM1291 .= (((g * (reproj (i,x))) . ((proj (i,m)) . x)) * (partdiff (f,x,i))) + (((f * (reproj (i,x))) . ((proj (i,m)) . x)) * (diff ((g * (reproj (i,x))),((proj (i,m)) . x)))) by P1, FDIFF_1:16 .= ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) by P6, P7, LMMMTh6 ; ::_thesis: verum end; theorem XXX1: :: PDIFF_9:65 for m being non empty Element of NAT for i being Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i implies ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) ) assume AS: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i ) ; ::_thesis: ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) P1: ( X c= dom f & X c= dom g ) by AS, PDIFF734; Q1: ( dom (f `partial| (X,i)) = X & dom (g `partial| (X,i)) = X ) by DefPDX, AS; dom (f + g) = (dom f) /\ (dom g) by VALUED_1:def_1; then P3: X c= dom (f + g) by P1, XBOOLE_1:19; XX1: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ (_f_+_g_is_partial_differentiable_in_x,i_&_partdiff_((f_+_g),x,i)_=_(partdiff_(f,x,i))_+_(partdiff_(g,x,i))_) let x be Element of REAL m; ::_thesis: ( x in X implies ( f + g is_partial_differentiable_in x,i & partdiff ((f + g),x,i) = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) assume x in X ; ::_thesis: ( f + g is_partial_differentiable_in x,i & partdiff ((f + g),x,i) = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) then ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i ) by AS, PDIFF734; hence ( f + g is_partial_differentiable_in x,i & partdiff ((f + g),x,i) = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) by PDIFF_1:29; ::_thesis: verum end; then P7: for x being Element of REAL m st x in X holds f + g is_partial_differentiable_in x,i ; then P8: f + g is_partial_differentiable_on X,i by P3, PDIFF734, AS; then P9: dom ((f + g) `partial| (X,i)) = X by DefPDX; P10: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ ((f_+_g)_`partial|_(X,i))_/._x_=_(partdiff_(f,x,i))_+_(partdiff_(g,x,i)) let x be Element of REAL m; ::_thesis: ( x in X implies ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) assume P10: x in X ; ::_thesis: ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) then ((f + g) `partial| (X,i)) /. x = partdiff ((f + g),x,i) by P8, DefPDX; hence ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) by XX1, P10; ::_thesis: verum end; P11: dom ((f `partial| (X,i)) + (g `partial| (X,i))) = (dom (f `partial| (X,i))) /\ (dom (g `partial| (X,i))) by VALUED_1:def_1; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ ((f_+_g)_`partial|_(X,i))_._x_=_((f_`partial|_(X,i))_+_(g_`partial|_(X,i)))_._x let x be Element of REAL m; ::_thesis: ( x in X implies ((f + g) `partial| (X,i)) . x = ((f `partial| (X,i)) + (g `partial| (X,i))) . x ) assume A1: x in X ; ::_thesis: ((f + g) `partial| (X,i)) . x = ((f `partial| (X,i)) + (g `partial| (X,i))) . x thus ((f + g) `partial| (X,i)) . x = ((f + g) `partial| (X,i)) /. x by A1, P9, PARTFUN1:def_6 .= (partdiff (f,x,i)) + (partdiff (g,x,i)) by P10, A1 .= ((f `partial| (X,i)) /. x) + (partdiff (g,x,i)) by A1, DefPDX, AS .= ((f `partial| (X,i)) /. x) + ((g `partial| (X,i)) /. x) by A1, DefPDX, AS .= ((f `partial| (X,i)) . x) + ((g `partial| (X,i)) /. x) by A1, Q1, PARTFUN1:def_6 .= ((f `partial| (X,i)) . x) + ((g `partial| (X,i)) . x) by A1, Q1, PARTFUN1:def_6 .= ((f `partial| (X,i)) + (g `partial| (X,i))) . x by A1, P11, Q1, VALUED_1:def_1 ; ::_thesis: verum end; hence ( f + g is_partial_differentiable_on X,i & (f + g) `partial| (X,i) = (f `partial| (X,i)) + (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f + g) `partial| (X,i)) /. x = (partdiff (f,x,i)) + (partdiff (g,x,i)) ) ) by P7, P3, PDIFF734, AS, P9, P10, P11, Q1, PARTFUN1:5; ::_thesis: verum end; theorem XXX2: :: PDIFF_9:66 for m being non empty Element of NAT for i being Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i implies ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) ) assume AS: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i ) ; ::_thesis: ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) P1: ( X c= dom f & X c= dom g ) by AS, PDIFF734; Q1: ( dom (f `partial| (X,i)) = X & dom (g `partial| (X,i)) = X ) by DefPDX, AS; dom (f - g) = (dom f) /\ (dom g) by VALUED_1:12; then P3: X c= dom (f - g) by P1, XBOOLE_1:19; XX1: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ (_f_-_g_is_partial_differentiable_in_x,i_&_partdiff_((f_-_g),x,i)_=_(partdiff_(f,x,i))_-_(partdiff_(g,x,i))_) let x be Element of REAL m; ::_thesis: ( x in X implies ( f - g is_partial_differentiable_in x,i & partdiff ((f - g),x,i) = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) assume x in X ; ::_thesis: ( f - g is_partial_differentiable_in x,i & partdiff ((f - g),x,i) = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) then ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i ) by AS, PDIFF734; hence ( f - g is_partial_differentiable_in x,i & partdiff ((f - g),x,i) = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) by PDIFF_1:31; ::_thesis: verum end; then P7: for x being Element of REAL m st x in X holds f - g is_partial_differentiable_in x,i ; then P8: f - g is_partial_differentiable_on X,i by P3, PDIFF734, AS; then B1: dom ((f - g) `partial| (X,i)) = X by DefPDX; P10: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ ((f_-_g)_`partial|_(X,i))_/._x_=_(partdiff_(f,x,i))_-_(partdiff_(g,x,i)) let x be Element of REAL m; ::_thesis: ( x in X implies ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) assume P10: x in X ; ::_thesis: ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) then ((f - g) `partial| (X,i)) /. x = partdiff ((f - g),x,i) by P8, DefPDX; hence ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) by XX1, P10; ::_thesis: verum end; B2: dom ((f `partial| (X,i)) - (g `partial| (X,i))) = (dom (f `partial| (X,i))) /\ (dom (g `partial| (X,i))) by VALUED_1:12; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ ((f_-_g)_`partial|_(X,i))_._x_=_((f_`partial|_(X,i))_-_(g_`partial|_(X,i)))_._x let x be Element of REAL m; ::_thesis: ( x in X implies ((f - g) `partial| (X,i)) . x = ((f `partial| (X,i)) - (g `partial| (X,i))) . x ) assume A1: x in X ; ::_thesis: ((f - g) `partial| (X,i)) . x = ((f `partial| (X,i)) - (g `partial| (X,i))) . x thus ((f - g) `partial| (X,i)) . x = ((f - g) `partial| (X,i)) /. x by A1, B1, PARTFUN1:def_6 .= (partdiff (f,x,i)) - (partdiff (g,x,i)) by P10, A1 .= ((f `partial| (X,i)) /. x) - (partdiff (g,x,i)) by A1, DefPDX, AS .= ((f `partial| (X,i)) /. x) - ((g `partial| (X,i)) /. x) by A1, DefPDX, AS .= ((f `partial| (X,i)) . x) - ((g `partial| (X,i)) /. x) by A1, Q1, PARTFUN1:def_6 .= ((f `partial| (X,i)) . x) - ((g `partial| (X,i)) . x) by A1, Q1, PARTFUN1:def_6 .= ((f `partial| (X,i)) - (g `partial| (X,i))) . x by A1, B2, Q1, VALUED_1:13 ; ::_thesis: verum end; hence ( f - g is_partial_differentiable_on X,i & (f - g) `partial| (X,i) = (f `partial| (X,i)) - (g `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((f - g) `partial| (X,i)) /. x = (partdiff (f,x,i)) - (partdiff (g,x,i)) ) ) by B1, B2, Q1, P7, P10, P3, PDIFF734, AS, PARTFUN1:5; ::_thesis: verum end; theorem XXX3: :: PDIFF_9:67 for m being non empty Element of NAT for i being Element of NAT for X being Subset of (REAL m) for r being Real for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for X being Subset of (REAL m) for r being Real for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m) for r being Real for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) let X be Subset of (REAL m); ::_thesis: for r being Real for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) let r be Real; ::_thesis: for f being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i holds ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) let f be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i implies ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) ) assume AS: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i ) ; ::_thesis: ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) Q1: dom (f `partial| (X,i)) = X by DefPDX, AS; dom (r (#) f) = dom f by VALUED_1:def_5; then P3: X c= dom (r (#) f) by AS, PDIFF734; XX1: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ (_r_(#)_f_is_partial_differentiable_in_x,i_&_partdiff_((r_(#)_f),x,i)_=_r_*_(partdiff_(f,x,i))_) let x be Element of REAL m; ::_thesis: ( x in X implies ( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) ) ) assume x in X ; ::_thesis: ( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) ) then f is_partial_differentiable_in x,i by AS, PDIFF734; hence ( r (#) f is_partial_differentiable_in x,i & partdiff ((r (#) f),x,i) = r * (partdiff (f,x,i)) ) by PDIFF_1:33; ::_thesis: verum end; then P7: for x being Element of REAL m st x in X holds r (#) f is_partial_differentiable_in x,i ; then P8: r (#) f is_partial_differentiable_on X,i by P3, PDIFF734, AS; then P9: dom ((r (#) f) `partial| (X,i)) = X by DefPDX; P10: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ ((r_(#)_f)_`partial|_(X,i))_/._x_=_r_*_(partdiff_(f,x,i)) let x be Element of REAL m; ::_thesis: ( x in X implies ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) assume P10: x in X ; ::_thesis: ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) then ((r (#) f) `partial| (X,i)) /. x = partdiff ((r (#) f),x,i) by P8, DefPDX; hence ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) by XX1, P10; ::_thesis: verum end; dom (r (#) (f `partial| (X,i))) = dom (f `partial| (X,i)) by VALUED_1:def_5; then P11: dom (r (#) (f `partial| (X,i))) = X by DefPDX, AS; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ ((r_(#)_f)_`partial|_(X,i))_._x_=_(r_(#)_(f_`partial|_(X,i)))_._x let x be Element of REAL m; ::_thesis: ( x in X implies ((r (#) f) `partial| (X,i)) . x = (r (#) (f `partial| (X,i))) . x ) assume A1: x in X ; ::_thesis: ((r (#) f) `partial| (X,i)) . x = (r (#) (f `partial| (X,i))) . x thus ((r (#) f) `partial| (X,i)) . x = ((r (#) f) `partial| (X,i)) /. x by A1, P9, PARTFUN1:def_6 .= r * (partdiff (f,x,i)) by P10, A1 .= r * ((f `partial| (X,i)) /. x) by A1, DefPDX, AS .= r * ((f `partial| (X,i)) . x) by A1, Q1, PARTFUN1:def_6 .= (r (#) (f `partial| (X,i))) . x by A1, P11, VALUED_1:def_5 ; ::_thesis: verum end; hence ( r (#) f is_partial_differentiable_on X,i & (r (#) f) `partial| (X,i) = r (#) (f `partial| (X,i)) & ( for x being Element of REAL m st x in X holds ((r (#) f) `partial| (X,i)) /. x = r * (partdiff (f,x,i)) ) ) by P9, P11, P7, P10, P3, PDIFF734, AS, PARTFUN1:5; ::_thesis: verum end; theorem XXX4: :: PDIFF_9:68 for m being non empty Element of NAT for i being Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) let X be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i holds ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i implies ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) ) assume AS: ( X is open & 1 <= i & i <= m & f is_partial_differentiable_on X,i & g is_partial_differentiable_on X,i ) ; ::_thesis: ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) P1: ( X c= dom f & X c= dom g ) by AS, PDIFF734; Q1: ( dom (f `partial| (X,i)) = X & dom (g `partial| (X,i)) = X ) by DefPDX, AS; dom (f (#) g) = (dom f) /\ (dom g) by VALUED_1:def_4; then P3: X c= dom (f (#) g) by P1, XBOOLE_1:19; XX1: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ (_f_(#)_g_is_partial_differentiable_in_x,i_&_partdiff_((f_(#)_g),x,i)_=_((partdiff_(f,x,i))_*_(g_._x))_+_((f_._x)_*_(partdiff_(g,x,i)))_) let x be Element of REAL m; ::_thesis: ( x in X implies ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) assume x in X ; ::_thesis: ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) then ( f is_partial_differentiable_in x,i & g is_partial_differentiable_in x,i ) by AS, PDIFF734; hence ( f (#) g is_partial_differentiable_in x,i & partdiff ((f (#) g),x,i) = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) by MPDIFF129; ::_thesis: verum end; then P7: for x being Element of REAL m st x in X holds f (#) g is_partial_differentiable_in x,i ; then P8: f (#) g is_partial_differentiable_on X,i by P3, PDIFF734, AS; then P9: dom ((f (#) g) `partial| (X,i)) = X by DefPDX; P10: now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ ((f_(#)_g)_`partial|_(X,i))_/._x_=_((partdiff_(f,x,i))_*_(g_._x))_+_((f_._x)_*_(partdiff_(g,x,i))) let x be Element of REAL m; ::_thesis: ( x in X implies ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) assume P10: x in X ; ::_thesis: ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) then ((f (#) g) `partial| (X,i)) /. x = partdiff ((f (#) g),x,i) by P8, DefPDX; hence ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) by XX1, P10; ::_thesis: verum end; ( dom ((f `partial| (X,i)) (#) g) = (dom (f `partial| (X,i))) /\ (dom g) & dom (f (#) (g `partial| (X,i))) = (dom f) /\ (dom (g `partial| (X,i))) ) by VALUED_1:def_4; then P12: ( dom ((f `partial| (X,i)) (#) g) = X & dom (f (#) (g `partial| (X,i))) = X ) by Q1, P1, XBOOLE_1:28; P14: dom (((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i)))) = (dom ((f `partial| (X,i)) (#) g)) /\ (dom (f (#) (g `partial| (X,i)))) by VALUED_1:def_1; now__::_thesis:_for_x_being_Element_of_REAL_m_st_x_in_X_holds_ ((f_(#)_g)_`partial|_(X,i))_._x_=_(((f_`partial|_(X,i))_(#)_g)_+_(f_(#)_(g_`partial|_(X,i))))_._x let x be Element of REAL m; ::_thesis: ( x in X implies ((f (#) g) `partial| (X,i)) . x = (((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i)))) . x ) assume A1: x in X ; ::_thesis: ((f (#) g) `partial| (X,i)) . x = (((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i)))) . x thus ((f (#) g) `partial| (X,i)) . x = ((f (#) g) `partial| (X,i)) /. x by A1, P9, PARTFUN1:def_6 .= ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) by P10, A1 .= (((f `partial| (X,i)) /. x) * (g . x)) + ((f . x) * (partdiff (g,x,i))) by A1, DefPDX, AS .= (((f `partial| (X,i)) /. x) * (g . x)) + ((f . x) * ((g `partial| (X,i)) /. x)) by A1, DefPDX, AS .= (((f `partial| (X,i)) . x) * (g . x)) + ((f . x) * ((g `partial| (X,i)) /. x)) by A1, Q1, PARTFUN1:def_6 .= (((f `partial| (X,i)) . x) * (g . x)) + ((f . x) * ((g `partial| (X,i)) . x)) by A1, Q1, PARTFUN1:def_6 .= (((f `partial| (X,i)) (#) g) . x) + ((f . x) * ((g `partial| (X,i)) . x)) by VALUED_1:5 .= (((f `partial| (X,i)) (#) g) . x) + ((f (#) (g `partial| (X,i))) . x) by VALUED_1:5 .= (((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i)))) . x by A1, P14, P12, VALUED_1:def_1 ; ::_thesis: verum end; hence ( f (#) g is_partial_differentiable_on X,i & (f (#) g) `partial| (X,i) = ((f `partial| (X,i)) (#) g) + (f (#) (g `partial| (X,i))) & ( for x being Element of REAL m st x in X holds ((f (#) g) `partial| (X,i)) /. x = ((partdiff (f,x,i)) * (g . x)) + ((f . x) * (partdiff (g,x,i))) ) ) by P7, P10, P3, PDIFF734, AS, P9, P12, P14, PARTFUN1:5; ::_thesis: verum end; begin definition let m be non empty Element of NAT ; let Z be set ; let I be FinSequence of NAT ; let f be PartFunc of (REAL m),REAL; func PartDiffSeq (f,Z,I) -> Functional_Sequence of (REAL m),REAL means :TDef5: :: PDIFF_9:def 7 ( it . 0 = f & ( for i being natural number holds it . (i + 1) = (it . i) `partial| (Z,(I /. (i + 1))) ) ); existence ex b1 being Functional_Sequence of (REAL m),REAL st ( b1 . 0 = f & ( for i being natural number holds b1 . (i + 1) = (b1 . i) `partial| (Z,(I /. (i + 1))) ) ) proof reconsider fZ = f as Element of PFuncs ((REAL m),REAL) by PARTFUN1:45; defpred S1[ set , set , set ] means ex k being Element of NAT ex h being PartFunc of (REAL m),REAL st ( \$1 = k & \$2 = h & \$3 = h `partial| (Z,(I /. (k + 1))) ); A1: for n being Element of NAT for x being Element of PFuncs ((REAL m),REAL) ex y being Element of PFuncs ((REAL m),REAL) st S1[n,x,y] proof let n be Element of NAT ; ::_thesis: for x being Element of PFuncs ((REAL m),REAL) ex y being Element of PFuncs ((REAL m),REAL) st S1[n,x,y] let x be Element of PFuncs ((REAL m),REAL); ::_thesis: ex y being Element of PFuncs ((REAL m),REAL) st S1[n,x,y] reconsider x9 = x as PartFunc of (REAL m),REAL by PARTFUN1:46; reconsider y = x9 `partial| (Z,(I /. (n + 1))) as Element of PFuncs ((REAL m),REAL) by PARTFUN1:45; ex h being PartFunc of (REAL m),REAL st ( x = h & y = h `partial| (Z,(I /. (n + 1))) ) ; hence ex y being Element of PFuncs ((REAL m),REAL) st S1[n,x,y] ; ::_thesis: verum end; consider g being Function of NAT,(PFuncs ((REAL m),REAL)) such that A2: ( g . 0 = fZ & ( for n being Element of NAT holds S1[n,g . n,g . (n + 1)] ) ) from RECDEF_1:sch_2(A1); reconsider g = g as Functional_Sequence of (REAL m),REAL ; take g ; ::_thesis: ( g . 0 = f & ( for i being natural number holds g . (i + 1) = (g . i) `partial| (Z,(I /. (i + 1))) ) ) thus g . 0 = f by A2; ::_thesis: for i being natural number holds g . (i + 1) = (g . i) `partial| (Z,(I /. (i + 1))) let i be natural number ; ::_thesis: g . (i + 1) = (g . i) `partial| (Z,(I /. (i + 1))) i is Element of NAT by ORDINAL1:def_12; then S1[i,g . i,g . (i + 1)] by A2; hence g . (i + 1) = (g . i) `partial| (Z,(I /. (i + 1))) ; ::_thesis: verum end; uniqueness for b1, b2 being Functional_Sequence of (REAL m),REAL st b1 . 0 = f & ( for i being natural number holds b1 . (i + 1) = (b1 . i) `partial| (Z,(I /. (i + 1))) ) & b2 . 0 = f & ( for i being natural number holds b2 . (i + 1) = (b2 . i) `partial| (Z,(I /. (i + 1))) ) holds b1 = b2 proof let seq1, seq2 be Functional_Sequence of (REAL m),REAL; ::_thesis: ( seq1 . 0 = f & ( for i being natural number holds seq1 . (i + 1) = (seq1 . i) `partial| (Z,(I /. (i + 1))) ) & seq2 . 0 = f & ( for i being natural number holds seq2 . (i + 1) = (seq2 . i) `partial| (Z,(I /. (i + 1))) ) implies seq1 = seq2 ) assume that A3: seq1 . 0 = f and A4: for n being natural number holds seq1 . (n + 1) = (seq1 . n) `partial| (Z,(I /. (n + 1))) and A5: seq2 . 0 = f and A6: for n being natural number holds seq2 . (n + 1) = (seq2 . n) `partial| (Z,(I /. (n + 1))) ; ::_thesis: seq1 = seq2 defpred S1[ Element of NAT ] means seq1 . \$1 = seq2 . \$1; A7: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A8: S1[k] ; ::_thesis: S1[k + 1] seq1 . (k + 1) = (seq1 . k) `partial| (Z,(I /. (k + 1))) by A4; hence seq1 . (k + 1) = seq2 . (k + 1) by A6, A8; ::_thesis: verum end; A9: S1[ 0 ] by A3, A5; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A9, A7); hence seq1 = seq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem TDef5 defines PartDiffSeq PDIFF_9:def_7_:_ for m being non empty Element of NAT for Z being set for I being FinSequence of NAT for f being PartFunc of (REAL m),REAL for b5 being Functional_Sequence of (REAL m),REAL holds ( b5 = PartDiffSeq (f,Z,I) iff ( b5 . 0 = f & ( for i being natural number holds b5 . (i + 1) = (b5 . i) `partial| (Z,(I /. (i + 1))) ) ) ); definition let m be non empty Element of NAT ; let Z be set ; let I be FinSequence of NAT ; let f be PartFunc of (REAL m),REAL; predf is_partial_differentiable_on Z,I means :TDef6: :: PDIFF_9:def 8 for i being Element of NAT st i <= (len I) - 1 holds (PartDiffSeq (f,Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1); end; :: deftheorem TDef6 defines is_partial_differentiable_on PDIFF_9:def_8_:_ for m being non empty Element of NAT for Z being set for I being FinSequence of NAT for f being PartFunc of (REAL m),REAL holds ( f is_partial_differentiable_on Z,I iff for i being Element of NAT st i <= (len I) - 1 holds (PartDiffSeq (f,Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) ); definition let m be non empty Element of NAT ; let Z be set ; let I be FinSequence of NAT ; let f be PartFunc of (REAL m),REAL; funcf `partial| (Z,I) -> PartFunc of (REAL m),REAL equals :: PDIFF_9:def 9 (PartDiffSeq (f,Z,I)) . (len I); correctness coherence (PartDiffSeq (f,Z,I)) . (len I) is PartFunc of (REAL m),REAL; ; end; :: deftheorem defines `partial| PDIFF_9:def_9_:_ for m being non empty Element of NAT for Z being set for I being FinSequence of NAT for f being PartFunc of (REAL m),REAL holds f `partial| (Z,I) = (PartDiffSeq (f,Z,I)) . (len I); XCWLM1: for i being Element of NAT for I being non empty FinSequence of NAT for X being set st 1 <= i & i <= len I & rng I c= X holds I /. i in X proof let i be Element of NAT ; ::_thesis: for I being non empty FinSequence of NAT for X being set st 1 <= i & i <= len I & rng I c= X holds I /. i in X let I be non empty FinSequence of NAT ; ::_thesis: for X being set st 1 <= i & i <= len I & rng I c= X holds I /. i in X let X be set ; ::_thesis: ( 1 <= i & i <= len I & rng I c= X implies I /. i in X ) assume AS: ( 1 <= i & i <= len I & rng I c= X ) ; ::_thesis: I /. i in X then i in Seg (len I) ; then X1: i in dom I by FINSEQ_1:def_3; then I . i in rng I by FUNCT_1:3; then I /. i in rng I by X1, PARTFUN1:def_6; hence I /. i in X by AS; ::_thesis: verum end; theorem XCW010: :: PDIFF_9:69 for m being non empty Element of NAT for X being Subset of (REAL m) for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f + g),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((f + g),X,I)) . i = ((PartDiffSeq (f,X,I)) . i) + ((PartDiffSeq (g,X,I)) . i) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f + g),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((f + g),X,I)) . i = ((PartDiffSeq (f,X,I)) . i) + ((PartDiffSeq (g,X,I)) . i) ) let Z be Subset of (REAL m); ::_thesis: for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) ) let I be non empty FinSequence of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I implies for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) ) ) assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) ; ::_thesis: for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) ) thus for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) ) ::_thesis: verum proof defpred S1[ Element of NAT ] means ( \$1 <= (len I) - 1 implies ( (PartDiffSeq ((f + g),Z,I)) . \$1 is_partial_differentiable_on Z,I /. (\$1 + 1) & (PartDiffSeq ((f + g),Z,I)) . \$1 = ((PartDiffSeq (f,Z,I)) . \$1) + ((PartDiffSeq (g,Z,I)) . \$1) ) ); reconsider Z0 = 0 as Element of NAT ; A9: S1[ 0 ] proof assume 0 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((f + g),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((f + g),Z,I)) . 0 = ((PartDiffSeq (f,Z,I)) . 0) + ((PartDiffSeq (g,Z,I)) . 0) ) then Q2: ( (PartDiffSeq (f,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) & (PartDiffSeq (g,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) ) by AS, TDef6; Q0: ( f = (PartDiffSeq (f,Z,I)) . Z0 & (PartDiffSeq ((f + g),Z,I)) . Z0 = f + g ) by TDef5; 1 <= len I by FINSEQ_1:20; then I /. 1 in Seg m by AS, XCWLM1; then ( 1 <= I /. 1 & I /. 1 <= m ) by FINSEQ_1:1; then ((PartDiffSeq (f,Z,I)) . Z0) + ((PartDiffSeq (g,Z,I)) . Z0) is_partial_differentiable_on Z,I /. (Z0 + 1) by AS, Q2, XXX1; hence ( (PartDiffSeq ((f + g),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((f + g),Z,I)) . 0 = ((PartDiffSeq (f,Z,I)) . 0) + ((PartDiffSeq (g,Z,I)) . 0) ) by Q0, TDef5; ::_thesis: verum end; A7: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A8: S1[k] ; ::_thesis: S1[k + 1] assume A81: k + 1 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((f + g),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq ((f + g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) + ((PartDiffSeq (g,Z,I)) . (k + 1)) ) A83: k <= k + 1 by NAT_1:11; then A82: k <= (len I) - 1 by A81, XXREAL_0:2; A84: ( (PartDiffSeq (f,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq (g,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) ) by A81, AS, TDef6; k + 1 <= ((len I) - 1) + 1 by A82, XREAL_1:6; then I /. (k + 1) in Seg m by AS, XCWLM1, NAT_1:11; then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1; A840: ( (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) & (PartDiffSeq (g,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) ) by A82, AS, TDef6; R1: (PartDiffSeq (f,Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5; (k + 1) + 1 <= ((len I) - 1) + 1 by A81, XREAL_1:6; then I /. ((k + 1) + 1) in Seg m by AS, XCWLM1, NAT_1:11; then Q5: ( 1 <= I /. ((k + 1) + 1) & I /. ((k + 1) + 1) <= m ) by FINSEQ_1:1; A86: (PartDiffSeq ((f + g),Z,I)) . (k + 1) = (((PartDiffSeq (f,Z,I)) . k) + ((PartDiffSeq (g,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by A83, A8, A81, TDef5, XXREAL_0:2 .= (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) + (((PartDiffSeq (g,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by A840, AS, Q4, XXX1 .= ((PartDiffSeq (f,Z,I)) . (k + 1)) + ((PartDiffSeq (g,Z,I)) . (k + 1)) by R1, TDef5 ; hence (PartDiffSeq ((f + g),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) by AS, A84, Q5, XXX1; ::_thesis: (PartDiffSeq ((f + g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) + ((PartDiffSeq (g,Z,I)) . (k + 1)) thus (PartDiffSeq ((f + g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) + ((PartDiffSeq (g,Z,I)) . (k + 1)) by A86; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A9, A7); hence for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f + g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) + ((PartDiffSeq (g,Z,I)) . i) ) ; ::_thesis: verum end; end; theorem XCW011: :: PDIFF_9:70 for m being non empty Element of NAT for X being Subset of (REAL m) for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds ( f + g is_partial_differentiable_on X,I & (f + g) `partial| (X,I) = (f `partial| (X,I)) + (g `partial| (X,I)) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds ( f + g is_partial_differentiable_on X,I & (f + g) `partial| (X,I) = (f `partial| (X,I)) + (g `partial| (X,I)) ) let Z be Subset of (REAL m); ::_thesis: for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds ( f + g is_partial_differentiable_on Z,I & (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) ) let I be non empty FinSequence of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds ( f + g is_partial_differentiable_on Z,I & (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I implies ( f + g is_partial_differentiable_on Z,I & (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) ) ) assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) ; ::_thesis: ( f + g is_partial_differentiable_on Z,I & (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) ) then for i being Element of NAT st i <= (len I) - 1 holds (PartDiffSeq ((f + g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) by XCW010; hence f + g is_partial_differentiable_on Z,I by TDef6; ::_thesis: (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) 1 <= len I by FINSEQ_1:20; then reconsider k = (len I) - 1 as Element of NAT by INT_1:5; P1: ( (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) & (PartDiffSeq (g,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) ) by AS, TDef6; 1 <= k + 1 by NAT_1:11; then I /. (k + 1) in Seg m by AS, XCWLM1; then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1; R1: (PartDiffSeq ((f + g),Z,I)) . (k + 1) = ((PartDiffSeq ((f + g),Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5 .= (((PartDiffSeq (f,Z,I)) . k) + ((PartDiffSeq (g,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by AS, XCW010 .= (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) + (((PartDiffSeq (g,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by P1, AS, Q4, XXX1 ; (PartDiffSeq (f,Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5; hence (f + g) `partial| (Z,I) = (f `partial| (Z,I)) + (g `partial| (Z,I)) by R1, TDef5; ::_thesis: verum end; theorem XCW020: :: PDIFF_9:71 for m being non empty Element of NAT for X being Subset of (REAL m) for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f - g),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((f - g),X,I)) . i = ((PartDiffSeq (f,X,I)) . i) - ((PartDiffSeq (g,X,I)) . i) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f - g),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((f - g),X,I)) . i = ((PartDiffSeq (f,X,I)) . i) - ((PartDiffSeq (g,X,I)) . i) ) let Z be Subset of (REAL m); ::_thesis: for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) ) let I be non empty FinSequence of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I implies for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) ) ) assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) ; ::_thesis: for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) ) defpred S1[ Element of NAT ] means ( \$1 <= (len I) - 1 implies ( (PartDiffSeq ((f - g),Z,I)) . \$1 is_partial_differentiable_on Z,I /. (\$1 + 1) & (PartDiffSeq ((f - g),Z,I)) . \$1 = ((PartDiffSeq (f,Z,I)) . \$1) - ((PartDiffSeq (g,Z,I)) . \$1) ) ); reconsider Z0 = 0 as Element of NAT ; A9: S1[ 0 ] proof assume 0 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((f - g),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((f - g),Z,I)) . 0 = ((PartDiffSeq (f,Z,I)) . 0) - ((PartDiffSeq (g,Z,I)) . 0) ) then Q2: ( (PartDiffSeq (f,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) & (PartDiffSeq (g,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) ) by AS, TDef6; ( f = (PartDiffSeq (f,Z,I)) . Z0 & f - g = (PartDiffSeq ((f - g),Z,I)) . Z0 ) by TDef5; then Q5: (PartDiffSeq ((f - g),Z,I)) . Z0 = ((PartDiffSeq (f,Z,I)) . Z0) - ((PartDiffSeq (g,Z,I)) . Z0) by TDef5; 1 <= len I by FINSEQ_1:20; then I /. 1 in Seg m by AS, XCWLM1; then ( 1 <= I /. 1 & I /. 1 <= m ) by FINSEQ_1:1; hence ( (PartDiffSeq ((f - g),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((f - g),Z,I)) . 0 = ((PartDiffSeq (f,Z,I)) . 0) - ((PartDiffSeq (g,Z,I)) . 0) ) by Q5, AS, Q2, XXX2; ::_thesis: verum end; A7: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A8: S1[k] ; ::_thesis: S1[k + 1] assume A81: k + 1 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((f - g),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) - ((PartDiffSeq (g,Z,I)) . (k + 1)) ) B1: k <= k + 1 by NAT_1:11; then A82: k <= (len I) - 1 by A81, XXREAL_0:2; A84: ( (PartDiffSeq (f,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq (g,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) ) by A81, AS, TDef6; k + 1 <= ((len I) - 1) + 1 by A82, XREAL_1:6; then I /. (k + 1) in Seg m by AS, XCWLM1, NAT_1:11; then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1; A85: ( (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) & (PartDiffSeq (g,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) ) by A82, AS, TDef6; R1: (PartDiffSeq (f,Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5; (k + 1) + 1 <= ((len I) - 1) + 1 by A81, XREAL_1:6; then I /. ((k + 1) + 1) in Seg m by AS, XCWLM1, NAT_1:11; then Q5: ( 1 <= I /. ((k + 1) + 1) & I /. ((k + 1) + 1) <= m ) by FINSEQ_1:1; A86: (PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq ((f - g),Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5 .= (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) - (((PartDiffSeq (g,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by A85, AS, Q4, XXX2, B1, A8, A81, XXREAL_0:2 .= ((PartDiffSeq (f,Z,I)) . (k + 1)) - ((PartDiffSeq (g,Z,I)) . (k + 1)) by R1, TDef5 ; hence (PartDiffSeq ((f - g),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) by AS, A84, Q5, XXX2; ::_thesis: (PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) - ((PartDiffSeq (g,Z,I)) . (k + 1)) thus (PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . (k + 1)) - ((PartDiffSeq (g,Z,I)) . (k + 1)) by A86; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A9, A7); hence for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((f - g),Z,I)) . i = ((PartDiffSeq (f,Z,I)) . i) - ((PartDiffSeq (g,Z,I)) . i) ) ; ::_thesis: verum end; theorem XCW021: :: PDIFF_9:72 for m being non empty Element of NAT for X being Subset of (REAL m) for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds ( f - g is_partial_differentiable_on X,I & (f - g) `partial| (X,I) = (f `partial| (X,I)) - (g `partial| (X,I)) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I & g is_partial_differentiable_on X,I holds ( f - g is_partial_differentiable_on X,I & (f - g) `partial| (X,I) = (f `partial| (X,I)) - (g `partial| (X,I)) ) let Z be Subset of (REAL m); ::_thesis: for I being non empty FinSequence of NAT for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds ( f - g is_partial_differentiable_on Z,I & (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) ) let I be non empty FinSequence of NAT ; ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I holds ( f - g is_partial_differentiable_on Z,I & (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I implies ( f - g is_partial_differentiable_on Z,I & (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) ) ) assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) ; ::_thesis: ( f - g is_partial_differentiable_on Z,I & (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) ) then for i being Element of NAT st i <= (len I) - 1 holds (PartDiffSeq ((f - g),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) by XCW020; hence f - g is_partial_differentiable_on Z,I by TDef6; ::_thesis: (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) 1 <= len I by FINSEQ_1:20; then reconsider k = (len I) - 1 as Element of NAT by INT_1:5; P1: ( (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) & (PartDiffSeq (g,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) ) by AS, TDef6; 1 <= k + 1 by NAT_1:11; then I /. (k + 1) in Seg m by AS, XCWLM1; then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1; (PartDiffSeq ((f - g),Z,I)) . (k + 1) = ((PartDiffSeq ((f - g),Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5 .= (((PartDiffSeq (f,Z,I)) . k) - ((PartDiffSeq (g,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by AS, XCW020 ; then R1: (PartDiffSeq ((f - g),Z,I)) . (k + 1) = (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) - (((PartDiffSeq (g,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by P1, AS, Q4, XXX2; (PartDiffSeq (f,Z,I)) . (k + 1) = ((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5; hence (f - g) `partial| (Z,I) = (f `partial| (Z,I)) - (g `partial| (Z,I)) by R1, TDef5; ::_thesis: verum end; theorem XCW030: :: PDIFF_9:73 for m being non empty Element of NAT for X being Subset of (REAL m) for r being Real for I being non empty FinSequence of NAT for f being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((r (#) f),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((r (#) f),X,I)) . i = r (#) ((PartDiffSeq (f,X,I)) . i) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for r being Real for I being non empty FinSequence of NAT for f being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((r (#) f),X,I)) . i is_partial_differentiable_on X,I /. (i + 1) & (PartDiffSeq ((r (#) f),X,I)) . i = r (#) ((PartDiffSeq (f,X,I)) . i) ) let Z be Subset of (REAL m); ::_thesis: for r being Real for I being non empty FinSequence of NAT for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) ) let r be Real; ::_thesis: for I being non empty FinSequence of NAT for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) ) let I be non empty FinSequence of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) ) let f be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I implies for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) ) ) assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I ) ; ::_thesis: for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) ) defpred S1[ Element of NAT ] means ( \$1 <= (len I) - 1 implies ( (PartDiffSeq ((r (#) f),Z,I)) . \$1 is_partial_differentiable_on Z,I /. (\$1 + 1) & (PartDiffSeq ((r (#) f),Z,I)) . \$1 = r (#) ((PartDiffSeq (f,Z,I)) . \$1) ) ); reconsider Z0 = 0 as Element of NAT ; A9: S1[ 0 ] proof assume 0 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((r (#) f),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((r (#) f),Z,I)) . 0 = r (#) ((PartDiffSeq (f,Z,I)) . 0) ) then Q2: (PartDiffSeq (f,Z,I)) . Z0 is_partial_differentiable_on Z,I /. (Z0 + 1) by AS, TDef6; (PartDiffSeq ((r (#) f),Z,I)) . Z0 = r (#) f by TDef5; then Q5: (PartDiffSeq ((r (#) f),Z,I)) . Z0 = r (#) ((PartDiffSeq (f,Z,I)) . Z0) by TDef5; 1 <= len I by FINSEQ_1:20; then I /. 1 in Seg m by AS, XCWLM1; then ( 1 <= I /. 1 & I /. 1 <= m ) by FINSEQ_1:1; hence ( (PartDiffSeq ((r (#) f),Z,I)) . 0 is_partial_differentiable_on Z,I /. (0 + 1) & (PartDiffSeq ((r (#) f),Z,I)) . 0 = r (#) ((PartDiffSeq (f,Z,I)) . 0) ) by Q5, AS, Q2, XXX3; ::_thesis: verum end; A7: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A8: S1[k] ; ::_thesis: S1[k + 1] assume A81: k + 1 <= (len I) - 1 ; ::_thesis: ( (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) & (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = r (#) ((PartDiffSeq (f,Z,I)) . (k + 1)) ) B1: k <= k + 1 by NAT_1:11; then A82: k <= (len I) - 1 by A81, XXREAL_0:2; A84: (PartDiffSeq (f,Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) by A81, AS, TDef6; k + 1 <= ((len I) - 1) + 1 by A82, XREAL_1:6; then I /. (k + 1) in Seg m by AS, XCWLM1, NAT_1:11; then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1; A85: (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) by A82, AS, TDef6; (k + 1) + 1 <= ((len I) - 1) + 1 by A81, XREAL_1:6; then I /. ((k + 1) + 1) in Seg m by AS, XCWLM1, NAT_1:11; then Q5: ( 1 <= I /. ((k + 1) + 1) & I /. ((k + 1) + 1) <= m ) by FINSEQ_1:1; A86: (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = (r (#) ((PartDiffSeq (f,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by B1, A81, A8, TDef5, XXREAL_0:2 .= r (#) (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by A85, AS, Q4, XXX3 .= r (#) ((PartDiffSeq (f,Z,I)) . (k + 1)) by TDef5 ; hence (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) is_partial_differentiable_on Z,I /. ((k + 1) + 1) by AS, A84, Q5, XXX3; ::_thesis: (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = r (#) ((PartDiffSeq (f,Z,I)) . (k + 1)) thus (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = r (#) ((PartDiffSeq (f,Z,I)) . (k + 1)) by A86; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A9, A7); hence for i being Element of NAT st i <= (len I) - 1 holds ( (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) & (PartDiffSeq ((r (#) f),Z,I)) . i = r (#) ((PartDiffSeq (f,Z,I)) . i) ) ; ::_thesis: verum end; theorem XCW031: :: PDIFF_9:74 for m being non empty Element of NAT for X being Subset of (REAL m) for r being Real for I being non empty FinSequence of NAT for f being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I holds ( r (#) f is_partial_differentiable_on X,I & (r (#) f) `partial| (X,I) = r (#) (f `partial| (X,I)) ) proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) for r being Real for I being non empty FinSequence of NAT for f being PartFunc of (REAL m),REAL st X is open & rng I c= Seg m & f is_partial_differentiable_on X,I holds ( r (#) f is_partial_differentiable_on X,I & (r (#) f) `partial| (X,I) = r (#) (f `partial| (X,I)) ) let Z be Subset of (REAL m); ::_thesis: for r being Real for I being non empty FinSequence of NAT for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds ( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) ) let r be Real; ::_thesis: for I being non empty FinSequence of NAT for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds ( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) ) let I be non empty FinSequence of NAT ; ::_thesis: for f being PartFunc of (REAL m),REAL st Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I holds ( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) ) let f be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I implies ( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) ) ) assume AS: ( Z is open & rng I c= Seg m & f is_partial_differentiable_on Z,I ) ; ::_thesis: ( r (#) f is_partial_differentiable_on Z,I & (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) ) then for i being Element of NAT st i <= (len I) - 1 holds (PartDiffSeq ((r (#) f),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) by XCW030; hence r (#) f is_partial_differentiable_on Z,I by TDef6; ::_thesis: (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) 1 <= len I by FINSEQ_1:20; then reconsider k = (len I) - 1 as Element of NAT by INT_1:5; PP1: (PartDiffSeq (f,Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) by AS, TDef6; 1 <= k + 1 by NAT_1:11; then I /. (k + 1) in Seg m by AS, XCWLM1; then Q4: ( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by FINSEQ_1:1; (PartDiffSeq ((r (#) f),Z,I)) . (k + 1) = ((PartDiffSeq ((r (#) f),Z,I)) . k) `partial| (Z,(I /. (k + 1))) by TDef5 .= (r (#) ((PartDiffSeq (f,Z,I)) . k)) `partial| (Z,(I /. (k + 1))) by AS, XCW030 .= r (#) (((PartDiffSeq (f,Z,I)) . k) `partial| (Z,(I /. (k + 1)))) by PP1, AS, Q4, XXX3 ; hence (r (#) f) `partial| (Z,I) = r (#) (f `partial| (Z,I)) by TDef5; ::_thesis: verum end; definition let m be non empty Element of NAT ; let f be PartFunc of (REAL m),REAL; let k be Element of NAT ; let Z be set ; predf is_partial_differentiable_up_to_order k,Z means :TDef9: :: PDIFF_9:def 10 for I being non empty FinSequence of NAT st len I <= k & rng I c= Seg m holds f is_partial_differentiable_on Z,I; end; :: deftheorem TDef9 defines is_partial_differentiable_up_to_order PDIFF_9:def_10_:_ for m being non empty Element of NAT for f being PartFunc of (REAL m),REAL for k being Element of NAT for Z being set holds ( f is_partial_differentiable_up_to_order k,Z iff for I being non empty FinSequence of NAT st len I <= k & rng I c= Seg m holds f is_partial_differentiable_on Z,I ); theorem XCW040: :: PDIFF_9:75 for m being non empty Element of NAT for Z being set for f being PartFunc of (REAL m),REAL for I, G being non empty FinSequence of NAT holds ( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) ) proof let m be non empty Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL for I, G being non empty FinSequence of NAT holds ( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) ) let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL for I, G being non empty FinSequence of NAT holds ( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) ) let f be PartFunc of (REAL m),REAL; ::_thesis: for I, G being non empty FinSequence of NAT holds ( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) ) let I, G be non empty FinSequence of NAT ; ::_thesis: ( f is_partial_differentiable_on Z,G ^ I iff ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) ) set g = f `partial| (Z,G); reconsider Z0 = 0 as Element of NAT ; S1: dom G c= dom (G ^ I) by FINSEQ_1:26; Y0: for i being Element of NAT st i <= (len G) - 1 holds (G ^ I) /. (i + 1) = G /. (i + 1) proof let i be Element of NAT ; ::_thesis: ( i <= (len G) - 1 implies (G ^ I) /. (i + 1) = G /. (i + 1) ) assume i <= (len G) - 1 ; ::_thesis: (G ^ I) /. (i + 1) = G /. (i + 1) then ( 1 <= i + 1 & i + 1 <= len G ) by NAT_1:11, XREAL_1:19; then D3: i + 1 in dom G by FINSEQ_3:25; then (G ^ I) /. (i + 1) = (G ^ I) . (i + 1) by S1, PARTFUN1:def_6; then (G ^ I) /. (i + 1) = G . (i + 1) by D3, FINSEQ_1:def_7; hence (G ^ I) /. (i + 1) = G /. (i + 1) by D3, PARTFUN1:def_6; ::_thesis: verum end; X2: len (G ^ I) = (len G) + (len I) by FINSEQ_1:22; X3: for i being Element of NAT st i <= (len I) - 1 holds (G ^ I) /. ((len G) + (i + 1)) = I /. (i + 1) proof let i be Element of NAT ; ::_thesis: ( i <= (len I) - 1 implies (G ^ I) /. ((len G) + (i + 1)) = I /. (i + 1) ) assume i <= (len I) - 1 ; ::_thesis: (G ^ I) /. ((len G) + (i + 1)) = I /. (i + 1) then D2: i + 1 <= len I by XREAL_1:19; 1 <= i + 1 by NAT_1:11; then D3: i + 1 in dom I by D2, FINSEQ_3:25; D9: 1 <= (len G) + (i + 1) by NAT_1:11, XREAL_1:38; (len G) + (i + 1) <= len (G ^ I) by X2, D2, XREAL_1:7; then (len G) + (i + 1) in dom (G ^ I) by D9, FINSEQ_3:25; hence (G ^ I) /. ((len G) + (i + 1)) = (G ^ I) . ((len G) + (i + 1)) by PARTFUN1:def_6 .= I . (i + 1) by D3, FINSEQ_1:def_7 .= I /. (i + 1) by D3, PARTFUN1:def_6 ; ::_thesis: verum end; defpred S1[ Element of NAT ] means ( \$1 <= (len G) - 1 implies (PartDiffSeq (f,Z,(G ^ I))) . \$1 = (PartDiffSeq (f,Z,G)) . \$1 ); B1: S1[ 0 ] proof assume 0 <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . 0 = (PartDiffSeq (f,Z,G)) . 0 (PartDiffSeq (f,Z,(G ^ I))) . 0 = f by TDef5; hence (PartDiffSeq (f,Z,(G ^ I))) . 0 = (PartDiffSeq (f,Z,G)) . 0 by TDef5; ::_thesis: verum end; B2: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume D1: S1[k] ; ::_thesis: S1[k + 1] assume D2: k + 1 <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . (k + 1) = (PartDiffSeq (f,Z,G)) . (k + 1) D20: k <= k + 1 by NAT_1:11; thus (PartDiffSeq (f,Z,(G ^ I))) . (k + 1) = ((PartDiffSeq (f,Z,(G ^ I))) . k) `partial| (Z,((G ^ I) /. (k + 1))) by TDef5 .= ((PartDiffSeq (f,Z,G)) . k) `partial| (Z,(G /. (k + 1))) by D20, Y0, D1, D2, XXREAL_0:2 .= (PartDiffSeq (f,Z,G)) . (k + 1) by TDef5 ; ::_thesis: verum end; Y1: for n being Element of NAT holds S1[n] from NAT_1:sch_1(B1, B2); 1 <= len G by FINSEQ_1:20; then reconsider j = (len G) - 1 as Element of NAT by INT_1:5; Y11: (PartDiffSeq (f,Z,(G ^ I))) . (len G) = ((PartDiffSeq (f,Z,(G ^ I))) . j) `partial| (Z,((G ^ I) /. (j + 1))) by TDef5 .= ((PartDiffSeq (f,Z,G)) . j) `partial| (Z,((G ^ I) /. (j + 1))) by Y1 .= ((PartDiffSeq (f,Z,G)) . j) `partial| (Z,(G /. (j + 1))) by Y0 .= (PartDiffSeq (f,Z,G)) . (len G) by TDef5 ; defpred S2[ Element of NAT ] means ( \$1 <= (len I) - 1 implies (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . \$1 = (PartDiffSeq (f,Z,(G ^ I))) . ((len G) + \$1) ); A1: S2[ 0 ] by Y11, TDef5; A2: for k being Element of NAT st S2[k] holds S2[k + 1] proof let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] ) assume A3: S2[k] ; ::_thesis: S2[k + 1] set i = (len G) + k; assume P0: k + 1 <= (len I) - 1 ; ::_thesis: (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . (k + 1) = (PartDiffSeq (f,Z,(G ^ I))) . ((len G) + (k + 1)) P1: k <= k + 1 by NAT_1:11; (G ^ I) /. (((len G) + k) + 1) = (G ^ I) /. ((len G) + (k + 1)) ; then P2: (G ^ I) /. (((len G) + k) + 1) = I /. (k + 1) by X3, P1, P0, XXREAL_0:2; (PartDiffSeq (f,Z,(G ^ I))) . ((len G) + (k + 1)) = ((PartDiffSeq (f,Z,(G ^ I))) . ((len G) + k)) `partial| (Z,((G ^ I) /. (((len G) + k) + 1))) by TDef5; hence (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . (k + 1) = (PartDiffSeq (f,Z,(G ^ I))) . ((len G) + (k + 1)) by P1, P0, A3, P2, TDef5, XXREAL_0:2; ::_thesis: verum end; X1: for n being Element of NAT holds S2[n] from NAT_1:sch_1(A1, A2); hereby ::_thesis: ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I implies f is_partial_differentiable_on Z,G ^ I ) assume P1: f is_partial_differentiable_on Z,G ^ I ; ::_thesis: ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) now__::_thesis:_for_i_being_Element_of_NAT_st_i_<=_(len_G)_-_1_holds_ (PartDiffSeq_(f,Z,G))_._i_is_partial_differentiable_on_Z,G_/._(i_+_1) let i be Element of NAT ; ::_thesis: ( i <= (len G) - 1 implies (PartDiffSeq (f,Z,G)) . i is_partial_differentiable_on Z,G /. (i + 1) ) assume D1: i <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,G)) . i is_partial_differentiable_on Z,G /. (i + 1) then i + Z0 <= ((len G) - 1) + (len I) by XREAL_1:7; then i <= (len (G ^ I)) - 1 by X2; then P2: (PartDiffSeq (f,Z,(G ^ I))) . i is_partial_differentiable_on Z,(G ^ I) /. (i + 1) by TDef6, P1; (G ^ I) /. (i + 1) = G /. (i + 1) by D1, Y0; hence (PartDiffSeq (f,Z,G)) . i is_partial_differentiable_on Z,G /. (i + 1) by P2, D1, Y1; ::_thesis: verum end; hence f is_partial_differentiable_on Z,G by TDef6; ::_thesis: f `partial| (Z,G) is_partial_differentiable_on Z,I now__::_thesis:_for_i_being_Element_of_NAT_st_i_<=_(len_I)_-_1_holds_ (PartDiffSeq_((f_`partial|_(Z,G)),Z,I))_._i_is_partial_differentiable_on_Z,I_/._(i_+_1) let i be Element of NAT ; ::_thesis: ( i <= (len I) - 1 implies (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) ) assume S1: i <= (len I) - 1 ; ::_thesis: (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) then (len G) + i <= (len G) + ((len I) - 1) by XREAL_1:6; then (len G) + i <= (len (G ^ I)) - 1 by X2; then X4: (PartDiffSeq (f,Z,(G ^ I))) . ((len G) + i) is_partial_differentiable_on Z,(G ^ I) /. (((len G) + i) + 1) by TDef6, P1; (G ^ I) /. (((len G) + i) + 1) = (G ^ I) /. ((len G) + (i + 1)) ; then (G ^ I) /. (((len G) + i) + 1) = I /. (i + 1) by S1, X3; hence (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . i is_partial_differentiable_on Z,I /. (i + 1) by X4, S1, X1; ::_thesis: verum end; hence f `partial| (Z,G) is_partial_differentiable_on Z,I by TDef6; ::_thesis: verum end; now__::_thesis:_(_f_is_partial_differentiable_on_Z,G_&_f_`partial|_(Z,G)_is_partial_differentiable_on_Z,I_implies_f_is_partial_differentiable_on_Z,G_^_I_) assume P0: ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I ) ; ::_thesis: f is_partial_differentiable_on Z,G ^ I now__::_thesis:_for_i_being_Element_of_NAT_st_i_<=_(len_(G_^_I))_-_1_holds_ (PartDiffSeq_(f,Z,(G_^_I)))_._i_is_partial_differentiable_on_Z,(G_^_I)_/._(i_+_1) let i be Element of NAT ; ::_thesis: ( i <= (len (G ^ I)) - 1 implies (PartDiffSeq (f,Z,(G ^ I))) . b1 is_partial_differentiable_on Z,(G ^ I) /. (b1 + 1) ) assume Q1: i <= (len (G ^ I)) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . b1 is_partial_differentiable_on Z,(G ^ I) /. (b1 + 1) percases ( i <= (len G) - 1 or not i <= (len G) - 1 ) ; supposeQ2: i <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . b1 is_partial_differentiable_on Z,(G ^ I) /. (b1 + 1) then Q3: (PartDiffSeq (f,Z,G)) . i is_partial_differentiable_on Z,G /. (i + 1) by TDef6, P0; G /. (i + 1) = (G ^ I) /. (i + 1) by Q2, Y0; hence (PartDiffSeq (f,Z,(G ^ I))) . i is_partial_differentiable_on Z,(G ^ I) /. (i + 1) by Q2, Y1, Q3; ::_thesis: verum end; suppose not i <= (len G) - 1 ; ::_thesis: (PartDiffSeq (f,Z,(G ^ I))) . b1 is_partial_differentiable_on Z,(G ^ I) /. (b1 + 1) then len G < i + 1 by XREAL_1:19; then len G <= i by NAT_1:13; then reconsider k = i - (len G) as Element of NAT by INT_1:5; R5: i - (len G) <= (((len G) + (len I)) - 1) - (len G) by Q1, X2, XREAL_1:9; then Q5: ( k <= (len I) - 1 & i = k + (len G) ) ; then Q6: (PartDiffSeq ((f `partial| (Z,G)),Z,I)) . k is_partial_differentiable_on Z,I /. (k + 1) by TDef6, P0; i + 1 = (k + 1) + (len G) ; then I /. (k + 1) = (G ^ I) /. (i + 1) by R5, X3; hence (PartDiffSeq (f,Z,(G ^ I))) . i is_partial_differentiable_on Z,(G ^ I) /. (i + 1) by Q6, Q5, X1; ::_thesis: verum end; end; end; hence f is_partial_differentiable_on Z,G ^ I by TDef6; ::_thesis: verum end; hence ( f is_partial_differentiable_on Z,G & f `partial| (Z,G) is_partial_differentiable_on Z,I implies f is_partial_differentiable_on Z,G ^ I ) ; ::_thesis: verum end; set Z0 = 0 ; theorem XCW041: :: PDIFF_9:76 for m being non empty Element of NAT for i being Element of NAT for Z being set for f being PartFunc of (REAL m),REAL holds ( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for Z being set for f being PartFunc of (REAL m),REAL holds ( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i ) let i be Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL holds ( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i ) let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL holds ( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i ) let f be PartFunc of (REAL m),REAL; ::_thesis: ( f is_partial_differentiable_on Z,<*i*> iff f is_partial_differentiable_on Z,i ) set I = <*i*>; P0: len <*i*> = 1 by FINSEQ_1:39; P1: (PartDiffSeq (f,Z,<*i*>)) . 0 = f by TDef5; 1 in Seg 1 ; then X2: 1 in dom <*i*> by FINSEQ_1:38; <*i*> /. (0 + 1) = <*i*> . 1 by X2, PARTFUN1:def_6; then Q1: <*i*> /. (0 + 1) = i by FINSEQ_1:40; hereby ::_thesis: ( f is_partial_differentiable_on Z,i implies f is_partial_differentiable_on Z,<*i*> ) assume PX1: f is_partial_differentiable_on Z,<*i*> ; ::_thesis: f is_partial_differentiable_on Z,i 0 <= (len <*i*>) - 1 by P0; hence f is_partial_differentiable_on Z,i by Q1, P1, PX1, TDef6; ::_thesis: verum end; assume P3: f is_partial_differentiable_on Z,i ; ::_thesis: f is_partial_differentiable_on Z,<*i*> now__::_thesis:_for_k_being_Element_of_NAT_st_k_<=_(len_<*i*>)_-_1_holds_ (PartDiffSeq_(f,Z,<*i*>))_._k_is_partial_differentiable_on_Z,<*i*>_/._(k_+_1) let k be Element of NAT ; ::_thesis: ( k <= (len <*i*>) - 1 implies (PartDiffSeq (f,Z,<*i*>)) . k is_partial_differentiable_on Z,<*i*> /. (k + 1) ) assume k <= (len <*i*>) - 1 ; ::_thesis: (PartDiffSeq (f,Z,<*i*>)) . k is_partial_differentiable_on Z,<*i*> /. (k + 1) then P5: k = 0 by P0; then <*i*> /. (k + 1) = <*i*> . 1 by X2, PARTFUN1:def_6; then <*i*> /. (k + 1) = i by FINSEQ_1:40; hence (PartDiffSeq (f,Z,<*i*>)) . k is_partial_differentiable_on Z,<*i*> /. (k + 1) by P3, P5, TDef5; ::_thesis: verum end; hence f is_partial_differentiable_on Z,<*i*> by TDef6; ::_thesis: verum end; theorem XCW042: :: PDIFF_9:77 for m being non empty Element of NAT for i being Element of NAT for Z being set for f being PartFunc of (REAL m),REAL holds f `partial| (Z,<*i*>) = f `partial| (Z,i) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for Z being set for f being PartFunc of (REAL m),REAL holds f `partial| (Z,<*i*>) = f `partial| (Z,i) let i be Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL holds f `partial| (Z,<*i*>) = f `partial| (Z,i) let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL holds f `partial| (Z,<*i*>) = f `partial| (Z,i) let f be PartFunc of (REAL m),REAL; ::_thesis: f `partial| (Z,<*i*>) = f `partial| (Z,i) set I = <*i*>; 1 in Seg 1 ; then 1 in dom <*i*> by FINSEQ_1:38; then <*i*> /. (0 + 1) = <*i*> . 1 by PARTFUN1:def_6; then Q1: <*i*> /. (0 + 1) = i by FINSEQ_1:40; thus f `partial| (Z,<*i*>) = (PartDiffSeq (f,Z,<*i*>)) . 1 by FINSEQ_1:39 .= ((PartDiffSeq (f,Z,<*i*>)) . 0) `partial| (Z,(<*i*> /. (0 + 1))) by TDef5 .= f `partial| (Z,i) by Q1, TDef5 ; ::_thesis: verum end; theorem XCW0400: :: PDIFF_9:78 for m being non empty Element of NAT for i, j being Element of NAT for Z being set for f being PartFunc of (REAL m),REAL for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z proof let m be non empty Element of NAT ; ::_thesis: for i, j being Element of NAT for Z being set for f being PartFunc of (REAL m),REAL for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z let i, j be Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z let f be PartFunc of (REAL m),REAL; ::_thesis: for I being non empty FinSequence of NAT st f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j holds f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z let I be non empty FinSequence of NAT ; ::_thesis: ( f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j implies f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z ) assume AS: ( f is_partial_differentiable_up_to_order i + j,Z & rng I c= Seg m & len I = j ) ; ::_thesis: f `partial| (Z,I) is_partial_differentiable_up_to_order i,Z let J be non empty FinSequence of NAT ; :: according to PDIFF_9:def_10 ::_thesis: ( len J <= i & rng J c= Seg m implies f `partial| (Z,I) is_partial_differentiable_on Z,J ) assume AS1: ( len J <= i & rng J c= Seg m ) ; ::_thesis: f `partial| (Z,I) is_partial_differentiable_on Z,J reconsider G = I ^ J as non empty FinSequence of NAT ; P1: rng G = (rng I) \/ (rng J) by FINSEQ_1:31; len G = (len I) + (len J) by FINSEQ_1:22; then ( len G <= i + j & rng G c= Seg m ) by AS1, P1, AS, XBOOLE_1:8, XREAL_1:6; then f is_partial_differentiable_on Z,G by AS, TDef9; hence f `partial| (Z,I) is_partial_differentiable_on Z,J by XCW040; ::_thesis: verum end; theorem XCW0410: :: PDIFF_9:79 for m being non empty Element of NAT for i, j being Element of NAT for Z being set for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & j <= i holds f is_partial_differentiable_up_to_order j,Z proof let m be non empty Element of NAT ; ::_thesis: for i, j being Element of NAT for Z being set for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & j <= i holds f is_partial_differentiable_up_to_order j,Z let i, j be Element of NAT ; ::_thesis: for Z being set for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & j <= i holds f is_partial_differentiable_up_to_order j,Z let Z be set ; ::_thesis: for f being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & j <= i holds f is_partial_differentiable_up_to_order j,Z let f be PartFunc of (REAL m),REAL; ::_thesis: ( f is_partial_differentiable_up_to_order i,Z & j <= i implies f is_partial_differentiable_up_to_order j,Z ) assume AS: ( f is_partial_differentiable_up_to_order i,Z & j <= i ) ; ::_thesis: f is_partial_differentiable_up_to_order j,Z let I be non empty FinSequence of NAT ; :: according to PDIFF_9:def_10 ::_thesis: ( len I <= j & rng I c= Seg m implies f is_partial_differentiable_on Z,I ) assume AS1: ( len I <= j & rng I c= Seg m ) ; ::_thesis: f is_partial_differentiable_on Z,I then len I <= i by AS, XXREAL_0:2; hence f is_partial_differentiable_on Z,I by AS, AS1, TDef9; ::_thesis: verum end; theorem :: PDIFF_9:80 for m being non empty Element of NAT for i being Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds ( f + g is_partial_differentiable_up_to_order i,X & f - g is_partial_differentiable_up_to_order i,X ) proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds ( f + g is_partial_differentiable_up_to_order i,X & f - g is_partial_differentiable_up_to_order i,X ) let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f, g being PartFunc of (REAL m),REAL st X is open & f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds ( f + g is_partial_differentiable_up_to_order i,X & f - g is_partial_differentiable_up_to_order i,X ) let Z be Subset of (REAL m); ::_thesis: for f, g being PartFunc of (REAL m),REAL st Z is open & f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z holds ( f + g is_partial_differentiable_up_to_order i,Z & f - g is_partial_differentiable_up_to_order i,Z ) let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( Z is open & f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z implies ( f + g is_partial_differentiable_up_to_order i,Z & f - g is_partial_differentiable_up_to_order i,Z ) ) assume AS: ( Z is open & f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z ) ; ::_thesis: ( f + g is_partial_differentiable_up_to_order i,Z & f - g is_partial_differentiable_up_to_order i,Z ) hereby :: according to PDIFF_9:def_10 ::_thesis: f - g is_partial_differentiable_up_to_order i,Z let I be non empty FinSequence of NAT ; ::_thesis: ( len I <= i & rng I c= Seg m implies f + g is_partial_differentiable_on Z,I ) assume P1: ( len I <= i & rng I c= Seg m ) ; ::_thesis: f + g is_partial_differentiable_on Z,I then ( f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) by AS, TDef9; hence f + g is_partial_differentiable_on Z,I by AS, P1, XCW011; ::_thesis: verum end; let I be non empty FinSequence of NAT ; :: according to PDIFF_9:def_10 ::_thesis: ( len I <= i & rng I c= Seg m implies f - g is_partial_differentiable_on Z,I ) assume P1: ( len I <= i & rng I c= Seg m ) ; ::_thesis: f - g is_partial_differentiable_on Z,I then ( f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) by AS, TDef9; hence f - g is_partial_differentiable_on Z,I by AS, P1, XCW021; ::_thesis: verum end; theorem :: PDIFF_9:81 for m being non empty Element of NAT for i being Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for r being Real st X is open & f is_partial_differentiable_up_to_order i,X holds r (#) f is_partial_differentiable_up_to_order i,X proof let m be non empty Element of NAT ; ::_thesis: for i being Element of NAT for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for r being Real st X is open & f is_partial_differentiable_up_to_order i,X holds r (#) f is_partial_differentiable_up_to_order i,X let i be Element of NAT ; ::_thesis: for X being Subset of (REAL m) for f being PartFunc of (REAL m),REAL for r being Real st X is open & f is_partial_differentiable_up_to_order i,X holds r (#) f is_partial_differentiable_up_to_order i,X let Z be Subset of (REAL m); ::_thesis: for f being PartFunc of (REAL m),REAL for r being Real st Z is open & f is_partial_differentiable_up_to_order i,Z holds r (#) f is_partial_differentiable_up_to_order i,Z let f be PartFunc of (REAL m),REAL; ::_thesis: for r being Real st Z is open & f is_partial_differentiable_up_to_order i,Z holds r (#) f is_partial_differentiable_up_to_order i,Z let r be Real; ::_thesis: ( Z is open & f is_partial_differentiable_up_to_order i,Z implies r (#) f is_partial_differentiable_up_to_order i,Z ) assume AS: ( Z is open & f is_partial_differentiable_up_to_order i,Z ) ; ::_thesis: r (#) f is_partial_differentiable_up_to_order i,Z let I be non empty FinSequence of NAT ; :: according to PDIFF_9:def_10 ::_thesis: ( len I <= i & rng I c= Seg m implies r (#) f is_partial_differentiable_on Z,I ) assume P1: ( len I <= i & rng I c= Seg m ) ; ::_thesis: r (#) f is_partial_differentiable_on Z,I then f is_partial_differentiable_on Z,I by AS, TDef9; hence r (#) f is_partial_differentiable_on Z,I by AS, P1, XCW031; ::_thesis: verum end; theorem :: PDIFF_9:82 for m being non empty Element of NAT for X being Subset of (REAL m) st X is open holds for i being Element of NAT for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds f (#) g is_partial_differentiable_up_to_order i,X proof let m be non empty Element of NAT ; ::_thesis: for X being Subset of (REAL m) st X is open holds for i being Element of NAT for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,X & g is_partial_differentiable_up_to_order i,X holds f (#) g is_partial_differentiable_up_to_order i,X let Z be Subset of (REAL m); ::_thesis: ( Z is open implies for i being Element of NAT for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z holds f (#) g is_partial_differentiable_up_to_order i,Z ) assume AS: Z is open ; ::_thesis: for i being Element of NAT for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z holds f (#) g is_partial_differentiable_up_to_order i,Z defpred S1[ Element of NAT ] means for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order \$1,Z & g is_partial_differentiable_up_to_order \$1,Z holds f (#) g is_partial_differentiable_up_to_order \$1,Z; P9: S1[ 0 ] proof let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( f is_partial_differentiable_up_to_order 0 ,Z & g is_partial_differentiable_up_to_order 0 ,Z implies f (#) g is_partial_differentiable_up_to_order 0 ,Z ) assume ( f is_partial_differentiable_up_to_order 0 ,Z & g is_partial_differentiable_up_to_order 0 ,Z ) ; ::_thesis: f (#) g is_partial_differentiable_up_to_order 0 ,Z for I being non empty FinSequence of NAT st len I <= 0 & rng I c= Seg m holds f (#) g is_partial_differentiable_on Z,I by FINSEQ_1:20; hence f (#) g is_partial_differentiable_up_to_order 0 ,Z by TDef9; ::_thesis: verum end; P7: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume P71: S1[k] ; ::_thesis: S1[k + 1] let f, g be PartFunc of (REAL m),REAL; ::_thesis: ( f is_partial_differentiable_up_to_order k + 1,Z & g is_partial_differentiable_up_to_order k + 1,Z implies f (#) g is_partial_differentiable_up_to_order k + 1,Z ) assume R71: ( f is_partial_differentiable_up_to_order k + 1,Z & g is_partial_differentiable_up_to_order k + 1,Z ) ; ::_thesis: f (#) g is_partial_differentiable_up_to_order k + 1,Z then R711: ( f is_partial_differentiable_up_to_order k,Z & g is_partial_differentiable_up_to_order k,Z ) by XCW0410, NAT_1:11; now__::_thesis:_for_I_being_non_empty_FinSequence_of_NAT_st_len_I_<=_k_+_1_&_rng_I_c=_Seg_m_holds_ f_(#)_g_is_partial_differentiable_on_Z,I let I be non empty FinSequence of NAT ; ::_thesis: ( len I <= k + 1 & rng I c= Seg m implies f (#) g is_partial_differentiable_on Z,b1 ) assume P72: ( len I <= k + 1 & rng I c= Seg m ) ; ::_thesis: f (#) g is_partial_differentiable_on Z,b1 then R721: ( f is_partial_differentiable_on Z,I & g is_partial_differentiable_on Z,I ) by R71, TDef9; RRRR: 1 <= len I by FINSEQ_1:20; then T1: 1 in dom I by FINSEQ_3:25; then T4: I /. 1 = I . 1 by PARTFUN1:def_6; T2: I . 1 in rng I by T1, FUNCT_1:3; then I . 1 in Seg m by P72; then reconsider i = I . 1 as Element of NAT ; P75: ( 1 <= i & i <= m ) by T2, P72, FINSEQ_1:1; percases ( 1 = len I or 1 <> len I ) ; suppose 1 = len I ; ::_thesis: f (#) g is_partial_differentiable_on Z,b1 then T3: I = <*(I /. 1)*> by FINSEQ_5:14; then ( f is_partial_differentiable_on Z,i & g is_partial_differentiable_on Z,i ) by XCW041, R721, T4; then f (#) g is_partial_differentiable_on Z,i by P75, XXX4, AS; hence f (#) g is_partial_differentiable_on Z,I by XCW041, T3, T4; ::_thesis: verum end; suppose 1 <> len I ; ::_thesis: f (#) g is_partial_differentiable_on Z,b1 then 1 < len I by RRRR, XXREAL_0:1; then 1 + 1 <= len I by NAT_1:13; then 1 <= (len I) - 1 by XREAL_1:19; then 1 <= len (I /^ 1) by RRRR, RFINSEQ:def_1; then reconsider J = I /^ 1 as non empty FinSequence of NAT by FINSEQ_1:20; set I1 = <*i*>; (len I) - 1 <= k by P72, XREAL_1:20; then P74: len J <= k by RRRR, RFINSEQ:def_1; V1: I = <*(I /. 1)*> ^ (I /^ 1) by FINSEQ_5:29; then U1: ( rng <*i*> c= rng I & rng J c= rng I ) by T4, FINSEQ_1:29, FINSEQ_1:30; then P76: rng J c= Seg m by P72, XBOOLE_1:1; I = <*i*> ^ J by T4, FINSEQ_5:29; then ( f is_partial_differentiable_on Z,<*i*> & g is_partial_differentiable_on Z,<*i*> ) by XCW040, R721; then P79: ( f is_partial_differentiable_on Z,i & g is_partial_differentiable_on Z,i ) by XCW041; then f (#) g is_partial_differentiable_on Z,i by P75, XXX4, AS; then P86: f (#) g is_partial_differentiable_on Z,<*i*> by XCW041; P87: (f (#) g) `partial| (Z,<*i*>) = (f (#) g) `partial| (Z,i) by XCW042 .= ((f `partial| (Z,i)) (#) g) + (f (#) (g `partial| (Z,i))) by P79, P75, XXX4, AS .= ((f `partial| (Z,<*i*>)) (#) g) + (f (#) (g `partial| (Z,i))) by XCW042 .= ((f `partial| (Z,<*i*>)) (#) g) + (f (#) (g `partial| (Z,<*i*>))) by XCW042 ; ( len <*i*> = 1 & rng <*i*> c= Seg m ) by U1, P72, FINSEQ_1:39, XBOOLE_1:1; then ( f `partial| (Z,<*i*>) is_partial_differentiable_up_to_order k,Z & g `partial| (Z,<*i*>) is_partial_differentiable_up_to_order k,Z ) by XCW0400, R71; then ( (f `partial| (Z,<*i*>)) (#) g is_partial_differentiable_up_to_order k,Z & f (#) (g `partial| (Z,<*i*>)) is_partial_differentiable_up_to_order k,Z ) by P71, R711; then ( (f `partial| (Z,<*i*>)) (#) g is_partial_differentiable_on Z,J & f (#) (g `partial| (Z,<*i*>)) is_partial_differentiable_on Z,J ) by P74, P76, TDef9; then ((f `partial| (Z,<*i*>)) (#) g) + (f (#) (g `partial| (Z,<*i*>))) is_partial_differentiable_on Z,J by AS, P76, XCW011; hence f (#) g is_partial_differentiable_on Z,I by P86, V1, T4, P87, XCW040; ::_thesis: verum end; end; end; hence f (#) g is_partial_differentiable_up_to_order k + 1,Z by TDef9; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(P9, P7); hence for i being Element of NAT for f, g being PartFunc of (REAL m),REAL st f is_partial_differentiable_up_to_order i,Z & g is_partial_differentiable_up_to_order i,Z holds f (#) g is_partial_differentiable_up_to_order i,Z ; ::_thesis: verum end;