:: VALUED_1 semantic presentation begin Lm1: for f being FinSequence for h being Function st dom h = dom f holds h is FinSequence proof let f be FinSequence; ::_thesis: for h being Function st dom h = dom f holds h is FinSequence let h be Function; ::_thesis: ( dom h = dom f implies h is FinSequence ) assume A1: dom h = dom f ; ::_thesis: h is FinSequence h is FinSequence-like proof take len f ; :: according to FINSEQ_1:def_2 ::_thesis: dom h = Seg (len f) thus dom h = Seg (len f) by A1, FINSEQ_1:def_3; ::_thesis: verum end; hence h is FinSequence ; ::_thesis: verum end; Lm2: for f, g being FinSequence for h being Function st dom h = (dom f) /\ (dom g) holds h is FinSequence proof let f, g be FinSequence; ::_thesis: for h being Function st dom h = (dom f) /\ (dom g) holds h is FinSequence let h be Function; ::_thesis: ( dom h = (dom f) /\ (dom g) implies h is FinSequence ) assume A1: dom h = (dom f) /\ (dom g) ; ::_thesis: h is FinSequence consider n being Nat such that A2: dom f = Seg n by FINSEQ_1:def_2; consider m being Nat such that A3: dom g = Seg m by FINSEQ_1:def_2; h is FinSequence-like proof percases ( n <= m or m <= n ) ; supposeA4: n <= m ; ::_thesis: h is FinSequence-like take n ; :: according to FINSEQ_1:def_2 ::_thesis: dom h = Seg n thus dom h = Seg n by A1, A2, A3, A4, FINSEQ_1:7; ::_thesis: verum end; supposeA5: m <= n ; ::_thesis: h is FinSequence-like take m ; :: according to FINSEQ_1:def_2 ::_thesis: dom h = Seg m thus dom h = Seg m by A1, A2, A3, A5, FINSEQ_1:7; ::_thesis: verum end; end; end; hence h is FinSequence ; ::_thesis: verum end; registration cluster Relation-like Function-like complex-valued FinSequence-like for set ; existence ex b1 being FinSequence st b1 is complex-valued proof take <*> COMPLEX ; ::_thesis: <*> COMPLEX is complex-valued thus <*> COMPLEX is complex-valued ; ::_thesis: verum end; end; registration let r be rational number ; cluster|.r.| -> rational ; coherence |.r.| is rational proof ( |.r.| = - r or |.r.| = r ) by COMPLEX1:71; hence |.r.| is rational ; ::_thesis: verum end; end; definition let f1, f2 be complex-valued Function; deffunc H1( set ) -> set = (f1 . \$1) + (f2 . \$1); set X = (dom f1) /\ (dom f2); funcf1 + f2 -> Function means :Def1: :: VALUED_1:def 1 ( dom it = (dom f1) /\ (dom f2) & ( for c being set st c in dom it holds it . c = (f1 . c) + (f2 . c) ) ); existence ex b1 being Function st ( dom b1 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b1 holds b1 . c = (f1 . c) + (f2 . c) ) ) proof ex f being Function st ( dom f = (dom f1) /\ (dom f2) & ( for x being set st x in (dom f1) /\ (dom f2) holds f . x = H1(x) ) ) from FUNCT_1:sch_3(); hence ex b1 being Function st ( dom b1 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b1 holds b1 . c = (f1 . c) + (f2 . c) ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Function st dom b1 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b1 holds b1 . c = (f1 . c) + (f2 . c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b2 holds b2 . c = (f1 . c) + (f2 . c) ) holds b1 = b2 proof let f, g be Function; ::_thesis: ( dom f = (dom f1) /\ (dom f2) & ( for c being set st c in dom f holds f . c = (f1 . c) + (f2 . c) ) & dom g = (dom f1) /\ (dom f2) & ( for c being set st c in dom g holds g . c = (f1 . c) + (f2 . c) ) implies f = g ) assume that A1: dom f = (dom f1) /\ (dom f2) and A2: for c being set st c in dom f holds f . c = H1(c) and A3: dom g = (dom f1) /\ (dom f2) and A4: for c being set st c in dom g holds g . c = H1(c) ; ::_thesis: f = g now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_ f_._x_=_g_._x let x be set ; ::_thesis: ( x in dom f implies f . x = g . x ) assume A5: x in dom f ; ::_thesis: f . x = g . x hence f . x = H1(x) by A2 .= g . x by A1, A3, A4, A5 ; ::_thesis: verum end; hence f = g by A1, A3, FUNCT_1:2; ::_thesis: verum end; commutativity for b1 being Function for f1, f2 being complex-valued Function st dom b1 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b1 holds b1 . c = (f1 . c) + (f2 . c) ) holds ( dom b1 = (dom f2) /\ (dom f1) & ( for c being set st c in dom b1 holds b1 . c = (f2 . c) + (f1 . c) ) ) ; end; :: deftheorem Def1 defines + VALUED_1:def_1_:_ for f1, f2 being complex-valued Function for b3 being Function holds ( b3 = f1 + f2 iff ( dom b3 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b3 holds b3 . c = (f1 . c) + (f2 . c) ) ) ); registration let f1, f2 be complex-valued Function; clusterf1 + f2 -> complex-valued ; coherence f1 + f2 is complex-valued proof let x be set ; :: according to VALUED_0:def_7 ::_thesis: ( not x in dom (f1 + f2) or (f1 + f2) . x is complex ) assume x in dom (f1 + f2) ; ::_thesis: (f1 + f2) . x is complex then (f1 + f2) . x = (f1 . x) + (f2 . x) by Def1; hence (f1 + f2) . x is complex ; ::_thesis: verum end; end; registration let f1, f2 be real-valued Function; clusterf1 + f2 -> real-valued ; coherence f1 + f2 is real-valued proof let x be set ; :: according to VALUED_0:def_9 ::_thesis: ( not x in dom (f1 + f2) or (f1 + f2) . x is real ) assume x in dom (f1 + f2) ; ::_thesis: (f1 + f2) . x is real then (f1 + f2) . x = (f1 . x) + (f2 . x) by Def1; hence (f1 + f2) . x is real ; ::_thesis: verum end; end; registration let f1, f2 be RAT -valued Function; clusterf1 + f2 -> RAT -valued ; coherence f1 + f2 is RAT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng (f1 + f2) or y in RAT ) assume y in rng (f1 + f2) ; ::_thesis: y in RAT then consider x being set such that A1: x in dom (f1 + f2) and A2: (f1 + f2) . x = y by FUNCT_1:def_3; (f1 + f2) . x = (f1 . x) + (f2 . x) by A1, Def1; hence y in RAT by A2, RAT_1:def_2; ::_thesis: verum end; end; registration let f1, f2 be INT -valued Function; clusterf1 + f2 -> INT -valued ; coherence f1 + f2 is INT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng (f1 + f2) or y in INT ) assume y in rng (f1 + f2) ; ::_thesis: y in INT then consider x being set such that A1: x in dom (f1 + f2) and A2: (f1 + f2) . x = y by FUNCT_1:def_3; (f1 + f2) . x = (f1 . x) + (f2 . x) by A1, Def1; hence y in INT by A2, INT_1:def_2; ::_thesis: verum end; end; registration let f1, f2 be natural-valued Function; clusterf1 + f2 -> natural-valued ; coherence f1 + f2 is natural-valued proof let x be set ; :: according to VALUED_0:def_12 ::_thesis: ( not x in dom (f1 + f2) or (f1 + f2) . x is natural ) assume x in dom (f1 + f2) ; ::_thesis: (f1 + f2) . x is natural then (f1 + f2) . x = (f1 . x) + (f2 . x) by Def1; hence (f1 + f2) . x is natural ; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be complex-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: + redefine funcf1 + f2 -> PartFunc of C,COMPLEX; coherence f1 + f2 is PartFunc of C,COMPLEX proof ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng (f1 + f2) c= COMPLEX ) by Def1, VALUED_0:def_1; hence f1 + f2 is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be real-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: + redefine funcf1 + f2 -> PartFunc of C,REAL; coherence f1 + f2 is PartFunc of C,REAL proof ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng (f1 + f2) c= REAL ) by Def1, VALUED_0:def_3; hence f1 + f2 is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be rational-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: + redefine funcf1 + f2 -> PartFunc of C,RAT; coherence f1 + f2 is PartFunc of C,RAT proof ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng (f1 + f2) c= RAT ) by Def1, RELAT_1:def_19; hence f1 + f2 is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be integer-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: + redefine funcf1 + f2 -> PartFunc of C,INT; coherence f1 + f2 is PartFunc of C,INT proof ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng (f1 + f2) c= INT ) by Def1, RELAT_1:def_19; hence f1 + f2 is PartFunc of C,INT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be natural-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: + redefine funcf1 + f2 -> PartFunc of C,NAT; coherence f1 + f2 is PartFunc of C,NAT proof ( dom (f1 + f2) = (dom f1) /\ (dom f2) & rng (f1 + f2) c= NAT ) by Def1, VALUED_0:def_6; hence f1 + f2 is PartFunc of C,NAT by RELSET_1:4; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty complex-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 + f2 -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = f1 + f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 + f2) = C /\ C by Def1 .= C ; hence for b1 being PartFunc of C,COMPLEX st b1 = f1 + f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty real-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 + f2 -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = f1 + f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 + f2) = C /\ C by Def1 .= C ; hence for b1 being PartFunc of C,REAL st b1 = f1 + f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty rational-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 + f2 -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = f1 + f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 + f2) = C /\ C by Def1 .= C ; hence for b1 being PartFunc of C,RAT st b1 = f1 + f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty integer-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 + f2 -> total for PartFunc of C,INT; coherence for b1 being PartFunc of C,INT st b1 = f1 + f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 + f2) = C /\ C by Def1 .= C ; hence for b1 being PartFunc of C,INT st b1 = f1 + f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty natural-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 + f2 -> total for PartFunc of C,NAT; coherence for b1 being PartFunc of C,NAT st b1 = f1 + f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 + f2) = C /\ C by Def1 .= C ; hence for b1 being PartFunc of C,NAT st b1 = f1 + f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; theorem :: VALUED_1:1 for C being set for D1, D2 being non empty complex-membered set for f1 being Function of C,D1 for f2 being Function of C,D2 for c being Element of C holds (f1 + f2) . c = (f1 . c) + (f2 . c) proof let C be set ; ::_thesis: for D1, D2 being non empty complex-membered set for f1 being Function of C,D1 for f2 being Function of C,D2 for c being Element of C holds (f1 + f2) . c = (f1 . c) + (f2 . c) let D1, D2 be non empty complex-membered set ; ::_thesis: for f1 being Function of C,D1 for f2 being Function of C,D2 for c being Element of C holds (f1 + f2) . c = (f1 . c) + (f2 . c) let f1 be Function of C,D1; ::_thesis: for f2 being Function of C,D2 for c being Element of C holds (f1 + f2) . c = (f1 . c) + (f2 . c) let f2 be Function of C,D2; ::_thesis: for c being Element of C holds (f1 + f2) . c = (f1 . c) + (f2 . c) A1: dom (f1 + f2) = C by FUNCT_2:def_1; let c be Element of C; ::_thesis: (f1 + f2) . c = (f1 . c) + (f2 . c) percases ( not C is empty or C is empty ) ; suppose not C is empty ; ::_thesis: (f1 + f2) . c = (f1 . c) + (f2 . c) hence (f1 + f2) . c = (f1 . c) + (f2 . c) by A1, Def1; ::_thesis: verum end; supposeA2: C is empty ; ::_thesis: (f1 + f2) . c = (f1 . c) + (f2 . c) then dom f1 = {} ; then f1 . c = 0 by FUNCT_1:def_2; hence (f1 + f2) . c = (f1 . c) + (f2 . c) by A2; ::_thesis: verum end; end; end; registration let f1, f2 be complex-valued FinSequence; clusterf1 + f2 -> FinSequence-like ; coherence f1 + f2 is FinSequence-like proof dom (f1 + f2) = (dom f1) /\ (dom f2) by Def1; hence f1 + f2 is FinSequence-like by Lm2; ::_thesis: verum end; end; begin definition let f be complex-valued Function; let r be complex number ; deffunc H1( set ) -> set = r + (f . \$1); funcr + f -> Function means :Def2: :: VALUED_1:def 2 ( dom it = dom f & ( for c being set st c in dom it holds it . c = r + (f . c) ) ); existence ex b1 being Function st ( dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = r + (f . c) ) ) proof ex g being Function st ( dom g = dom f & ( for x being set st x in dom f holds g . x = H1(x) ) ) from FUNCT_1:sch_3(); hence ex b1 being Function st ( dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = r + (f . c) ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Function st dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = r + (f . c) ) & dom b2 = dom f & ( for c being set st c in dom b2 holds b2 . c = r + (f . c) ) holds b1 = b2 proof let h, g be Function; ::_thesis: ( dom h = dom f & ( for c being set st c in dom h holds h . c = r + (f . c) ) & dom g = dom f & ( for c being set st c in dom g holds g . c = r + (f . c) ) implies h = g ) assume that A1: dom h = dom f and A2: for c being set st c in dom h holds h . c = H1(c) and A3: dom g = dom f and A4: for c being set st c in dom g holds g . c = H1(c) ; ::_thesis: h = g now__::_thesis:_for_x_being_set_st_x_in_dom_h_holds_ h_._x_=_g_._x let x be set ; ::_thesis: ( x in dom h implies h . x = g . x ) assume A5: x in dom h ; ::_thesis: h . x = g . x hence h . x = H1(x) by A2 .= g . x by A1, A3, A4, A5 ; ::_thesis: verum end; hence h = g by A1, A3, FUNCT_1:2; ::_thesis: verum end; end; :: deftheorem Def2 defines + VALUED_1:def_2_:_ for f being complex-valued Function for r being complex number for b3 being Function holds ( b3 = r + f iff ( dom b3 = dom f & ( for c being set st c in dom b3 holds b3 . c = r + (f . c) ) ) ); notation let f be complex-valued Function; let r be complex number ; synonym f + r for r + f; end; registration let f be complex-valued Function; let r be complex number ; clusterr + f -> complex-valued ; coherence r + f is complex-valued proof let x be set ; :: according to VALUED_0:def_7 ::_thesis: ( not x in dom (r + f) or (r + f) . x is complex ) assume x in dom (r + f) ; ::_thesis: (r + f) . x is complex then (r + f) . x = r + (f . x) by Def2; hence (r + f) . x is complex ; ::_thesis: verum end; end; registration let f be real-valued Function; let r be real number ; clusterr + f -> real-valued ; coherence r + f is real-valued proof let x be set ; :: according to VALUED_0:def_9 ::_thesis: ( not x in dom (r + f) or (r + f) . x is real ) assume x in dom (r + f) ; ::_thesis: (r + f) . x is real then (r + f) . x = r + (f . x) by Def2; hence (r + f) . x is real ; ::_thesis: verum end; end; registration let f be RAT -valued Function; let r be rational number ; clusterr + f -> RAT -valued ; coherence r + f is RAT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng (r + f) or y in RAT ) assume y in rng (r + f) ; ::_thesis: y in RAT then consider x being set such that A1: x in dom (r + f) and A2: (r + f) . x = y by FUNCT_1:def_3; (r + f) . x = r + (f . x) by A1, Def2; hence y in RAT by A2, RAT_1:def_2; ::_thesis: verum end; end; registration let f be INT -valued Function; let r be integer number ; clusterr + f -> INT -valued ; coherence r + f is INT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng (r + f) or y in INT ) assume y in rng (r + f) ; ::_thesis: y in INT then consider x being set such that A1: x in dom (r + f) and A2: (r + f) . x = y by FUNCT_1:def_3; (r + f) . x = r + (f . x) by A1, Def2; hence y in INT by A2, INT_1:def_2; ::_thesis: verum end; end; registration let f be natural-valued Function; let r be Nat; clusterr + f -> natural-valued ; coherence r + f is natural-valued proof let x be set ; :: according to VALUED_0:def_12 ::_thesis: ( not x in dom (r + f) or (r + f) . x is natural ) assume x in dom (r + f) ; ::_thesis: (r + f) . x is natural then (r + f) . x = r + (f . x) by Def2; hence (r + f) . x is natural ; ::_thesis: verum end; end; definition let C be set ; let D be complex-membered set ; let f be PartFunc of C,D; let r be complex number ; :: original: + redefine funcr + f -> PartFunc of C,COMPLEX; coherence r + f is PartFunc of C,COMPLEX proof ( dom (r + f) = dom f & rng (r + f) c= COMPLEX ) by Def2, VALUED_0:def_1; hence r + f is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be real-membered set ; let f be PartFunc of C,D; let r be real number ; :: original: + redefine funcr + f -> PartFunc of C,REAL; coherence r + f is PartFunc of C,REAL proof ( dom (r + f) = dom f & rng (r + f) c= REAL ) by Def2, VALUED_0:def_3; hence r + f is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be rational-membered set ; let f be PartFunc of C,D; let r be rational number ; :: original: + redefine funcr + f -> PartFunc of C,RAT; coherence r + f is PartFunc of C,RAT proof ( dom (r + f) = dom f & rng (r + f) c= RAT ) by Def2, RELAT_1:def_19; hence r + f is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be integer-membered set ; let f be PartFunc of C,D; let r be integer number ; :: original: + redefine funcr + f -> PartFunc of C,INT; coherence r + f is PartFunc of C,INT proof ( dom (r + f) = dom f & rng (r + f) c= INT ) by Def2, RELAT_1:def_19; hence r + f is PartFunc of C,INT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be natural-membered set ; let f be PartFunc of C,D; let r be Nat; :: original: + redefine funcr + f -> PartFunc of C,NAT; coherence r + f is PartFunc of C,NAT proof ( dom (r + f) = dom f & rng (r + f) c= NAT ) by Def2, VALUED_0:def_6; hence r + f is PartFunc of C,NAT by RELSET_1:4; ::_thesis: verum end; end; registration let C be set ; let D be non empty complex-membered set ; let f be Function of C,D; let r be complex number ; clusterr + f -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = r + f holds b1 is total proof dom (r + f) = dom f by Def2 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,COMPLEX st b1 = r + f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty real-membered set ; let f be Function of C,D; let r be real number ; clusterr + f -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = r + f holds b1 is total proof dom (r + f) = dom f by Def2 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,REAL st b1 = r + f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty rational-membered set ; let f be Function of C,D; let r be rational number ; clusterr + f -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = r + f holds b1 is total proof dom (r + f) = dom f by Def2 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,RAT st b1 = r + f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty integer-membered set ; let f be Function of C,D; let r be integer number ; clusterr + f -> total for PartFunc of C,INT; coherence for b1 being PartFunc of C,INT st b1 = r + f holds b1 is total proof dom (r + f) = dom f by Def2 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,INT st b1 = r + f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty natural-membered set ; let f be Function of C,D; let r be Nat; clusterr + f -> total for PartFunc of C,NAT; coherence for b1 being PartFunc of C,NAT st b1 = r + f holds b1 is total proof dom (r + f) = dom f by Def2 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,NAT st b1 = r + f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; theorem :: VALUED_1:2 for C being non empty set for D being non empty complex-membered set for f being Function of C,D for r being complex number for c being Element of C holds (r + f) . c = r + (f . c) proof let C be non empty set ; ::_thesis: for D being non empty complex-membered set for f being Function of C,D for r being complex number for c being Element of C holds (r + f) . c = r + (f . c) let D be non empty complex-membered set ; ::_thesis: for f being Function of C,D for r being complex number for c being Element of C holds (r + f) . c = r + (f . c) let f be Function of C,D; ::_thesis: for r being complex number for c being Element of C holds (r + f) . c = r + (f . c) let r be complex number ; ::_thesis: for c being Element of C holds (r + f) . c = r + (f . c) dom (r + f) = C by FUNCT_2:def_1; hence for c being Element of C holds (r + f) . c = r + (f . c) by Def2; ::_thesis: verum end; registration let f be complex-valued FinSequence; let r be complex number ; clusterr + f -> FinSequence-like ; coherence r + f is FinSequence-like proof dom (r + f) = dom f by Def2; hence r + f is FinSequence-like by Lm1; ::_thesis: verum end; end; begin definition let f be complex-valued Function; let r be complex number ; funcf - r -> Function equals :: VALUED_1:def 3 (- r) + f; coherence (- r) + f is Function ; end; :: deftheorem defines - VALUED_1:def_3_:_ for f being complex-valued Function for r being complex number holds f - r = (- r) + f; theorem :: VALUED_1:3 for f being complex-valued Function for r being complex number holds ( dom (f - r) = dom f & ( for c being set st c in dom f holds (f - r) . c = (f . c) - r ) ) proof let f be complex-valued Function; ::_thesis: for r being complex number holds ( dom (f - r) = dom f & ( for c being set st c in dom f holds (f - r) . c = (f . c) - r ) ) let r be complex number ; ::_thesis: ( dom (f - r) = dom f & ( for c being set st c in dom f holds (f - r) . c = (f . c) - r ) ) dom (f - r) = dom f by Def2; hence ( dom (f - r) = dom f & ( for c being set st c in dom f holds (f - r) . c = (f . c) - r ) ) by Def2; ::_thesis: verum end; registration let f be complex-valued Function; let r be complex number ; clusterf - r -> complex-valued ; coherence f - r is complex-valued ; end; registration let f be real-valued Function; let r be real number ; clusterf - r -> real-valued ; coherence f - r is real-valued ; end; registration let f be RAT -valued Function; let r be rational number ; clusterf - r -> RAT -valued ; coherence f - r is RAT -valued ; end; registration let f be INT -valued Function; let r be integer number ; clusterf - r -> INT -valued ; coherence f - r is INT -valued ; end; definition let C be set ; let D be complex-membered set ; let f be PartFunc of C,D; let r be complex number ; :: original: - redefine funcf - r -> PartFunc of C,COMPLEX; coherence f - r is PartFunc of C,COMPLEX proof ( dom (f - r) = dom f & rng (f - r) c= COMPLEX ) by Def2, VALUED_0:def_1; hence f - r is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be real-membered set ; let f be PartFunc of C,D; let r be real number ; :: original: - redefine funcf - r -> PartFunc of C,REAL; coherence f - r is PartFunc of C,REAL proof ( dom (f - r) = dom f & rng (f - r) c= REAL ) by Def2, VALUED_0:def_3; hence f - r is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be rational-membered set ; let f be PartFunc of C,D; let r be rational number ; :: original: - redefine funcf - r -> PartFunc of C,RAT; coherence f - r is PartFunc of C,RAT proof ( dom (f - r) = dom f & rng (f - r) c= RAT ) by Def2, RELAT_1:def_19; hence f - r is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be integer-membered set ; let f be PartFunc of C,D; let r be integer number ; :: original: - redefine funcf - r -> PartFunc of C,INT; coherence f - r is PartFunc of C,INT proof ( dom (f - r) = dom f & rng (f - r) c= INT ) by Def2, RELAT_1:def_19; hence f - r is PartFunc of C,INT by RELSET_1:4; ::_thesis: verum end; end; registration let C be set ; let D be non empty complex-membered set ; let f be Function of C,D; let r be complex number ; clusterf - r -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = f - r holds b1 is total ; end; registration let C be set ; let D be non empty real-membered set ; let f be Function of C,D; let r be real number ; clusterf - r -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = f - r holds b1 is total ; end; registration let C be set ; let D be non empty rational-membered set ; let f be Function of C,D; let r be rational number ; clusterf - r -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = f - r holds b1 is total ; end; registration let C be set ; let D be non empty integer-membered set ; let f be Function of C,D; let r be integer number ; clusterf - r -> total for PartFunc of C,INT; coherence for b1 being PartFunc of C,INT st b1 = f - r holds b1 is total ; end; theorem :: VALUED_1:4 for C being non empty set for D being non empty complex-membered set for f being Function of C,D for r being complex number for c being Element of C holds (f - r) . c = (f . c) - r proof let C be non empty set ; ::_thesis: for D being non empty complex-membered set for f being Function of C,D for r being complex number for c being Element of C holds (f - r) . c = (f . c) - r let D be non empty complex-membered set ; ::_thesis: for f being Function of C,D for r being complex number for c being Element of C holds (f - r) . c = (f . c) - r let f be Function of C,D; ::_thesis: for r being complex number for c being Element of C holds (f - r) . c = (f . c) - r let r be complex number ; ::_thesis: for c being Element of C holds (f - r) . c = (f . c) - r dom (f - r) = C by FUNCT_2:def_1; hence for c being Element of C holds (f - r) . c = (f . c) - r by Def2; ::_thesis: verum end; registration let f be complex-valued FinSequence; let r be complex number ; clusterf - r -> FinSequence-like ; coherence f - r is FinSequence-like ; end; begin definition let f1, f2 be complex-valued Function; deffunc H1( set ) -> set = (f1 . \$1) * (f2 . \$1); set X = (dom f1) /\ (dom f2); funcf1 (#) f2 -> Function means :Def4: :: VALUED_1:def 4 ( dom it = (dom f1) /\ (dom f2) & ( for c being set st c in dom it holds it . c = (f1 . c) * (f2 . c) ) ); existence ex b1 being Function st ( dom b1 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b1 holds b1 . c = (f1 . c) * (f2 . c) ) ) proof ex f being Function st ( dom f = (dom f1) /\ (dom f2) & ( for x being set st x in (dom f1) /\ (dom f2) holds f . x = H1(x) ) ) from FUNCT_1:sch_3(); hence ex b1 being Function st ( dom b1 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b1 holds b1 . c = (f1 . c) * (f2 . c) ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Function st dom b1 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b1 holds b1 . c = (f1 . c) * (f2 . c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b2 holds b2 . c = (f1 . c) * (f2 . c) ) holds b1 = b2 proof let f, g be Function; ::_thesis: ( dom f = (dom f1) /\ (dom f2) & ( for c being set st c in dom f holds f . c = (f1 . c) * (f2 . c) ) & dom g = (dom f1) /\ (dom f2) & ( for c being set st c in dom g holds g . c = (f1 . c) * (f2 . c) ) implies f = g ) assume that A1: dom f = (dom f1) /\ (dom f2) and A2: for c being set st c in dom f holds f . c = H1(c) and A3: dom g = (dom f1) /\ (dom f2) and A4: for c being set st c in dom g holds g . c = H1(c) ; ::_thesis: f = g now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_ f_._x_=_g_._x let x be set ; ::_thesis: ( x in dom f implies f . x = g . x ) assume A5: x in dom f ; ::_thesis: f . x = g . x hence f . x = H1(x) by A2 .= g . x by A1, A3, A4, A5 ; ::_thesis: verum end; hence f = g by A1, A3, FUNCT_1:2; ::_thesis: verum end; commutativity for b1 being Function for f1, f2 being complex-valued Function st dom b1 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b1 holds b1 . c = (f1 . c) * (f2 . c) ) holds ( dom b1 = (dom f2) /\ (dom f1) & ( for c being set st c in dom b1 holds b1 . c = (f2 . c) * (f1 . c) ) ) ; end; :: deftheorem Def4 defines (#) VALUED_1:def_4_:_ for f1, f2 being complex-valued Function for b3 being Function holds ( b3 = f1 (#) f2 iff ( dom b3 = (dom f1) /\ (dom f2) & ( for c being set st c in dom b3 holds b3 . c = (f1 . c) * (f2 . c) ) ) ); theorem :: VALUED_1:5 for f1, f2 being complex-valued Function for c being set holds (f1 (#) f2) . c = (f1 . c) * (f2 . c) proof let f1, f2 be complex-valued Function; ::_thesis: for c being set holds (f1 (#) f2) . c = (f1 . c) * (f2 . c) let c be set ; ::_thesis: (f1 (#) f2) . c = (f1 . c) * (f2 . c) A1: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by Def4; percases ( c in dom (f1 (#) f2) or not c in dom (f1 (#) f2) ) ; suppose c in dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) . c = (f1 . c) * (f2 . c) hence (f1 (#) f2) . c = (f1 . c) * (f2 . c) by Def4; ::_thesis: verum end; supposeA2: not c in dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) . c = (f1 . c) * (f2 . c) then ( not c in dom f1 or not c in dom f2 ) by A1, XBOOLE_0:def_4; then ( f1 . c = 0 or f2 . c = 0 ) by FUNCT_1:def_2; hence (f1 (#) f2) . c = (f1 . c) * (f2 . c) by A2, FUNCT_1:def_2; ::_thesis: verum end; end; end; registration let f1, f2 be complex-valued Function; clusterf1 (#) f2 -> complex-valued ; coherence f1 (#) f2 is complex-valued proof let x be set ; :: according to VALUED_0:def_7 ::_thesis: ( not x in dom (f1 (#) f2) or (f1 (#) f2) . x is complex ) assume x in dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) . x is complex then (f1 (#) f2) . x = (f1 . x) * (f2 . x) by Def4; hence (f1 (#) f2) . x is complex ; ::_thesis: verum end; end; registration let f1, f2 be real-valued Function; clusterf1 (#) f2 -> real-valued ; coherence f1 (#) f2 is real-valued proof let x be set ; :: according to VALUED_0:def_9 ::_thesis: ( not x in dom (f1 (#) f2) or (f1 (#) f2) . x is real ) assume x in dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) . x is real then (f1 (#) f2) . x = (f1 . x) * (f2 . x) by Def4; hence (f1 (#) f2) . x is real ; ::_thesis: verum end; end; registration let f1, f2 be RAT -valued Function; clusterf1 (#) f2 -> RAT -valued ; coherence f1 (#) f2 is RAT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng (f1 (#) f2) or y in RAT ) assume y in rng (f1 (#) f2) ; ::_thesis: y in RAT then consider x being set such that A1: x in dom (f1 (#) f2) and A2: (f1 (#) f2) . x = y by FUNCT_1:def_3; (f1 (#) f2) . x = (f1 . x) * (f2 . x) by A1, Def4; hence y in RAT by A2, RAT_1:def_2; ::_thesis: verum end; end; registration let f1, f2 be INT -valued Function; clusterf1 (#) f2 -> INT -valued ; coherence f1 (#) f2 is INT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng (f1 (#) f2) or y in INT ) assume y in rng (f1 (#) f2) ; ::_thesis: y in INT then consider x being set such that A1: x in dom (f1 (#) f2) and A2: (f1 (#) f2) . x = y by FUNCT_1:def_3; (f1 (#) f2) . x = (f1 . x) * (f2 . x) by A1, Def4; hence y in INT by A2, INT_1:def_2; ::_thesis: verum end; end; registration let f1, f2 be natural-valued Function; clusterf1 (#) f2 -> natural-valued ; coherence f1 (#) f2 is natural-valued proof let x be set ; :: according to VALUED_0:def_12 ::_thesis: ( not x in dom (f1 (#) f2) or (f1 (#) f2) . x is natural ) assume x in dom (f1 (#) f2) ; ::_thesis: (f1 (#) f2) . x is natural then (f1 (#) f2) . x = (f1 . x) * (f2 . x) by Def4; hence (f1 (#) f2) . x is natural ; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be complex-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: (#) redefine funcf1 (#) f2 -> PartFunc of C,COMPLEX; coherence f1 (#) f2 is PartFunc of C,COMPLEX proof ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng (f1 (#) f2) c= COMPLEX ) by Def4, VALUED_0:def_1; hence f1 (#) f2 is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be real-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: (#) redefine funcf1 (#) f2 -> PartFunc of C,REAL; coherence f1 (#) f2 is PartFunc of C,REAL proof ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng (f1 (#) f2) c= REAL ) by Def4, VALUED_0:def_3; hence f1 (#) f2 is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be rational-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: (#) redefine funcf1 (#) f2 -> PartFunc of C,RAT; coherence f1 (#) f2 is PartFunc of C,RAT proof ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng (f1 (#) f2) c= RAT ) by Def4, RELAT_1:def_19; hence f1 (#) f2 is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be integer-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: (#) redefine funcf1 (#) f2 -> PartFunc of C,INT; coherence f1 (#) f2 is PartFunc of C,INT proof ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng (f1 (#) f2) c= INT ) by Def4, RELAT_1:def_19; hence f1 (#) f2 is PartFunc of C,INT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be natural-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: (#) redefine funcf1 (#) f2 -> PartFunc of C,NAT; coherence f1 (#) f2 is PartFunc of C,NAT proof ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng (f1 (#) f2) c= NAT ) by Def4, VALUED_0:def_6; hence f1 (#) f2 is PartFunc of C,NAT by RELSET_1:4; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty complex-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 (#) f2 -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = f1 (#) f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 (#) f2) = C /\ C by Def4 .= C ; hence for b1 being PartFunc of C,COMPLEX st b1 = f1 (#) f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty real-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 (#) f2 -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = f1 (#) f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 (#) f2) = C /\ C by Def4 .= C ; hence for b1 being PartFunc of C,REAL st b1 = f1 (#) f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty rational-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 (#) f2 -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = f1 (#) f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 (#) f2) = C /\ C by Def4 .= C ; hence for b1 being PartFunc of C,RAT st b1 = f1 (#) f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty integer-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 (#) f2 -> total for PartFunc of C,INT; coherence for b1 being PartFunc of C,INT st b1 = f1 (#) f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 (#) f2) = C /\ C by Def4 .= C ; hence for b1 being PartFunc of C,INT st b1 = f1 (#) f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty natural-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 (#) f2 -> total for PartFunc of C,NAT; coherence for b1 being PartFunc of C,NAT st b1 = f1 (#) f2 holds b1 is total proof ( dom f1 = C & dom f2 = C ) by FUNCT_2:def_1; then dom (f1 (#) f2) = C /\ C by Def4 .= C ; hence for b1 being PartFunc of C,NAT st b1 = f1 (#) f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let f1, f2 be complex-valued FinSequence; clusterf1 (#) f2 -> FinSequence-like ; coherence f1 (#) f2 is FinSequence-like proof dom (f1 (#) f2) = (dom f1) /\ (dom f2) by Def4; hence f1 (#) f2 is FinSequence-like by Lm2; ::_thesis: verum end; end; begin definition let f be complex-valued Function; let r be complex number ; deffunc H1( set ) -> set = r * (f . \$1); funcr (#) f -> Function means :Def5: :: VALUED_1:def 5 ( dom it = dom f & ( for c being set st c in dom it holds it . c = r * (f . c) ) ); existence ex b1 being Function st ( dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = r * (f . c) ) ) proof ex g being Function st ( dom g = dom f & ( for x being set st x in dom f holds g . x = H1(x) ) ) from FUNCT_1:sch_3(); hence ex b1 being Function st ( dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = r * (f . c) ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Function st dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = r * (f . c) ) & dom b2 = dom f & ( for c being set st c in dom b2 holds b2 . c = r * (f . c) ) holds b1 = b2 proof let h, g be Function; ::_thesis: ( dom h = dom f & ( for c being set st c in dom h holds h . c = r * (f . c) ) & dom g = dom f & ( for c being set st c in dom g holds g . c = r * (f . c) ) implies h = g ) assume that A1: dom h = dom f and A2: for c being set st c in dom h holds h . c = H1(c) and A3: dom g = dom f and A4: for c being set st c in dom g holds g . c = H1(c) ; ::_thesis: h = g now__::_thesis:_for_x_being_set_st_x_in_dom_h_holds_ h_._x_=_g_._x let x be set ; ::_thesis: ( x in dom h implies h . x = g . x ) assume A5: x in dom h ; ::_thesis: h . x = g . x hence h . x = H1(x) by A2 .= g . x by A1, A3, A4, A5 ; ::_thesis: verum end; hence h = g by A1, A3, FUNCT_1:2; ::_thesis: verum end; end; :: deftheorem Def5 defines (#) VALUED_1:def_5_:_ for f being complex-valued Function for r being complex number for b3 being Function holds ( b3 = r (#) f iff ( dom b3 = dom f & ( for c being set st c in dom b3 holds b3 . c = r * (f . c) ) ) ); notation let f be complex-valued Function; let r be complex number ; synonym f (#) r for r (#) f; end; theorem Th6: :: VALUED_1:6 for f being complex-valued Function for r being complex number for c being set holds (r (#) f) . c = r * (f . c) proof let f be complex-valued Function; ::_thesis: for r being complex number for c being set holds (r (#) f) . c = r * (f . c) let r be complex number ; ::_thesis: for c being set holds (r (#) f) . c = r * (f . c) let c be set ; ::_thesis: (r (#) f) . c = r * (f . c) A1: dom f = dom (r (#) f) by Def5; percases ( c in dom f or not c in dom f ) ; suppose c in dom f ; ::_thesis: (r (#) f) . c = r * (f . c) hence (r (#) f) . c = r * (f . c) by A1, Def5; ::_thesis: verum end; supposeA2: not c in dom f ; ::_thesis: (r (#) f) . c = r * (f . c) hence (r (#) f) . c = r * 0 by A1, FUNCT_1:def_2 .= r * (f . c) by A2, FUNCT_1:def_2 ; ::_thesis: verum end; end; end; registration let f be complex-valued Function; let r be complex number ; clusterr (#) f -> complex-valued ; coherence r (#) f is complex-valued proof let x be set ; :: according to VALUED_0:def_7 ::_thesis: ( not x in dom (r (#) f) or (r (#) f) . x is complex ) assume x in dom (r (#) f) ; ::_thesis: (r (#) f) . x is complex then (r (#) f) . x = r * (f . x) by Def5; hence (r (#) f) . x is complex ; ::_thesis: verum end; end; registration let f be real-valued Function; let r be real number ; clusterr (#) f -> real-valued ; coherence r (#) f is real-valued proof let x be set ; :: according to VALUED_0:def_9 ::_thesis: ( not x in dom (r (#) f) or (r (#) f) . x is real ) assume x in dom (r (#) f) ; ::_thesis: (r (#) f) . x is real then (r (#) f) . x = r * (f . x) by Def5; hence (r (#) f) . x is real ; ::_thesis: verum end; end; registration let f be RAT -valued Function; let r be rational number ; clusterr (#) f -> RAT -valued ; coherence r (#) f is RAT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng (r (#) f) or y in RAT ) assume y in rng (r (#) f) ; ::_thesis: y in RAT then consider x being set such that A1: x in dom (r (#) f) and A2: (r (#) f) . x = y by FUNCT_1:def_3; (r (#) f) . x = r * (f . x) by A1, Def5; hence y in RAT by A2, RAT_1:def_2; ::_thesis: verum end; end; registration let f be INT -valued Function; let r be integer number ; clusterr (#) f -> INT -valued ; coherence r (#) f is INT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng (r (#) f) or y in INT ) assume y in rng (r (#) f) ; ::_thesis: y in INT then consider x being set such that A1: x in dom (r (#) f) and A2: (r (#) f) . x = y by FUNCT_1:def_3; (r (#) f) . x = r * (f . x) by A1, Def5; hence y in INT by A2, INT_1:def_2; ::_thesis: verum end; end; registration let f be natural-valued Function; let r be Nat; clusterr (#) f -> natural-valued ; coherence r (#) f is natural-valued proof let x be set ; :: according to VALUED_0:def_12 ::_thesis: ( not x in dom (r (#) f) or (r (#) f) . x is natural ) assume x in dom (r (#) f) ; ::_thesis: (r (#) f) . x is natural then (r (#) f) . x = r * (f . x) by Def5; hence (r (#) f) . x is natural ; ::_thesis: verum end; end; definition let C be set ; let D be complex-membered set ; let f be PartFunc of C,D; let r be complex number ; :: original: (#) redefine funcr (#) f -> PartFunc of C,COMPLEX; coherence r (#) f is PartFunc of C,COMPLEX proof ( dom (r (#) f) = dom f & rng (r (#) f) c= COMPLEX ) by Def5, VALUED_0:def_1; hence r (#) f is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be real-membered set ; let f be PartFunc of C,D; let r be real number ; :: original: (#) redefine funcr (#) f -> PartFunc of C,REAL; coherence r (#) f is PartFunc of C,REAL proof ( dom (r (#) f) = dom f & rng (r (#) f) c= REAL ) by Def5, VALUED_0:def_3; hence r (#) f is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be rational-membered set ; let f be PartFunc of C,D; let r be rational number ; :: original: (#) redefine funcr (#) f -> PartFunc of C,RAT; coherence r (#) f is PartFunc of C,RAT proof ( dom (r (#) f) = dom f & rng (r (#) f) c= RAT ) by Def5, RELAT_1:def_19; hence r (#) f is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be integer-membered set ; let f be PartFunc of C,D; let r be integer number ; :: original: (#) redefine funcr (#) f -> PartFunc of C,INT; coherence r (#) f is PartFunc of C,INT proof ( dom (r (#) f) = dom f & rng (r (#) f) c= INT ) by Def5, RELAT_1:def_19; hence r (#) f is PartFunc of C,INT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be natural-membered set ; let f be PartFunc of C,D; let r be Nat; :: original: (#) redefine funcr (#) f -> PartFunc of C,NAT; coherence r (#) f is PartFunc of C,NAT proof ( dom (r (#) f) = dom f & rng (r (#) f) c= NAT ) by Def5, VALUED_0:def_6; hence r (#) f is PartFunc of C,NAT by RELSET_1:4; ::_thesis: verum end; end; registration let C be set ; let D be non empty complex-membered set ; let f be Function of C,D; let r be complex number ; clusterr (#) f -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = r (#) f holds b1 is total proof dom (r (#) f) = dom f by Def5 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,COMPLEX st b1 = r (#) f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty real-membered set ; let f be Function of C,D; let r be real number ; clusterr (#) f -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = r (#) f holds b1 is total proof dom (r (#) f) = dom f by Def5 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,REAL st b1 = r (#) f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty rational-membered set ; let f be Function of C,D; let r be rational number ; clusterr (#) f -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = r (#) f holds b1 is total proof dom (r (#) f) = dom f by Def5 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,RAT st b1 = r (#) f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty integer-membered set ; let f be Function of C,D; let r be integer number ; clusterr (#) f -> total for PartFunc of C,INT; coherence for b1 being PartFunc of C,INT st b1 = r (#) f holds b1 is total proof dom (r (#) f) = dom f by Def5 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,INT st b1 = r (#) f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty natural-membered set ; let f be Function of C,D; let r be Nat; clusterr (#) f -> total for PartFunc of C,NAT; coherence for b1 being PartFunc of C,NAT st b1 = r (#) f holds b1 is total proof dom (r (#) f) = dom f by Def5 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,NAT st b1 = r (#) f holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; theorem :: VALUED_1:7 for C being non empty set for D being non empty complex-membered set for f being Function of C,D for r being complex number for g being Function of C,COMPLEX st ( for c being Element of C holds g . c = r * (f . c) ) holds g = r (#) f proof let C be non empty set ; ::_thesis: for D being non empty complex-membered set for f being Function of C,D for r being complex number for g being Function of C,COMPLEX st ( for c being Element of C holds g . c = r * (f . c) ) holds g = r (#) f let D be non empty complex-membered set ; ::_thesis: for f being Function of C,D for r being complex number for g being Function of C,COMPLEX st ( for c being Element of C holds g . c = r * (f . c) ) holds g = r (#) f let f be Function of C,D; ::_thesis: for r being complex number for g being Function of C,COMPLEX st ( for c being Element of C holds g . c = r * (f . c) ) holds g = r (#) f let r be complex number ; ::_thesis: for g being Function of C,COMPLEX st ( for c being Element of C holds g . c = r * (f . c) ) holds g = r (#) f let g be Function of C,COMPLEX; ::_thesis: ( ( for c being Element of C holds g . c = r * (f . c) ) implies g = r (#) f ) assume A1: for c being Element of C holds g . c = r * (f . c) ; ::_thesis: g = r (#) f let x be Element of C; :: according to FUNCT_2:def_8 ::_thesis: g . x = (r (#) f) . x thus g . x = r * (f . x) by A1 .= (r (#) f) . x by Th6 ; ::_thesis: verum end; registration let f be complex-valued FinSequence; let r be complex number ; clusterr (#) f -> FinSequence-like ; coherence r (#) f is FinSequence-like proof dom (r (#) f) = dom f by Def5; hence r (#) f is FinSequence-like by Lm1; ::_thesis: verum end; end; begin definition let f be complex-valued Function; func - f -> complex-valued Function equals :: VALUED_1:def 6 (- 1) (#) f; coherence (- 1) (#) f is complex-valued Function ; involutiveness for b1, b2 being complex-valued Function st b1 = (- 1) (#) b2 holds b2 = (- 1) (#) b1 proof let r, h be complex-valued Function; ::_thesis: ( r = (- 1) (#) h implies h = (- 1) (#) r ) assume A1: r = (- 1) (#) h ; ::_thesis: h = (- 1) (#) r thus dom ((- 1) (#) r) = dom r by Def5 .= dom h by A1, Def5 ; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom h or h . b1 = ((- 1) (#) r) . b1 ) let c be set ; ::_thesis: ( not c in dom h or h . c = ((- 1) (#) r) . c ) assume c in dom h ; ::_thesis: h . c = ((- 1) (#) r) . c reconsider a = (- 1) * (h . c) as complex number ; thus h . c = (- 1) * a .= (- 1) * (r . c) by A1, Th6 .= ((- 1) (#) r) . c by Th6 ; ::_thesis: verum end; end; :: deftheorem defines - VALUED_1:def_6_:_ for f being complex-valued Function holds - f = (- 1) (#) f; theorem Th8: :: VALUED_1:8 for f being complex-valued Function holds ( dom (- f) = dom f & ( for c being set holds (- f) . c = - (f . c) ) ) proof let f be complex-valued Function; ::_thesis: ( dom (- f) = dom f & ( for c being set holds (- f) . c = - (f . c) ) ) thus A1: dom (- f) = dom f by Def5; ::_thesis: for c being set holds (- f) . c = - (f . c) let c be set ; ::_thesis: (- f) . c = - (f . c) percases ( c in dom f or not c in dom f ) ; suppose c in dom f ; ::_thesis: (- f) . c = - (f . c) hence (- f) . c = (- 1) * (f . c) by A1, Def5 .= - (f . c) ; ::_thesis: verum end; supposeA2: not c in dom f ; ::_thesis: (- f) . c = - (f . c) hence (- f) . c = - 0 by A1, FUNCT_1:def_2 .= - (f . c) by A2, FUNCT_1:def_2 ; ::_thesis: verum end; end; end; theorem :: VALUED_1:9 for f being complex-valued Function for g being Function st dom f = dom g & ( for c being set st c in dom f holds g . c = - (f . c) ) holds g = - f proof let f be complex-valued Function; ::_thesis: for g being Function st dom f = dom g & ( for c being set st c in dom f holds g . c = - (f . c) ) holds g = - f let g be Function; ::_thesis: ( dom f = dom g & ( for c being set st c in dom f holds g . c = - (f . c) ) implies g = - f ) assume that A1: dom f = dom g and A2: for c being set st c in dom f holds g . c = - (f . c) ; ::_thesis: g = - f thus dom (- f) = dom g by A1, Def5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom g or g . b1 = (- f) . b1 ) let c be set ; ::_thesis: ( not c in dom g or g . c = (- f) . c ) assume A3: c in dom g ; ::_thesis: g . c = (- f) . c thus (- f) . c = - (f . c) by Th8 .= g . c by A1, A2, A3 ; ::_thesis: verum end; registration let f be complex-valued Function; cluster - f -> complex-valued ; coherence - f is complex-valued ; end; registration let f be real-valued Function; cluster - f -> complex-valued real-valued ; coherence - f is real-valued ; end; registration let f be RAT -valued Function; cluster - f -> RAT -valued complex-valued ; coherence - f is RAT -valued ; end; registration let f be INT -valued Function; cluster - f -> INT -valued complex-valued ; coherence - f is INT -valued ; end; definition let C be set ; let D be complex-membered set ; let f be PartFunc of C,D; :: original: - redefine func - f -> PartFunc of C,COMPLEX; coherence - f is PartFunc of C,COMPLEX proof ( dom (- f) = dom f & rng (- f) c= COMPLEX ) by Def5, VALUED_0:def_1; hence - f is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be real-membered set ; let f be PartFunc of C,D; :: original: - redefine func - f -> PartFunc of C,REAL; coherence - f is PartFunc of C,REAL proof ( dom (- f) = dom f & rng (- f) c= REAL ) by Def5, VALUED_0:def_3; hence - f is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be rational-membered set ; let f be PartFunc of C,D; :: original: - redefine func - f -> PartFunc of C,RAT; coherence - f is PartFunc of C,RAT proof ( dom (- f) = dom f & rng (- f) c= RAT ) by Def5, RELAT_1:def_19; hence - f is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be integer-membered set ; let f be PartFunc of C,D; :: original: - redefine func - f -> PartFunc of C,INT; coherence - f is PartFunc of C,INT proof ( dom (- f) = dom f & rng (- f) c= INT ) by Def5, RELAT_1:def_19; hence - f is PartFunc of C,INT by RELSET_1:4; ::_thesis: verum end; end; registration let C be set ; let D be non empty complex-membered set ; let f be Function of C,D; cluster - f -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = - f holds b1 is total ; end; registration let C be set ; let D be non empty real-membered set ; let f be Function of C,D; cluster - f -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = - f holds b1 is total ; end; registration let C be set ; let D be non empty rational-membered set ; let f be Function of C,D; cluster - f -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = - f holds b1 is total ; end; registration let C be set ; let D be non empty integer-membered set ; let f be Function of C,D; cluster - f -> total for PartFunc of C,INT; coherence for b1 being PartFunc of C,INT st b1 = - f holds b1 is total ; end; registration let f be complex-valued FinSequence; cluster - f -> complex-valued FinSequence-like ; coherence - f is FinSequence-like ; end; begin definition let f be complex-valued Function; deffunc H1( set ) -> set = (f . \$1) " ; funcf " -> complex-valued Function means :Def7: :: VALUED_1:def 7 ( dom it = dom f & ( for c being set st c in dom it holds it . c = (f . c) " ) ); existence ex b1 being complex-valued Function st ( dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = (f . c) " ) ) proof consider g being Function such that A1: ( dom g = dom f & ( for x being set st x in dom f holds g . x = H1(x) ) ) from FUNCT_1:sch_3(); g is complex-valued proof let x be set ; :: according to VALUED_0:def_7 ::_thesis: ( not x in dom g or g . x is complex ) assume x in dom g ; ::_thesis: g . x is complex then g . x = (f . x) " by A1; hence g . x is complex ; ::_thesis: verum end; hence ex b1 being complex-valued Function st ( dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = (f . c) " ) ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being complex-valued Function st dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = (f . c) " ) & dom b2 = dom f & ( for c being set st c in dom b2 holds b2 . c = (f . c) " ) holds b1 = b2 proof let h, g be complex-valued Function; ::_thesis: ( dom h = dom f & ( for c being set st c in dom h holds h . c = (f . c) " ) & dom g = dom f & ( for c being set st c in dom g holds g . c = (f . c) " ) implies h = g ) assume that A2: dom h = dom f and A3: for c being set st c in dom h holds h . c = H1(c) and A4: dom g = dom f and A5: for c being set st c in dom g holds g . c = H1(c) ; ::_thesis: h = g now__::_thesis:_for_x_being_set_st_x_in_dom_h_holds_ h_._x_=_g_._x let x be set ; ::_thesis: ( x in dom h implies h . x = g . x ) assume A6: x in dom h ; ::_thesis: h . x = g . x hence h . x = H1(x) by A3 .= g . x by A2, A4, A5, A6 ; ::_thesis: verum end; hence h = g by A2, A4, FUNCT_1:2; ::_thesis: verum end; involutiveness for b1, b2 being complex-valued Function st dom b1 = dom b2 & ( for c being set st c in dom b1 holds b1 . c = (b2 . c) " ) holds ( dom b2 = dom b1 & ( for c being set st c in dom b2 holds b2 . c = (b1 . c) " ) ) proof let r, h be complex-valued Function; ::_thesis: ( dom r = dom h & ( for c being set st c in dom r holds r . c = (h . c) " ) implies ( dom h = dom r & ( for c being set st c in dom h holds h . c = (r . c) " ) ) ) assume that A7: dom r = dom h and A8: for c being set st c in dom r holds r . c = (h . c) " ; ::_thesis: ( dom h = dom r & ( for c being set st c in dom h holds h . c = (r . c) " ) ) thus dom r = dom h by A7; ::_thesis: for c being set st c in dom h holds h . c = (r . c) " let c be set ; ::_thesis: ( c in dom h implies h . c = (r . c) " ) assume A9: c in dom h ; ::_thesis: h . c = (r . c) " thus h . c = ((h . c) ") " .= (r . c) " by A7, A8, A9 ; ::_thesis: verum end; end; :: deftheorem Def7 defines " VALUED_1:def_7_:_ for f, b2 being complex-valued Function holds ( b2 = f " iff ( dom b2 = dom f & ( for c being set st c in dom b2 holds b2 . c = (f . c) " ) ) ); theorem Th10: :: VALUED_1:10 for f being complex-valued Function for c being set holds (f ") . c = (f . c) " proof let f be complex-valued Function; ::_thesis: for c being set holds (f ") . c = (f . c) " let c be set ; ::_thesis: (f ") . c = (f . c) " A1: dom (f ") = dom f by Def7; percases ( c in dom f or not c in dom f ) ; suppose c in dom f ; ::_thesis: (f ") . c = (f . c) " hence (f ") . c = (f . c) " by A1, Def7; ::_thesis: verum end; supposeA2: not c in dom f ; ::_thesis: (f ") . c = (f . c) " hence (f ") . c = 0 " by A1, FUNCT_1:def_2 .= (f . c) " by A2, FUNCT_1:def_2 ; ::_thesis: verum end; end; end; registration let f be real-valued Function; clusterf " -> complex-valued real-valued ; coherence f " is real-valued proof let x be set ; :: according to VALUED_0:def_9 ::_thesis: ( not x in dom (f ") or (f ") . x is real ) assume x in dom (f ") ; ::_thesis: (f ") . x is real then (f ") . x = (f . x) " by Def7; hence (f ") . x is real ; ::_thesis: verum end; end; registration let f be RAT -valued Function; clusterf " -> RAT -valued complex-valued ; coherence f " is RAT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng (f ") or y in RAT ) assume y in rng (f ") ; ::_thesis: y in RAT then consider x being set such that A1: x in dom (f ") and A2: (f ") . x = y by FUNCT_1:def_3; (f ") . x = (f . x) " by A1, Def7; hence y in RAT by A2, RAT_1:def_2; ::_thesis: verum end; end; definition let C be set ; let D be complex-membered set ; let f be PartFunc of C,D; :: original: " redefine funcf " -> PartFunc of C,COMPLEX; coherence f " is PartFunc of C,COMPLEX proof ( dom (f ") = dom f & rng (f ") c= COMPLEX ) by Def7, VALUED_0:def_1; hence f " is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be real-membered set ; let f be PartFunc of C,D; :: original: " redefine funcf " -> PartFunc of C,REAL; coherence f " is PartFunc of C,REAL proof ( dom (f ") = dom f & rng (f ") c= REAL ) by Def7, VALUED_0:def_3; hence f " is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be rational-membered set ; let f be PartFunc of C,D; :: original: " redefine funcf " -> PartFunc of C,RAT; coherence f " is PartFunc of C,RAT proof ( dom (f ") = dom f & rng (f ") c= RAT ) by Def7, RELAT_1:def_19; hence f " is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; registration let C be set ; let D be non empty complex-membered set ; let f be Function of C,D; clusterf " -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = f " holds b1 is total proof dom (f ") = dom f by Def7 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,COMPLEX st b1 = f " holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty real-membered set ; let f be Function of C,D; clusterf " -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = f " holds b1 is total proof dom (f ") = dom f by Def7 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,REAL st b1 = f " holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty rational-membered set ; let f be Function of C,D; clusterf " -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = f " holds b1 is total proof dom (f ") = dom f by Def7 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,RAT st b1 = f " holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let f be complex-valued FinSequence; clusterf " -> complex-valued FinSequence-like ; coherence f " is FinSequence-like proof dom (f ") = dom f by Def7; hence f " is FinSequence-like by Lm1; ::_thesis: verum end; end; begin definition let f be complex-valued Function; funcf ^2 -> Function equals :: VALUED_1:def 8 f (#) f; coherence f (#) f is Function ; end; :: deftheorem defines ^2 VALUED_1:def_8_:_ for f being complex-valued Function holds f ^2 = f (#) f; theorem Th11: :: VALUED_1:11 for f being complex-valued Function holds ( dom (f ^2) = dom f & ( for c being set holds (f ^2) . c = (f . c) ^2 ) ) proof let f be complex-valued Function; ::_thesis: ( dom (f ^2) = dom f & ( for c being set holds (f ^2) . c = (f . c) ^2 ) ) thus A1: dom (f ^2) = (dom f) /\ (dom f) by Def4 .= dom f ; ::_thesis: for c being set holds (f ^2) . c = (f . c) ^2 let c be set ; ::_thesis: (f ^2) . c = (f . c) ^2 percases ( c in dom f or not c in dom f ) ; suppose c in dom f ; ::_thesis: (f ^2) . c = (f . c) ^2 hence (f ^2) . c = (f . c) ^2 by A1, Def4; ::_thesis: verum end; supposeA2: not c in dom f ; ::_thesis: (f ^2) . c = (f . c) ^2 hence (f ^2) . c = 0 ^2 by A1, FUNCT_1:def_2 .= (f . c) ^2 by A2, FUNCT_1:def_2 ; ::_thesis: verum end; end; end; registration let f be complex-valued Function; clusterf ^2 -> complex-valued ; coherence f ^2 is complex-valued ; end; registration let f be real-valued Function; clusterf ^2 -> real-valued ; coherence f ^2 is real-valued ; end; registration let f be RAT -valued Function; clusterf ^2 -> RAT -valued ; coherence f ^2 is RAT -valued ; end; registration let f be INT -valued Function; clusterf ^2 -> INT -valued ; coherence f ^2 is INT -valued ; end; registration let f be natural-valued Function; clusterf ^2 -> natural-valued ; coherence f ^2 is natural-valued ; end; definition let C be set ; let D be complex-membered set ; let f be PartFunc of C,D; :: original: ^2 redefine funcf ^2 -> PartFunc of C,COMPLEX; coherence f ^2 is PartFunc of C,COMPLEX proof ( dom (f ^2) = dom f & rng (f ^2) c= COMPLEX ) by Th11, VALUED_0:def_1; hence f ^2 is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be real-membered set ; let f be PartFunc of C,D; :: original: ^2 redefine funcf ^2 -> PartFunc of C,REAL; coherence f ^2 is PartFunc of C,REAL proof ( dom (f ^2) = dom f & rng (f ^2) c= REAL ) by Th11, VALUED_0:def_3; hence f ^2 is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be rational-membered set ; let f be PartFunc of C,D; :: original: ^2 redefine funcf ^2 -> PartFunc of C,RAT; coherence f ^2 is PartFunc of C,RAT proof ( dom (f ^2) = dom f & rng (f ^2) c= RAT ) by Th11, RELAT_1:def_19; hence f ^2 is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be integer-membered set ; let f be PartFunc of C,D; :: original: ^2 redefine funcf ^2 -> PartFunc of C,INT; coherence f ^2 is PartFunc of C,INT proof ( dom (f ^2) = dom f & rng (f ^2) c= INT ) by Th11, RELAT_1:def_19; hence f ^2 is PartFunc of C,INT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be natural-membered set ; let f be PartFunc of C,D; :: original: ^2 redefine funcf ^2 -> PartFunc of C,NAT; coherence f ^2 is PartFunc of C,NAT proof ( dom (f ^2) = dom f & rng (f ^2) c= NAT ) by Th11, VALUED_0:def_6; hence f ^2 is PartFunc of C,NAT by RELSET_1:4; ::_thesis: verum end; end; registration let C be set ; let D be non empty complex-membered set ; let f be Function of C,D; clusterf ^2 -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = f ^2 holds b1 is total ; end; registration let C be set ; let D be non empty real-membered set ; let f be Function of C,D; clusterf ^2 -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = f ^2 holds b1 is total ; end; registration let C be set ; let D be non empty rational-membered set ; let f be Function of C,D; clusterf ^2 -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = f ^2 holds b1 is total ; end; registration let C be set ; let D be non empty integer-membered set ; let f be Function of C,D; clusterf ^2 -> total for PartFunc of C,INT; coherence for b1 being PartFunc of C,INT st b1 = f ^2 holds b1 is total ; end; registration let C be set ; let D be non empty natural-membered set ; let f be Function of C,D; clusterf ^2 -> total for PartFunc of C,NAT; coherence for b1 being PartFunc of C,NAT st b1 = f ^2 holds b1 is total ; end; registration let f be complex-valued FinSequence; clusterf ^2 -> FinSequence-like ; coherence f ^2 is FinSequence-like ; end; begin definition let f1, f2 be complex-valued Function; funcf1 - f2 -> Function equals :: VALUED_1:def 9 f1 + (- f2); coherence f1 + (- f2) is Function ; end; :: deftheorem defines - VALUED_1:def_9_:_ for f1, f2 being complex-valued Function holds f1 - f2 = f1 + (- f2); registration let f1, f2 be complex-valued Function; clusterf1 - f2 -> complex-valued ; coherence f1 - f2 is complex-valued ; end; registration let f1, f2 be real-valued Function; clusterf1 - f2 -> real-valued ; coherence f1 - f2 is real-valued ; end; registration let f1, f2 be RAT -valued Function; clusterf1 - f2 -> RAT -valued ; coherence f1 - f2 is RAT -valued ; end; registration let f1, f2 be INT -valued Function; clusterf1 - f2 -> INT -valued ; coherence f1 - f2 is INT -valued ; end; theorem Th12: :: VALUED_1:12 for f1, f2 being complex-valued Function holds dom (f1 - f2) = (dom f1) /\ (dom f2) proof let f1, f2 be complex-valued Function; ::_thesis: dom (f1 - f2) = (dom f1) /\ (dom f2) thus dom (f1 - f2) = (dom f1) /\ (dom (- f2)) by Def1 .= (dom f1) /\ (dom f2) by Def5 ; ::_thesis: verum end; theorem :: VALUED_1:13 for f1, f2 being complex-valued Function for c being set st c in dom (f1 - f2) holds (f1 - f2) . c = (f1 . c) - (f2 . c) proof let f1, f2 be complex-valued Function; ::_thesis: for c being set st c in dom (f1 - f2) holds (f1 - f2) . c = (f1 . c) - (f2 . c) let c be set ; ::_thesis: ( c in dom (f1 - f2) implies (f1 - f2) . c = (f1 . c) - (f2 . c) ) assume c in dom (f1 - f2) ; ::_thesis: (f1 - f2) . c = (f1 . c) - (f2 . c) hence (f1 - f2) . c = (f1 . c) + ((- f2) . c) by Def1 .= (f1 . c) - (f2 . c) by Th8 ; ::_thesis: verum end; theorem :: VALUED_1:14 for f1, f2 being complex-valued Function for f being Function st dom f = dom (f1 - f2) & ( for c being set st c in dom f holds f . c = (f1 . c) - (f2 . c) ) holds f = f1 - f2 proof let f1, f2 be complex-valued Function; ::_thesis: for f being Function st dom f = dom (f1 - f2) & ( for c being set st c in dom f holds f . c = (f1 . c) - (f2 . c) ) holds f = f1 - f2 let f be Function; ::_thesis: ( dom f = dom (f1 - f2) & ( for c being set st c in dom f holds f . c = (f1 . c) - (f2 . c) ) implies f = f1 - f2 ) assume that A1: dom f = dom (f1 - f2) and A2: for c being set st c in dom f holds f . c = (f1 . c) - (f2 . c) ; ::_thesis: f = f1 - f2 thus dom f = dom (f1 - f2) by A1; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom f or f . b1 = (f1 - f2) . b1 ) let c be set ; ::_thesis: ( not c in dom f or f . c = (f1 - f2) . c ) assume A3: c in dom f ; ::_thesis: f . c = (f1 - f2) . c hence f . c = (f1 . c) - (f2 . c) by A2 .= (f1 . c) + ((- f2) . c) by Th8 .= (f1 - f2) . c by A1, A3, Def1 ; ::_thesis: verum end; definition let C be set ; let D1, D2 be complex-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: - redefine funcf1 - f2 -> PartFunc of C,COMPLEX; coherence f1 - f2 is PartFunc of C,COMPLEX proof ( dom (f1 - f2) = (dom f1) /\ (dom f2) & rng (f1 - f2) c= COMPLEX ) by Th12, VALUED_0:def_1; hence f1 - f2 is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be real-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: - redefine funcf1 - f2 -> PartFunc of C,REAL; coherence f1 - f2 is PartFunc of C,REAL proof ( dom (f1 - f2) = (dom f1) /\ (dom f2) & rng (f1 - f2) c= REAL ) by Th12, VALUED_0:def_3; hence f1 - f2 is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be rational-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: - redefine funcf1 - f2 -> PartFunc of C,RAT; coherence f1 - f2 is PartFunc of C,RAT proof ( dom (f1 - f2) = (dom f1) /\ (dom f2) & rng (f1 - f2) c= RAT ) by Th12, RELAT_1:def_19; hence f1 - f2 is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be integer-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: - redefine funcf1 - f2 -> PartFunc of C,INT; coherence f1 - f2 is PartFunc of C,INT proof ( dom (f1 - f2) = (dom f1) /\ (dom f2) & rng (f1 - f2) c= INT ) by Th12, RELAT_1:def_19; hence f1 - f2 is PartFunc of C,INT by RELSET_1:4; ::_thesis: verum end; end; Lm3: for C being set for D1, D2 being non empty complex-membered set for f1 being Function of C,D1 for f2 being Function of C,D2 holds dom (f1 - f2) = C proof let C be set ; ::_thesis: for D1, D2 being non empty complex-membered set for f1 being Function of C,D1 for f2 being Function of C,D2 holds dom (f1 - f2) = C let D1, D2 be non empty complex-membered set ; ::_thesis: for f1 being Function of C,D1 for f2 being Function of C,D2 holds dom (f1 - f2) = C let f1 be Function of C,D1; ::_thesis: for f2 being Function of C,D2 holds dom (f1 - f2) = C let f2 be Function of C,D2; ::_thesis: dom (f1 - f2) = C thus dom (f1 - f2) = (dom f1) /\ (dom (- f2)) by Def1 .= C /\ (dom (- f2)) by FUNCT_2:def_1 .= C /\ C by FUNCT_2:def_1 .= C ; ::_thesis: verum end; registration let C be set ; let D1, D2 be non empty complex-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 - f2 -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = f1 - f2 holds b1 is total proof dom (f1 - f2) = C by Lm3; hence for b1 being PartFunc of C,COMPLEX st b1 = f1 - f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty real-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 - f2 -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = f1 - f2 holds b1 is total proof dom (f1 - f2) = C by Lm3; hence for b1 being PartFunc of C,REAL st b1 = f1 - f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty rational-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 - f2 -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = f1 - f2 holds b1 is total proof dom (f1 - f2) = C by Lm3; hence for b1 being PartFunc of C,RAT st b1 = f1 - f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty integer-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 - f2 -> total for PartFunc of C,INT; coherence for b1 being PartFunc of C,INT st b1 = f1 - f2 holds b1 is total proof dom (f1 - f2) = C by Lm3; hence for b1 being PartFunc of C,INT st b1 = f1 - f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; theorem :: VALUED_1:15 for C being set for D1, D2 being non empty complex-membered set for f1 being Function of C,D1 for f2 being Function of C,D2 for c being Element of C holds (f1 - f2) . c = (f1 . c) - (f2 . c) proof let C be set ; ::_thesis: for D1, D2 being non empty complex-membered set for f1 being Function of C,D1 for f2 being Function of C,D2 for c being Element of C holds (f1 - f2) . c = (f1 . c) - (f2 . c) let D1, D2 be non empty complex-membered set ; ::_thesis: for f1 being Function of C,D1 for f2 being Function of C,D2 for c being Element of C holds (f1 - f2) . c = (f1 . c) - (f2 . c) let f1 be Function of C,D1; ::_thesis: for f2 being Function of C,D2 for c being Element of C holds (f1 - f2) . c = (f1 . c) - (f2 . c) let f2 be Function of C,D2; ::_thesis: for c being Element of C holds (f1 - f2) . c = (f1 . c) - (f2 . c) let c be Element of C; ::_thesis: (f1 - f2) . c = (f1 . c) - (f2 . c) A1: dom (f1 - f2) = C by FUNCT_2:def_1; percases ( not C is empty or C is empty ) ; suppose not C is empty ; ::_thesis: (f1 - f2) . c = (f1 . c) - (f2 . c) hence (f1 - f2) . c = (f1 . c) + ((- f2) . c) by A1, Def1 .= (f1 . c) - (f2 . c) by Th8 ; ::_thesis: verum end; supposeA2: C is empty ; ::_thesis: (f1 - f2) . c = (f1 . c) - (f2 . c) then dom f2 = {} ; then f2 . c = 0 by FUNCT_1:def_2; hence (f1 - f2) . c = (f1 . c) - (f2 . c) by A2; ::_thesis: verum end; end; end; registration let f1, f2 be complex-valued FinSequence; clusterf1 - f2 -> FinSequence-like ; coherence f1 - f2 is FinSequence-like ; end; begin definition let f1, f2 be complex-valued Function; funcf1 /" f2 -> Function equals :: VALUED_1:def 10 f1 (#) (f2 "); coherence f1 (#) (f2 ") is Function ; end; :: deftheorem defines /" VALUED_1:def_10_:_ for f1, f2 being complex-valued Function holds f1 /" f2 = f1 (#) (f2 "); theorem Th16: :: VALUED_1:16 for f1, f2 being complex-valued Function holds dom (f1 /" f2) = (dom f1) /\ (dom f2) proof let f1, f2 be complex-valued Function; ::_thesis: dom (f1 /" f2) = (dom f1) /\ (dom f2) thus dom (f1 /" f2) = (dom f1) /\ (dom (f2 ")) by Def4 .= (dom f1) /\ (dom f2) by Def7 ; ::_thesis: verum end; theorem :: VALUED_1:17 for f1, f2 being complex-valued Function for c being set holds (f1 /" f2) . c = (f1 . c) / (f2 . c) proof let f1, f2 be complex-valued Function; ::_thesis: for c being set holds (f1 /" f2) . c = (f1 . c) / (f2 . c) let c be set ; ::_thesis: (f1 /" f2) . c = (f1 . c) / (f2 . c) A1: dom (f1 /" f2) = (dom f1) /\ (dom f2) by Th16; percases ( c in dom (f1 /" f2) or not c in dom (f1 /" f2) ) ; suppose c in dom (f1 /" f2) ; ::_thesis: (f1 /" f2) . c = (f1 . c) / (f2 . c) hence (f1 /" f2) . c = (f1 . c) * ((f2 ") . c) by Def4 .= (f1 . c) / (f2 . c) by Th10 ; ::_thesis: verum end; supposeA2: not c in dom (f1 /" f2) ; ::_thesis: (f1 /" f2) . c = (f1 . c) / (f2 . c) then ( not c in dom f1 or not c in dom f2 ) by A1, XBOOLE_0:def_4; then A3: ( f1 . c = 0 or f2 . c = 0 ) by FUNCT_1:def_2; thus (f1 /" f2) . c = 0 / 0 by A2, FUNCT_1:def_2 .= (f1 . c) / (f2 . c) by A3 ; ::_thesis: verum end; end; end; registration let f1, f2 be complex-valued Function; clusterf1 /" f2 -> complex-valued ; coherence f1 /" f2 is complex-valued ; end; registration let f1, f2 be real-valued Function; clusterf1 /" f2 -> real-valued ; coherence f1 /" f2 is real-valued ; end; registration let f1, f2 be RAT -valued Function; clusterf1 /" f2 -> RAT -valued ; coherence f1 /" f2 is RAT -valued ; end; definition let C be set ; let D1, D2 be complex-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: /" redefine funcf1 /" f2 -> PartFunc of C,COMPLEX; coherence f1 /" f2 is PartFunc of C,COMPLEX proof ( dom (f1 /" f2) = (dom f1) /\ (dom f2) & rng (f1 /" f2) c= COMPLEX ) by Th16, VALUED_0:def_1; hence f1 /" f2 is PartFunc of C,COMPLEX by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be real-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: /" redefine funcf1 /" f2 -> PartFunc of C,REAL; coherence f1 /" f2 is PartFunc of C,REAL proof ( dom (f1 /" f2) = (dom f1) /\ (dom f2) & rng (f1 /" f2) c= REAL ) by Th16, VALUED_0:def_3; hence f1 /" f2 is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D1, D2 be rational-membered set ; let f1 be PartFunc of C,D1; let f2 be PartFunc of C,D2; :: original: /" redefine funcf1 /" f2 -> PartFunc of C,RAT; coherence f1 /" f2 is PartFunc of C,RAT proof ( dom (f1 /" f2) = (dom f1) /\ (dom f2) & rng (f1 /" f2) c= RAT ) by Th16, RELAT_1:def_19; hence f1 /" f2 is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; Lm4: for C being set for D1, D2 being non empty complex-membered set for f1 being Function of C,D1 for f2 being Function of C,D2 holds dom (f1 /" f2) = C proof let C be set ; ::_thesis: for D1, D2 being non empty complex-membered set for f1 being Function of C,D1 for f2 being Function of C,D2 holds dom (f1 /" f2) = C let D1, D2 be non empty complex-membered set ; ::_thesis: for f1 being Function of C,D1 for f2 being Function of C,D2 holds dom (f1 /" f2) = C let f1 be Function of C,D1; ::_thesis: for f2 being Function of C,D2 holds dom (f1 /" f2) = C let f2 be Function of C,D2; ::_thesis: dom (f1 /" f2) = C thus dom (f1 /" f2) = (dom f1) /\ (dom f2) by Th16 .= C /\ (dom f2) by FUNCT_2:def_1 .= C /\ C by FUNCT_2:def_1 .= C ; ::_thesis: verum end; registration let C be set ; let D1, D2 be non empty complex-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 /" f2 -> total for PartFunc of C,COMPLEX; coherence for b1 being PartFunc of C,COMPLEX st b1 = f1 /" f2 holds b1 is total proof dom (f1 /" f2) = C by Lm4; hence for b1 being PartFunc of C,COMPLEX st b1 = f1 /" f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty real-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 /" f2 -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = f1 /" f2 holds b1 is total proof dom (f1 /" f2) = C by Lm4; hence for b1 being PartFunc of C,REAL st b1 = f1 /" f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D1, D2 be non empty rational-membered set ; let f1 be Function of C,D1; let f2 be Function of C,D2; clusterf1 /" f2 -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = f1 /" f2 holds b1 is total proof dom (f1 /" f2) = C by Lm4; hence for b1 being PartFunc of C,RAT st b1 = f1 /" f2 holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let f1, f2 be complex-valued FinSequence; clusterf1 /" f2 -> FinSequence-like ; coherence f1 /" f2 is FinSequence-like ; end; begin definition let f be complex-valued Function; deffunc H1( set ) -> Element of REAL = |.(f . \$1).|; func|.f.| -> real-valued Function means :Def11: :: VALUED_1:def 11 ( dom it = dom f & ( for c being set st c in dom it holds it . c = |.(f . c).| ) ); existence ex b1 being real-valued Function st ( dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = |.(f . c).| ) ) proof consider g being Function such that A1: ( dom g = dom f & ( for x being set st x in dom f holds g . x = H1(x) ) ) from FUNCT_1:sch_3(); g is real-valued proof let x be set ; :: according to VALUED_0:def_9 ::_thesis: ( not x in dom g or g . x is real ) assume x in dom g ; ::_thesis: g . x is real then g . x = |.(f . x).| by A1; hence g . x is real ; ::_thesis: verum end; hence ex b1 being real-valued Function st ( dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = |.(f . c).| ) ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being real-valued Function st dom b1 = dom f & ( for c being set st c in dom b1 holds b1 . c = |.(f . c).| ) & dom b2 = dom f & ( for c being set st c in dom b2 holds b2 . c = |.(f . c).| ) holds b1 = b2 proof let h, g be real-valued Function; ::_thesis: ( dom h = dom f & ( for c being set st c in dom h holds h . c = |.(f . c).| ) & dom g = dom f & ( for c being set st c in dom g holds g . c = |.(f . c).| ) implies h = g ) assume that A2: dom h = dom f and A3: for c being set st c in dom h holds h . c = H1(c) and A4: dom g = dom f and A5: for c being set st c in dom g holds g . c = H1(c) ; ::_thesis: h = g now__::_thesis:_for_x_being_set_st_x_in_dom_h_holds_ h_._x_=_g_._x let x be set ; ::_thesis: ( x in dom h implies h . x = g . x ) assume A6: x in dom h ; ::_thesis: h . x = g . x hence h . x = H1(x) by A3 .= g . x by A2, A4, A5, A6 ; ::_thesis: verum end; hence h = g by A2, A4, FUNCT_1:2; ::_thesis: verum end; projectivity for b1 being real-valued Function for b2 being complex-valued Function st dom b1 = dom b2 & ( for c being set st c in dom b1 holds b1 . c = |.(b2 . c).| ) holds ( dom b1 = dom b1 & ( for c being set st c in dom b1 holds b1 . c = |.(b1 . c).| ) ) proof let r be real-valued Function; ::_thesis: for b1 being complex-valued Function st dom r = dom b1 & ( for c being set st c in dom r holds r . c = |.(b1 . c).| ) holds ( dom r = dom r & ( for c being set st c in dom r holds r . c = |.(r . c).| ) ) let h be complex-valued Function; ::_thesis: ( dom r = dom h & ( for c being set st c in dom r holds r . c = |.(h . c).| ) implies ( dom r = dom r & ( for c being set st c in dom r holds r . c = |.(r . c).| ) ) ) assume that dom r = dom h and A7: for c being set st c in dom r holds r . c = |.(h . c).| ; ::_thesis: ( dom r = dom r & ( for c being set st c in dom r holds r . c = |.(r . c).| ) ) thus dom r = dom r ; ::_thesis: for c being set st c in dom r holds r . c = |.(r . c).| let c be set ; ::_thesis: ( c in dom r implies r . c = |.(r . c).| ) assume A8: c in dom r ; ::_thesis: r . c = |.(r . c).| hence r . c = |.|.(h . c).|.| by A7 .= |.(r . c).| by A7, A8 ; ::_thesis: verum end; end; :: deftheorem Def11 defines |. VALUED_1:def_11_:_ for f being complex-valued Function for b2 being real-valued Function holds ( b2 = |.f.| iff ( dom b2 = dom f & ( for c being set st c in dom b2 holds b2 . c = |.(f . c).| ) ) ); notation let f be complex-valued Function; synonym abs f for |.f.|; end; theorem :: VALUED_1:18 for f being complex-valued Function for c being set holds |.f.| . c = |.(f . c).| proof let f be complex-valued Function; ::_thesis: for c being set holds |.f.| . c = |.(f . c).| let c be set ; ::_thesis: |.f.| . c = |.(f . c).| A1: dom |.f.| = dom f by Def11; percases ( c in dom f or not c in dom f ) ; suppose c in dom f ; ::_thesis: |.f.| . c = |.(f . c).| hence |.f.| . c = |.(f . c).| by A1, Def11; ::_thesis: verum end; supposeA2: not c in dom f ; ::_thesis: |.f.| . c = |.(f . c).| hence |.f.| . c = |.0.| by A1, COMPLEX1:44, FUNCT_1:def_2 .= |.(f . c).| by A2, FUNCT_1:def_2 ; ::_thesis: verum end; end; end; registration let f be RAT -valued Function; cluster|.f.| -> RAT -valued real-valued ; coherence |.f.| is RAT -valued proof let y be set ; :: according to TARSKI:def_3,RELAT_1:def_19 ::_thesis: ( not y in rng |.f.| or y in RAT ) assume y in rng |.f.| ; ::_thesis: y in RAT then consider x being set such that A1: x in dom |.f.| and A2: |.f.| . x = y by FUNCT_1:def_3; |.f.| . x = |.(f . x).| by A1, Def11; hence y in RAT by A2, RAT_1:def_2; ::_thesis: verum end; end; registration let f be INT -valued Function; cluster|.f.| -> real-valued natural-valued ; coherence |.f.| is natural-valued proof let x be set ; :: according to VALUED_0:def_12 ::_thesis: ( not x in dom |.f.| or |.f.| . x is natural ) abs (f . x) is natural ; hence ( not x in dom |.f.| or |.f.| . x is natural ) by Def11; ::_thesis: verum end; end; definition let C be set ; let D be complex-membered set ; let f be PartFunc of C,D; :: original: |. redefine func|.f.| -> PartFunc of C,REAL; coherence |.f.| is PartFunc of C,REAL proof ( dom |.f.| = dom f & rng |.f.| c= REAL ) by Def11, VALUED_0:def_3; hence |.f.| is PartFunc of C,REAL by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be complex-membered set ; let f be PartFunc of C,D; :: original: |. redefine func abs f -> PartFunc of C,REAL; coherence |.f.| is PartFunc of C,REAL proof abs f = |.f.| ; hence |.f.| is PartFunc of C,REAL ; ::_thesis: verum end; end; definition let C be set ; let D be rational-membered set ; let f be PartFunc of C,D; :: original: |. redefine func|.f.| -> PartFunc of C,RAT; coherence |.f.| is PartFunc of C,RAT proof ( dom |.f.| = dom f & rng |.f.| c= RAT ) by Def11, RELAT_1:def_19; hence |.f.| is PartFunc of C,RAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be rational-membered set ; let f be PartFunc of C,D; :: original: |. redefine func abs f -> PartFunc of C,RAT; coherence |.f.| is PartFunc of C,RAT proof abs f = |.f.| ; hence |.f.| is PartFunc of C,RAT ; ::_thesis: verum end; end; definition let C be set ; let D be integer-membered set ; let f be PartFunc of C,D; :: original: |. redefine func|.f.| -> PartFunc of C,NAT; coherence |.f.| is PartFunc of C,NAT proof ( dom |.f.| = dom f & rng |.f.| c= NAT ) by Def11, VALUED_0:def_6; hence |.f.| is PartFunc of C,NAT by RELSET_1:4; ::_thesis: verum end; end; definition let C be set ; let D be integer-membered set ; let f be PartFunc of C,D; :: original: |. redefine func abs f -> PartFunc of C,NAT; coherence |.f.| is PartFunc of C,NAT proof abs f = |.f.| ; hence |.f.| is PartFunc of C,NAT ; ::_thesis: verum end; end; registration let C be set ; let D be non empty complex-membered set ; let f be Function of C,D; cluster|.f.| -> total for PartFunc of C,REAL; coherence for b1 being PartFunc of C,REAL st b1 = |.f.| holds b1 is total proof dom |.f.| = dom f by Def11 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,REAL st b1 = |.f.| holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty rational-membered set ; let f be Function of C,D; cluster|.f.| -> total for PartFunc of C,RAT; coherence for b1 being PartFunc of C,RAT st b1 = |.f.| holds b1 is total proof dom |.f.| = dom f by Def11 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,RAT st b1 = |.f.| holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let C be set ; let D be non empty integer-membered set ; let f be Function of C,D; cluster|.f.| -> total for PartFunc of C,NAT; coherence for b1 being PartFunc of C,NAT st b1 = |.f.| holds b1 is total proof dom |.f.| = dom f by Def11 .= C by FUNCT_2:def_1 ; hence for b1 being PartFunc of C,NAT st b1 = |.f.| holds b1 is total by PARTFUN1:def_2; ::_thesis: verum end; end; registration let f be complex-valued FinSequence; cluster|.f.| -> real-valued FinSequence-like ; coherence |.f.| is FinSequence-like proof dom (abs f) = dom f by Def11; hence |.f.| is FinSequence-like by Lm1; ::_thesis: verum end; end; theorem :: VALUED_1:19 for f, g being FinSequence for h being Function st dom h = (dom f) /\ (dom g) holds h is FinSequence by Lm2; begin definition let p be Function; let k be Nat; func Shift (p,k) -> Function means :Def12: :: VALUED_1:def 12 ( dom it = { (m + k) where m is Element of NAT : m in dom p } & ( for m being Element of NAT st m in dom p holds it . (m + k) = p . m ) ); existence ex b1 being Function st ( dom b1 = { (m + k) where m is Element of NAT : m in dom p } & ( for m being Element of NAT st m in dom p holds b1 . (m + k) = p . m ) ) proof defpred S1[ set , set ] means ex m being Element of NAT st ( \$1 = m + k & \$2 = p . m ); set A = { (m + k) where m is Element of NAT : m in dom p } ; A1: for e being set st e in { (m + k) where m is Element of NAT : m in dom p } holds ex u being set st S1[e,u] proof let e be set ; ::_thesis: ( e in { (m + k) where m is Element of NAT : m in dom p } implies ex u being set st S1[e,u] ) assume e in { (m + k) where m is Element of NAT : m in dom p } ; ::_thesis: ex u being set st S1[e,u] then consider m being Element of NAT such that A2: e = m + k and m in dom p ; take p . m ; ::_thesis: S1[e,p . m] thus S1[e,p . m] by A2; ::_thesis: verum end; consider f being Function such that A3: dom f = { (m + k) where m is Element of NAT : m in dom p } and A4: for e being set st e in { (m + k) where m is Element of NAT : m in dom p } holds S1[e,f . e] from CLASSES1:sch_1(A1); take f ; ::_thesis: ( dom f = { (m + k) where m is Element of NAT : m in dom p } & ( for m being Element of NAT st m in dom p holds f . (m + k) = p . m ) ) thus dom f = { (m + k) where m is Element of NAT : m in dom p } by A3; ::_thesis: for m being Element of NAT st m in dom p holds f . (m + k) = p . m let m be Element of NAT ; ::_thesis: ( m in dom p implies f . (m + k) = p . m ) assume m in dom p ; ::_thesis: f . (m + k) = p . m then m + k in { (m + k) where m is Element of NAT : m in dom p } ; then ex j being Element of NAT st ( m + k = j + k & f . (m + k) = p . j ) by A4; hence f . (m + k) = p . m ; ::_thesis: verum end; uniqueness for b1, b2 being Function st dom b1 = { (m + k) where m is Element of NAT : m in dom p } & ( for m being Element of NAT st m in dom p holds b1 . (m + k) = p . m ) & dom b2 = { (m + k) where m is Element of NAT : m in dom p } & ( for m being Element of NAT st m in dom p holds b2 . (m + k) = p . m ) holds b1 = b2 proof let IT1, IT2 be Function; ::_thesis: ( dom IT1 = { (m + k) where m is Element of NAT : m in dom p } & ( for m being Element of NAT st m in dom p holds IT1 . (m + k) = p . m ) & dom IT2 = { (m + k) where m is Element of NAT : m in dom p } & ( for m being Element of NAT st m in dom p holds IT2 . (m + k) = p . m ) implies IT1 = IT2 ) assume that A5: dom IT1 = { (m + k) where m is Element of NAT : m in dom p } and A6: for m being Element of NAT st m in dom p holds IT1 . (m + k) = p . m and A7: dom IT2 = { (m + k) where m is Element of NAT : m in dom p } and A8: for m being Element of NAT st m in dom p holds IT2 . (m + k) = p . m ; ::_thesis: IT1 = IT2 for x being set st x in dom IT1 holds IT1 . x = IT2 . x proof let x be set ; ::_thesis: ( x in dom IT1 implies IT1 . x = IT2 . x ) assume x in dom IT1 ; ::_thesis: IT1 . x = IT2 . x then consider m being Element of NAT such that A9: ( x = m + k & m in dom p ) by A5; thus IT1 . x = p . m by A6, A9 .= IT2 . x by A8, A9 ; ::_thesis: verum end; hence IT1 = IT2 by A5, A7, FUNCT_1:2; ::_thesis: verum end; end; :: deftheorem Def12 defines Shift VALUED_1:def_12_:_ for p being Function for k being Nat for b3 being Function holds ( b3 = Shift (p,k) iff ( dom b3 = { (m + k) where m is Element of NAT : m in dom p } & ( for m being Element of NAT st m in dom p holds b3 . (m + k) = p . m ) ) ); registration let p be Function; let k be Nat; cluster Shift (p,k) -> NAT -defined ; coherence Shift (p,k) is NAT -defined proof A1: dom (Shift (p,k)) = { (m + k) where m is Element of NAT : m in dom p } by Def12; Shift (p,k) is NAT -defined proof let x be set ; :: according to TARSKI:def_3,RELAT_1:def_18 ::_thesis: ( not x in dom (Shift (p,k)) or x in NAT ) assume x in dom (Shift (p,k)) ; ::_thesis: x in NAT then ex m being Element of NAT st ( x = m + k & m in dom p ) by A1; hence x in NAT ; ::_thesis: verum end; hence Shift (p,k) is NAT -defined ; ::_thesis: verum end; end; theorem :: VALUED_1:20 for P, Q being Function for k being Nat st P c= Q holds Shift (P,k) c= Shift (Q,k) proof let P, Q be Function; ::_thesis: for k being Nat st P c= Q holds Shift (P,k) c= Shift (Q,k) let k be Nat; ::_thesis: ( P c= Q implies Shift (P,k) c= Shift (Q,k) ) assume A1: P c= Q ; ::_thesis: Shift (P,k) c= Shift (Q,k) then A2: dom P c= dom Q by GRFUNC_1:2; A3: dom (Shift (P,k)) = { (m + k) where m is Element of NAT : m in dom P } by Def12; A4: dom (Shift (Q,k)) = { (m + k) where m is Element of NAT : m in dom Q } by Def12; now__::_thesis:_for_x_being_set_st_x_in_dom_(Shift_(P,k))_holds_ x_in_dom_(Shift_(Q,k)) let x be set ; ::_thesis: ( x in dom (Shift (P,k)) implies x in dom (Shift (Q,k)) ) assume x in dom (Shift (P,k)) ; ::_thesis: x in dom (Shift (Q,k)) then ex m1 being Element of NAT st ( x = m1 + k & m1 in dom P ) by A3; hence x in dom (Shift (Q,k)) by A2, A4; ::_thesis: verum end; then A5: dom (Shift (P,k)) c= dom (Shift (Q,k)) by TARSKI:def_3; now__::_thesis:_for_x_being_set_st_x_in_dom_(Shift_(P,k))_holds_ (Shift_(P,k))_._x_=_(Shift_(Q,k))_._x let x be set ; ::_thesis: ( x in dom (Shift (P,k)) implies (Shift (P,k)) . x = (Shift (Q,k)) . x ) assume x in dom (Shift (P,k)) ; ::_thesis: (Shift (P,k)) . x = (Shift (Q,k)) . x then consider m1 being Element of NAT such that A6: x = m1 + k and A7: m1 in dom P by A3; thus (Shift (P,k)) . x = (Shift (P,k)) . (m1 + k) by A6 .= P . m1 by A7, Def12 .= Q . m1 by A1, A7, GRFUNC_1:2 .= (Shift (Q,k)) . (m1 + k) by A2, A7, Def12 .= (Shift (Q,k)) . x by A6 ; ::_thesis: verum end; hence Shift (P,k) c= Shift (Q,k) by A5, GRFUNC_1:2; ::_thesis: verum end; theorem :: VALUED_1:21 for n, m being Nat for I being Function holds Shift ((Shift (I,m)),n) = Shift (I,(m + n)) proof let n, m be Nat; ::_thesis: for I being Function holds Shift ((Shift (I,m)),n) = Shift (I,(m + n)) let I be Function; ::_thesis: Shift ((Shift (I,m)),n) = Shift (I,(m + n)) set A = { (l + m) where l is Element of NAT : l in dom I } ; A1: dom (Shift (I,m)) = { (l + m) where l is Element of NAT : l in dom I } by Def12; A2: now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_I_holds_ (Shift_((Shift_(I,m)),n))_._(l_+_(m_+_n))_=_I_._l let l be Element of NAT ; ::_thesis: ( l in dom I implies (Shift ((Shift (I,m)),n)) . (l + (m + n)) = I . l ) assume A3: l in dom I ; ::_thesis: (Shift ((Shift (I,m)),n)) . (l + (m + n)) = I . l then A4: l + m in dom (Shift (I,m)) by A1; thus (Shift ((Shift (I,m)),n)) . (l + (m + n)) = (Shift ((Shift (I,m)),n)) . ((l + m) + n) .= (Shift (I,m)) . (l + m) by A4, Def12 .= I . l by A3, Def12 ; ::_thesis: verum end; { (p + n) where p is Element of NAT : p in { (l + m) where l is Element of NAT : l in dom I } } = { (q + (m + n)) where q is Element of NAT : q in dom I } proof thus { (p + n) where p is Element of NAT : p in { (l + m) where l is Element of NAT : l in dom I } } c= { (q + (m + n)) where q is Element of NAT : q in dom I } :: according to XBOOLE_0:def_10 ::_thesis: { (q + (m + n)) where q is Element of NAT : q in dom I } c= { (p + n) where p is Element of NAT : p in { (l + m) where l is Element of NAT : l in dom I } } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (p + n) where p is Element of NAT : p in { (l + m) where l is Element of NAT : l in dom I } } or x in { (q + (m + n)) where q is Element of NAT : q in dom I } ) assume x in { (p + n) where p is Element of NAT : p in { (l + m) where l is Element of NAT : l in dom I } } ; ::_thesis: x in { (q + (m + n)) where q is Element of NAT : q in dom I } then consider p being Element of NAT such that A5: x = p + n and A6: p in { (l + m) where l is Element of NAT : l in dom I } ; consider l being Element of NAT such that A7: p = l + m and A8: l in dom I by A6; x = l + (m + n) by A5, A7; hence x in { (q + (m + n)) where q is Element of NAT : q in dom I } by A8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (q + (m + n)) where q is Element of NAT : q in dom I } or x in { (p + n) where p is Element of NAT : p in { (l + m) where l is Element of NAT : l in dom I } } ) assume x in { (q + (m + n)) where q is Element of NAT : q in dom I } ; ::_thesis: x in { (p + n) where p is Element of NAT : p in { (l + m) where l is Element of NAT : l in dom I } } then consider q being Element of NAT such that A9: ( x = q + (m + n) & q in dom I ) ; ( x = (q + m) + n & q + m in { (l + m) where l is Element of NAT : l in dom I } ) by A9; hence x in { (p + n) where p is Element of NAT : p in { (l + m) where l is Element of NAT : l in dom I } } ; ::_thesis: verum end; then dom (Shift ((Shift (I,m)),n)) = { (l + (m + n)) where l is Element of NAT : l in dom I } by A1, Def12; hence Shift ((Shift (I,m)),n) = Shift (I,(m + n)) by A2, Def12; ::_thesis: verum end; theorem :: VALUED_1:22 for s, f being Function for n being Nat holds Shift ((f * s),n) = f * (Shift (s,n)) proof let s, f be Function; ::_thesis: for n being Nat holds Shift ((f * s),n) = f * (Shift (s,n)) let n be Nat; ::_thesis: Shift ((f * s),n) = f * (Shift (s,n)) A1: dom (f * s) c= dom s by RELAT_1:25; A2: dom (Shift (s,n)) = { (m + n) where m is Element of NAT : m in dom s } by Def12; now__::_thesis:_for_e_being_set_holds_ (_(_e_in__{__(m_+_n)_where_m_is_Element_of_NAT_:_m_in_dom_(f_*_s)__}__implies_(_e_in_dom_(Shift_(s,n))_&_(Shift_(s,n))_._e_in_dom_f_)_)_&_(_e_in_dom_(Shift_(s,n))_&_(Shift_(s,n))_._e_in_dom_f_implies_e_in__{__(m_+_n)_where_m_is_Element_of_NAT_:_m_in_dom_(f_*_s)__}__)_) let e be set ; ::_thesis: ( ( e in { (m + n) where m is Element of NAT : m in dom (f * s) } implies ( e in dom (Shift (s,n)) & (Shift (s,n)) . e in dom f ) ) & ( e in dom (Shift (s,n)) & (Shift (s,n)) . e in dom f implies e in { (m + n) where m is Element of NAT : m in dom (f * s) } ) ) thus ( e in { (m + n) where m is Element of NAT : m in dom (f * s) } implies ( e in dom (Shift (s,n)) & (Shift (s,n)) . e in dom f ) ) ::_thesis: ( e in dom (Shift (s,n)) & (Shift (s,n)) . e in dom f implies e in { (m + n) where m is Element of NAT : m in dom (f * s) } ) proof assume e in { (m + n) where m is Element of NAT : m in dom (f * s) } ; ::_thesis: ( e in dom (Shift (s,n)) & (Shift (s,n)) . e in dom f ) then consider m being Element of NAT such that A3: e = m + n and A4: m in dom (f * s) ; thus e in dom (Shift (s,n)) by A2, A1, A3, A4; ::_thesis: (Shift (s,n)) . e in dom f (Shift (s,n)) . e = s . m by A1, A3, A4, Def12; hence (Shift (s,n)) . e in dom f by A4, FUNCT_1:11; ::_thesis: verum end; assume e in dom (Shift (s,n)) ; ::_thesis: ( (Shift (s,n)) . e in dom f implies e in { (m + n) where m is Element of NAT : m in dom (f * s) } ) then consider m0 being Element of NAT such that A5: e = m0 + n and A6: m0 in dom s by A2; assume (Shift (s,n)) . e in dom f ; ::_thesis: e in { (m + n) where m is Element of NAT : m in dom (f * s) } then s . m0 in dom f by A5, A6, Def12; then m0 in dom (f * s) by A6, FUNCT_1:11; hence e in { (m + n) where m is Element of NAT : m in dom (f * s) } by A5; ::_thesis: verum end; then (Shift (s,n)) " (dom f) = { (m + n) where m is Element of NAT : m in dom (f * s) } by FUNCT_1:def_7; then A7: dom (f * (Shift (s,n))) = { (m + n) where m is Element of NAT : m in dom (f * s) } by RELAT_1:147; now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_(f_*_s)_holds_ (f_*_(Shift_(s,n)))_._(m_+_n)_=_(f_*_s)_._m let m be Element of NAT ; ::_thesis: ( m in dom (f * s) implies (f * (Shift (s,n))) . (m + n) = (f * s) . m ) assume A8: m in dom (f * s) ; ::_thesis: (f * (Shift (s,n))) . (m + n) = (f * s) . m then m + n in dom (Shift (s,n)) by A2, A1; hence (f * (Shift (s,n))) . (m + n) = f . ((Shift (s,n)) . (m + n)) by FUNCT_1:13 .= f . (s . m) by A1, A8, Def12 .= (f * s) . m by A8, FUNCT_1:12 ; ::_thesis: verum end; hence Shift ((f * s),n) = f * (Shift (s,n)) by A7, Def12; ::_thesis: verum end; theorem :: VALUED_1:23 for I, J being Function for n being Nat holds Shift ((I +* J),n) = (Shift (I,n)) +* (Shift (J,n)) proof let I, J be Function; ::_thesis: for n being Nat holds Shift ((I +* J),n) = (Shift (I,n)) +* (Shift (J,n)) let n be Nat; ::_thesis: Shift ((I +* J),n) = (Shift (I,n)) +* (Shift (J,n)) A1: dom (Shift (J,n)) = { (m + n) where m is Element of NAT : m in dom J } by Def12; A2: now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_(I_+*_J)_holds_ ((Shift_(I,n))_+*_(Shift_(J,n)))_._(m_+_n)_=_(I_+*_J)_._m let m be Element of NAT ; ::_thesis: ( m in dom (I +* J) implies ((Shift (I,n)) +* (Shift (J,n))) . (b1 + n) = (I +* J) . b1 ) assume A3: m in dom (I +* J) ; ::_thesis: ((Shift (I,n)) +* (Shift (J,n))) . (b1 + n) = (I +* J) . b1 percases ( m in dom J or not m in dom J ) ; supposeA4: m in dom J ; ::_thesis: ((Shift (I,n)) +* (Shift (J,n))) . (b1 + n) = (I +* J) . b1 then m + n in dom (Shift (J,n)) by A1; hence ((Shift (I,n)) +* (Shift (J,n))) . (m + n) = (Shift (J,n)) . (m + n) by FUNCT_4:13 .= J . m by A4, Def12 .= (I +* J) . m by A4, FUNCT_4:13 ; ::_thesis: verum end; supposeA5: not m in dom J ; ::_thesis: ((Shift (I,n)) +* (Shift (J,n))) . (b1 + n) = (I +* J) . b1 m in (dom I) \/ (dom J) by A3, FUNCT_4:def_1; then A6: m in dom I by A5, XBOOLE_0:def_3; for l being Element of NAT holds ( not m + n = l + n or not l in dom J ) by A5; then not m + n in dom (Shift (J,n)) by A1; hence ((Shift (I,n)) +* (Shift (J,n))) . (m + n) = (Shift (I,n)) . (m + n) by FUNCT_4:11 .= I . m by A6, Def12 .= (I +* J) . m by A5, FUNCT_4:11 ; ::_thesis: verum end; end; end; A7: dom (Shift (I,n)) = { (m + n) where m is Element of NAT : m in dom I } by Def12; A8: (dom (Shift (I,n))) \/ (dom (Shift (J,n))) = { (m + n) where m is Element of NAT : m in (dom I) \/ (dom J) } proof hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: { (m + n) where m is Element of NAT : m in (dom I) \/ (dom J) } c= (dom (Shift (I,n))) \/ (dom (Shift (J,n))) let x be set ; ::_thesis: ( x in (dom (Shift (I,n))) \/ (dom (Shift (J,n))) implies x in { (l + n) where l is Element of NAT : l in (dom I) \/ (dom J) } ) assume x in (dom (Shift (I,n))) \/ (dom (Shift (J,n))) ; ::_thesis: x in { (l + n) where l is Element of NAT : l in (dom I) \/ (dom J) } then ( x in dom (Shift (I,n)) or x in dom (Shift (J,n)) ) by XBOOLE_0:def_3; then consider m being Element of NAT such that A9: ( ( x = m + n & m in dom J ) or ( x = m + n & m in dom I ) ) by A1, A7; m in (dom I) \/ (dom J) by A9, XBOOLE_0:def_3; hence x in { (l + n) where l is Element of NAT : l in (dom I) \/ (dom J) } by A9; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (m + n) where m is Element of NAT : m in (dom I) \/ (dom J) } or x in (dom (Shift (I,n))) \/ (dom (Shift (J,n))) ) assume x in { (m + n) where m is Element of NAT : m in (dom I) \/ (dom J) } ; ::_thesis: x in (dom (Shift (I,n))) \/ (dom (Shift (J,n))) then consider m being Element of NAT such that A10: x = m + n and A11: m in (dom I) \/ (dom J) ; ( m in dom I or m in dom J ) by A11, XBOOLE_0:def_3; then ( x in dom (Shift (I,n)) or x in dom (Shift (J,n)) ) by A1, A7, A10; hence x in (dom (Shift (I,n))) \/ (dom (Shift (J,n))) by XBOOLE_0:def_3; ::_thesis: verum end; dom (I +* J) = (dom I) \/ (dom J) by FUNCT_4:def_1; then dom ((Shift (I,n)) +* (Shift (J,n))) = { (m + n) where m is Element of NAT : m in dom (I +* J) } by A8, FUNCT_4:def_1; hence Shift ((I +* J),n) = (Shift (I,n)) +* (Shift (J,n)) by A2, Def12; ::_thesis: verum end; theorem Th24: :: VALUED_1:24 for p being Function for k, il being Nat st il in dom p holds il + k in dom (Shift (p,k)) proof let p be Function; ::_thesis: for k, il being Nat st il in dom p holds il + k in dom (Shift (p,k)) let k, il be Nat; ::_thesis: ( il in dom p implies il + k in dom (Shift (p,k)) ) assume A1: il in dom p ; ::_thesis: il + k in dom (Shift (p,k)) A2: dom (Shift (p,k)) = { (loc + k) where loc is Element of NAT : loc in dom p } by Def12; reconsider loc = il as Element of NAT by ORDINAL1:def_12; loc + k in dom (Shift (p,k)) by A2, A1; hence il + k in dom (Shift (p,k)) ; ::_thesis: verum end; theorem Th25: :: VALUED_1:25 for p being Function for k being Nat holds rng (Shift (p,k)) c= rng p proof let p be Function; ::_thesis: for k being Nat holds rng (Shift (p,k)) c= rng p let k be Nat; ::_thesis: rng (Shift (p,k)) c= rng p let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Shift (p,k)) or y in rng p ) assume y in rng (Shift (p,k)) ; ::_thesis: y in rng p then consider x being set such that A1: x in dom (Shift (p,k)) and A2: y = (Shift (p,k)) . x by FUNCT_1:def_3; x in { (m + k) where m is Element of NAT : m in dom p } by A1, Def12; then consider m being Element of NAT such that A3: x = m + k and A4: m in dom p ; p . m = (Shift (p,k)) . x by A3, A4, Def12; hence y in rng p by A2, A4, FUNCT_1:def_3; ::_thesis: verum end; theorem :: VALUED_1:26 for p being Function st dom p c= NAT holds for k being Nat holds rng (Shift (p,k)) = rng p proof let p be Function; ::_thesis: ( dom p c= NAT implies for k being Nat holds rng (Shift (p,k)) = rng p ) assume A1: dom p c= NAT ; ::_thesis: for k being Nat holds rng (Shift (p,k)) = rng p let k be Nat; ::_thesis: rng (Shift (p,k)) = rng p thus rng (Shift (p,k)) c= rng p by Th25; :: according to XBOOLE_0:def_10 ::_thesis: rng p c= rng (Shift (p,k)) let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng p or y in rng (Shift (p,k)) ) assume y in rng p ; ::_thesis: y in rng (Shift (p,k)) then consider x being set such that A2: x in dom p and A3: y = p . x by FUNCT_1:def_3; reconsider x = x as Element of NAT by A1, A2; ( x + k in dom (Shift (p,k)) & (Shift (p,k)) . (x + k) = y ) by A2, A3, Def12, Th24; hence y in rng (Shift (p,k)) by FUNCT_1:def_3; ::_thesis: verum end; registration let p be finite Function; let k be Nat; cluster Shift (p,k) -> finite ; coherence Shift (p,k) is finite proof deffunc H1( Element of NAT ) -> Element of NAT = p + k; A1: dom p is finite ; { H1(w) where w is Element of NAT : w in dom p } is finite from FRAENKEL:sch_21(A1); then dom (Shift (p,k)) is finite by Def12; hence Shift (p,k) is finite by FINSET_1:10; ::_thesis: verum end; end; definition let X be non empty ext-real-membered set ; let s be sequence of X; :: original: increasing redefine attrs is increasing means :: VALUED_1:def 13 for n being Nat holds s . n < s . (n + 1); compatibility ( s is increasing iff for n being Nat holds s . n < s . (n + 1) ) proof thus ( s is increasing implies for n being Nat holds s . n < s . (n + 1) ) ::_thesis: ( ( for n being Nat holds s . n < s . (n + 1) ) implies s is increasing ) proof assume A1: s is increasing ; ::_thesis: for n being Nat holds s . n < s . (n + 1) let n be Nat; ::_thesis: s . n < s . (n + 1) A2: n < n + 1 by NAT_1:13; ( dom s = NAT & n in NAT ) by FUNCT_2:def_1, ORDINAL1:def_12; hence s . n < s . (n + 1) by A1, A2, VALUED_0:def_13; ::_thesis: verum end; assume A3: for n being Nat holds s . n < s . (n + 1) ; ::_thesis: s is increasing let e1 be ext-real number ; :: according to VALUED_0:def_13 ::_thesis: for b1 being set holds ( not e1 in dom s or not b1 in dom s or b1 <= e1 or not s . b1 <= s . e1 ) let e2 be ext-real number ; ::_thesis: ( not e1 in dom s or not e2 in dom s or e2 <= e1 or not s . e2 <= s . e1 ) assume ( e1 in dom s & e2 in dom s ) ; ::_thesis: ( e2 <= e1 or not s . e2 <= s . e1 ) then reconsider m = e1, n = e2 as Nat ; defpred S1[ Nat] means ( m < \$1 implies s . m < s . \$1 ); A4: for j being Nat st S1[j] holds S1[j + 1] proof let j be Nat; ::_thesis: ( S1[j] implies S1[j + 1] ) assume A5: S1[j] ; ::_thesis: S1[j + 1] assume m < j + 1 ; ::_thesis: s . m < s . (j + 1) then m <= j by NAT_1:13; then ( s . m < s . j or m = j ) by A5, XXREAL_0:1; hence s . m < s . (j + 1) by A3, XXREAL_0:2; ::_thesis: verum end; assume A6: e1 < e2 ; ::_thesis: not s . e2 <= s . e1 A7: S1[ 0 ] ; for j being Nat holds S1[j] from NAT_1:sch_2(A7, A4); then s . m < s . n by A6; hence not s . e2 <= s . e1 ; ::_thesis: verum end; :: original: decreasing redefine attrs is decreasing means :: VALUED_1:def 14 for n being Nat holds s . n > s . (n + 1); compatibility ( s is decreasing iff for n being Nat holds s . n > s . (n + 1) ) proof thus ( s is decreasing implies for n being Nat holds s . n > s . (n + 1) ) ::_thesis: ( ( for n being Nat holds s . n > s . (n + 1) ) implies s is decreasing ) proof assume A8: s is decreasing ; ::_thesis: for n being Nat holds s . n > s . (n + 1) let n be Nat; ::_thesis: s . n > s . (n + 1) A9: n < n + 1 by NAT_1:13; ( dom s = NAT & n in NAT ) by FUNCT_2:def_1, ORDINAL1:def_12; hence s . n > s . (n + 1) by A8, A9, VALUED_0:def_14; ::_thesis: verum end; assume A10: for n being Nat holds s . n > s . (n + 1) ; ::_thesis: s is decreasing let e1 be ext-real number ; :: according to VALUED_0:def_14 ::_thesis: for b1 being set holds ( not e1 in dom s or not b1 in dom s or b1 <= e1 or not s . e1 <= s . b1 ) let e2 be ext-real number ; ::_thesis: ( not e1 in dom s or not e2 in dom s or e2 <= e1 or not s . e1 <= s . e2 ) assume ( e1 in dom s & e2 in dom s ) ; ::_thesis: ( e2 <= e1 or not s . e1 <= s . e2 ) then reconsider m = e1, n = e2 as Nat ; defpred S1[ Nat] means ( m < \$1 implies s . m > s . \$1 ); A11: for j being Nat st S1[j] holds S1[j + 1] proof let j be Nat; ::_thesis: ( S1[j] implies S1[j + 1] ) assume A12: S1[j] ; ::_thesis: S1[j + 1] assume m < j + 1 ; ::_thesis: s . m > s . (j + 1) then m <= j by NAT_1:13; then ( s . m > s . j or m = j ) by A12, XXREAL_0:1; hence s . m > s . (j + 1) by A10, XXREAL_0:2; ::_thesis: verum end; assume A13: e1 < e2 ; ::_thesis: not s . e1 <= s . e2 A14: S1[ 0 ] ; for j being Nat holds S1[j] from NAT_1:sch_2(A14, A11); then s . m > s . n by A13; hence not s . e1 <= s . e2 ; ::_thesis: verum end; :: original: non-decreasing redefine attrs is non-decreasing means :: VALUED_1:def 15 for n being Nat holds s . n <= s . (n + 1); compatibility ( s is non-decreasing iff for n being Nat holds s . n <= s . (n + 1) ) proof thus ( s is non-decreasing implies for n being Nat holds s . n <= s . (n + 1) ) ::_thesis: ( ( for n being Nat holds s . n <= s . (n + 1) ) implies s is non-decreasing ) proof assume A15: s is non-decreasing ; ::_thesis: for n being Nat holds s . n <= s . (n + 1) let n be Nat; ::_thesis: s . n <= s . (n + 1) A16: n < n + 1 by NAT_1:13; ( dom s = NAT & n in NAT ) by FUNCT_2:def_1, ORDINAL1:def_12; hence s . n <= s . (n + 1) by A15, A16, VALUED_0:def_15; ::_thesis: verum end; assume A17: for n being Nat holds s . n <= s . (n + 1) ; ::_thesis: s is non-decreasing let e1 be ext-real number ; :: according to VALUED_0:def_15 ::_thesis: for b1 being set holds ( not e1 in dom s or not b1 in dom s or not e1 <= b1 or s . e1 <= s . b1 ) let e2 be ext-real number ; ::_thesis: ( not e1 in dom s or not e2 in dom s or not e1 <= e2 or s . e1 <= s . e2 ) assume ( e1 in dom s & e2 in dom s ) ; ::_thesis: ( not e1 <= e2 or s . e1 <= s . e2 ) then reconsider m = e1, n = e2 as Nat ; defpred S1[ Nat] means ( m <= \$1 implies s . m <= s . \$1 ); A18: for j being Nat st S1[j] holds S1[j + 1] proof let j be Nat; ::_thesis: ( S1[j] implies S1[j + 1] ) assume A19: S1[j] ; ::_thesis: S1[j + 1] assume m <= j + 1 ; ::_thesis: s . m <= s . (j + 1) then A20: ( m < j + 1 or m = j + 1 ) by XXREAL_0:1; s . j <= s . (j + 1) by A17; hence s . m <= s . (j + 1) by A19, A20, NAT_1:13, XXREAL_0:2; ::_thesis: verum end; assume A21: e1 <= e2 ; ::_thesis: s . e1 <= s . e2 A22: S1[ 0 ] ; for j being Nat holds S1[j] from NAT_1:sch_2(A22, A18); then s . m <= s . n by A21; hence s . e1 <= s . e2 ; ::_thesis: verum end; :: original: non-increasing redefine attrs is non-increasing means :: VALUED_1:def 16 for n being Nat holds s . n >= s . (n + 1); compatibility ( s is non-increasing iff for n being Nat holds s . n >= s . (n + 1) ) proof thus ( s is non-increasing implies for n being Nat holds s . n >= s . (n + 1) ) ::_thesis: ( ( for n being Nat holds s . n >= s . (n + 1) ) implies s is non-increasing ) proof assume A23: s is non-increasing ; ::_thesis: for n being Nat holds s . n >= s . (n + 1) let n be Nat; ::_thesis: s . n >= s . (n + 1) A24: n < n + 1 by NAT_1:13; ( dom s = NAT & n in NAT ) by FUNCT_2:def_1, ORDINAL1:def_12; hence s . n >= s . (n + 1) by A23, A24, VALUED_0:def_16; ::_thesis: verum end; assume A25: for n being Nat holds s . n >= s . (n + 1) ; ::_thesis: s is non-increasing let e1 be ext-real number ; :: according to VALUED_0:def_16 ::_thesis: for b1 being set holds ( not e1 in dom s or not b1 in dom s or not e1 <= b1 or s . b1 <= s . e1 ) let e2 be ext-real number ; ::_thesis: ( not e1 in dom s or not e2 in dom s or not e1 <= e2 or s . e2 <= s . e1 ) assume ( e1 in dom s & e2 in dom s ) ; ::_thesis: ( not e1 <= e2 or s . e2 <= s . e1 ) then reconsider m = e1, n = e2 as Nat ; defpred S1[ Nat] means ( m <= \$1 implies s . m >= s . \$1 ); A26: for j being Nat st S1[j] holds S1[j + 1] proof let j be Nat; ::_thesis: ( S1[j] implies S1[j + 1] ) assume A27: S1[j] ; ::_thesis: S1[j + 1] assume m <= j + 1 ; ::_thesis: s . m >= s . (j + 1) then A28: ( m < j + 1 or m = j + 1 ) by XXREAL_0:1; s . j >= s . (j + 1) by A25; hence s . m >= s . (j + 1) by A27, A28, NAT_1:13, XXREAL_0:2; ::_thesis: verum end; assume A29: e1 <= e2 ; ::_thesis: s . e2 <= s . e1 A30: S1[ 0 ] ; for j being Nat holds S1[j] from NAT_1:sch_2(A30, A26); then s . m >= s . n by A29; hence s . e2 <= s . e1 ; ::_thesis: verum end; end; :: deftheorem defines increasing VALUED_1:def_13_:_ for X being non empty ext-real-membered set for s being sequence of X holds ( s is increasing iff for n being Nat holds s . n < s . (n + 1) ); :: deftheorem defines decreasing VALUED_1:def_14_:_ for X being non empty ext-real-membered set for s being sequence of X holds ( s is decreasing iff for n being Nat holds s . n > s . (n + 1) ); :: deftheorem defines non-decreasing VALUED_1:def_15_:_ for X being non empty ext-real-membered set for s being sequence of X holds ( s is non-decreasing iff for n being Nat holds s . n <= s . (n + 1) ); :: deftheorem defines non-increasing VALUED_1:def_16_:_ for X being non empty ext-real-membered set for s being sequence of X holds ( s is non-increasing iff for n being Nat holds s . n >= s . (n + 1) ); scheme :: VALUED_1:sch 1 SubSeqChoice{ F1() -> non empty set , F2() -> sequence of F1(), P1[ set ] } : ex S1 being subsequence of F2() st for n being Element of NAT holds P1[S1 . n] provided A1: for n being Element of NAT ex m being Element of NAT st ( n <= m & P1[F2() . m] ) proof defpred S1[ set , set , set ] means ( \$3 in NAT & ( for m, k being Element of NAT st m = \$2 & k = \$3 holds ( m < k & P1[F2() . k] ) ) ); consider n0 being Element of NAT such that 0 <= n0 and A2: P1[F2() . n0] by A1; A3: for n being Element of NAT for x being set ex y being set st S1[n,x,y] proof let n be Element of NAT ; ::_thesis: for x being set ex y being set st S1[n,x,y] let x be set ; ::_thesis: ex y being set st S1[n,x,y] percases ( x in NAT or not x in NAT ) ; suppose x in NAT ; ::_thesis: ex y being set st S1[n,x,y] then reconsider mx = x as Element of NAT ; consider my being Element of NAT such that A4: ( mx + 1 <= my & P1[F2() . my] ) by A1; take my ; ::_thesis: S1[n,x,my] thus my in NAT ; ::_thesis: for m, k being Element of NAT st m = x & k = my holds ( m < k & P1[F2() . k] ) thus for m, k being Element of NAT st m = x & k = my holds ( m < k & P1[F2() . k] ) by A4, NAT_1:13; ::_thesis: verum end; supposeA5: not x in NAT ; ::_thesis: ex y being set st S1[n,x,y] take 0 ; ::_thesis: S1[n,x, 0 ] set y = 0 ; thus 0 in NAT ; ::_thesis: for m, k being Element of NAT st m = x & k = 0 holds ( m < k & P1[F2() . k] ) let m, k be Element of NAT ; ::_thesis: ( m = x & k = 0 implies ( m < k & P1[F2() . k] ) ) assume that A6: m = x and k = 0 ; ::_thesis: ( m < k & P1[F2() . k] ) thus ( m < k & P1[F2() . k] ) by A5, A6; ::_thesis: verum end; end; end; consider g being Function such that A7: dom g = NAT and A8: g . 0 = n0 and A9: for n being Element of NAT holds S1[n,g . n,g . (n + 1)] from RECDEF_1:sch_1(A3); rng g c= NAT proof defpred S2[ Element of NAT ] means g . \$1 in NAT ; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng g or y in NAT ) assume y in rng g ; ::_thesis: y in NAT then consider x being set such that A10: x in dom g and A11: g . x = y by FUNCT_1:def_3; reconsider n = x as Element of NAT by A7, A10; A12: for k being Element of NAT st S2[k] holds S2[k + 1] by A9; A13: S2[ 0 ] by A8; for k being Element of NAT holds S2[k] from NAT_1:sch_1(A13, A12); then g . n in NAT ; hence y in NAT by A11; ::_thesis: verum end; then reconsider g = g as sequence of NAT by A7, FUNCT_2:2; g is V39() proof let n be Nat; :: according to VALUED_1:def_13 ::_thesis: g . n < g . (n + 1) reconsider n = n as Element of NAT by ORDINAL1:def_12; g . n < g . (n + 1) by A9; hence g . n < g . (n + 1) ; ::_thesis: verum end; then reconsider g = g as V39() sequence of NAT ; reconsider S1 = F2() * g as sequence of F1() ; A14: dom S1 = NAT by FUNCT_2:def_1; reconsider S1 = S1 as subsequence of F2() ; take S1 ; ::_thesis: for n being Element of NAT holds P1[S1 . n] thus for n being Element of NAT holds P1[S1 . n] ::_thesis: verum proof let n be Element of NAT ; ::_thesis: P1[S1 . n] percases ( n = 0 or n <> 0 ) ; suppose n = 0 ; ::_thesis: P1[S1 . n] hence P1[S1 . n] by A2, A8, A14, FUNCT_1:12; ::_thesis: verum end; suppose n <> 0 ; ::_thesis: P1[S1 . n] then n >= 0 + 1 by NAT_1:13; then reconsider n9 = n - 1 as Element of NAT by INT_1:5; reconsider k = g . (n9 + 1) as Element of NAT ; for m, k being Element of NAT st m = g . n9 & k = g . (n9 + 1) holds P1[F2() . k] by A9; then P1[F2() . k] ; hence P1[S1 . n] by A14, FUNCT_1:12; ::_thesis: verum end; end; end; end; theorem :: VALUED_1:27 for k being Nat for F being NAT -defined Function holds dom F, dom (Shift (F,k)) are_equipotent proof let k be Nat; ::_thesis: for F being NAT -defined Function holds dom F, dom (Shift (F,k)) are_equipotent let F be NAT -defined Function; ::_thesis: dom F, dom (Shift (F,k)) are_equipotent defpred S1[ set , set ] means ex il being Element of NAT st ( \$1 = il & \$2 = k + il ); A1: for e being set st e in dom F holds ex u being set st S1[e,u] proof let e be set ; ::_thesis: ( e in dom F implies ex u being set st S1[e,u] ) assume e in dom F ; ::_thesis: ex u being set st S1[e,u] then reconsider e = e as Element of NAT ; take k + e ; ::_thesis: S1[e,k + e] take e ; ::_thesis: ( e = e & k + e = k + e ) thus ( e = e & k + e = k + e ) ; ::_thesis: verum end; consider f being Function such that A2: dom f = dom F and A3: for x being set st x in dom F holds S1[x,f . x] from CLASSES1:sch_1(A1); take f ; :: according to WELLORD2:def_4 ::_thesis: ( f is one-to-one & dom f = dom F & rng f = dom (Shift (F,k)) ) hereby :: according to FUNCT_1:def_4 ::_thesis: ( dom f = dom F & rng f = dom (Shift (F,k)) ) let x1, x2 be set ; ::_thesis: ( x1 in dom f & x2 in dom f & f . x1 = f . x2 implies x1 = x2 ) assume that A4: x1 in dom f and A5: x2 in dom f and A6: f . x1 = f . x2 ; ::_thesis: x1 = x2 consider i1 being Element of NAT such that A7: x1 = i1 and A8: f . x1 = k + i1 by A2, A3, A4; consider i2 being Element of NAT such that A9: x2 = i2 and A10: f . x2 = k + i2 by A2, A3, A5; thus x1 = x2 by A7, A6, A8, A10, A9; ::_thesis: verum end; thus dom f = dom F by A2; ::_thesis: rng f = dom (Shift (F,k)) A11: dom (Shift (F,k)) = { (m + k) where m is Element of NAT : m in dom F } by Def12; hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: dom (Shift (F,k)) c= rng f let y be set ; ::_thesis: ( y in rng f implies y in dom (Shift (F,k)) ) assume y in rng f ; ::_thesis: y in dom (Shift (F,k)) then consider x being set such that A12: x in dom f and A13: f . x = y by FUNCT_1:def_3; consider il being Element of NAT such that A14: x = il and A15: f . x = k + il by A2, A3, A12; thus y in dom (Shift (F,k)) by A2, A11, A12, A13, A14, A15; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in dom (Shift (F,k)) or y in rng f ) assume y in dom (Shift (F,k)) ; ::_thesis: y in rng f then consider m being Element of NAT such that A16: y = m + k and A17: m in dom F by A11; consider il being Element of NAT such that A18: m = il and A19: f . m = k + il by A3, A17; thus y in rng f by A2, A16, A17, A18, A19, FUNCT_1:def_3; ::_thesis: verum end; registration let F be NAT -defined Function; reduce Shift (F,0) to F; reducibility Shift (F,0) = F proof A1: dom F = { (m + 0) where m is Element of NAT : m in dom F } proof hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: { (m + 0) where m is Element of NAT : m in dom F } c= dom F let a be set ; ::_thesis: ( a in dom F implies a in { (m + 0) where m is Element of NAT : m in dom F } ) assume A2: a in dom F ; ::_thesis: a in { (m + 0) where m is Element of NAT : m in dom F } then reconsider l = a as Element of NAT ; a = l + 0 ; hence a in { (m + 0) where m is Element of NAT : m in dom F } by A2; ::_thesis: verum end; let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { (m + 0) where m is Element of NAT : m in dom F } or a in dom F ) assume a in { (m + 0) where m is Element of NAT : m in dom F } ; ::_thesis: a in dom F then ex m being Element of NAT st ( a = m + 0 & m in dom F ) ; hence a in dom F ; ::_thesis: verum end; for m being Element of NAT st m in dom F holds F . (m + 0) = F . m ; hence Shift (F,0) = F by A1, Def12; ::_thesis: verum end; end; theorem :: VALUED_1:28 for F being NAT -defined Function holds Shift (F,0) = F ; registration let X be non empty set ; let F be X -valued Function; let k be Nat; cluster Shift (F,k) -> X -valued ; coherence Shift (F,k) is X -valued proof A1: dom (Shift (F,k)) = { (m + k) where m is Element of NAT : m in dom F } by Def12; thus rng (Shift (F,k)) c= X :: according to RELAT_1:def_19 ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (Shift (F,k)) or x in X ) assume x in rng (Shift (F,k)) ; ::_thesis: x in X then consider y being set such that A2: y in dom (Shift (F,k)) and A3: x = (Shift (F,k)) . y by FUNCT_1:def_3; consider m being Element of NAT such that A4: y = m + k and A5: m in dom F by A2, A1; (Shift (F,k)) . (m + k) = F . m by A5, Def12 .= F /. m by A5, PARTFUN1:def_6 ; hence x in X by A3, A4; ::_thesis: verum end; end; end; registration cluster Relation-like NAT -defined Function-like non empty for set ; existence ex b1 being Function st ( not b1 is empty & b1 is NAT -defined ) proof take id NAT ; ::_thesis: ( not id NAT is empty & id NAT is NAT -defined ) thus ( not id NAT is empty & id NAT is NAT -defined ) ; ::_thesis: verum end; end; registration let F be empty Function; let k be Nat; cluster Shift (F,k) -> empty ; coherence Shift (F,k) is empty proof A1: dom (Shift (F,k)) = { (m + k) where m is Element of NAT : m in dom F } by Def12; assume not Shift (F,k) is empty ; ::_thesis: contradiction then reconsider f = Shift (F,k) as non empty Function ; not dom f is empty ; then consider a being set such that A2: a in dom (Shift (F,k)) by XBOOLE_0:def_1; ex m being Element of NAT st ( a = m + k & m in dom F ) by A1, A2; hence contradiction ; ::_thesis: verum end; end; registration let F be NAT -defined non empty Function; let k be Nat; cluster Shift (F,k) -> non empty ; coherence not Shift (F,k) is empty proof A1: dom (Shift (F,k)) = { (m + k) where m is Element of NAT : m in dom F } by Def12; consider a being set such that A2: a in dom F by XBOOLE_0:def_1; reconsider a = a as Element of NAT by A2; consider m being Nat such that A3: a = m ; reconsider m = m as Element of NAT by ORDINAL1:def_12; m + k in dom (Shift (F,k)) by A1, A2, A3; hence not Shift (F,k) is empty ; ::_thesis: verum end; end; theorem :: VALUED_1:29 for F being Function for k being Nat st k > 0 holds not 0 in dom (Shift (F,k)) proof let F be Function; ::_thesis: for k being Nat st k > 0 holds not 0 in dom (Shift (F,k)) let k be Nat; ::_thesis: ( k > 0 implies not 0 in dom (Shift (F,k)) ) assume that A1: k > 0 and A2: 0 in dom (Shift (F,k)) ; ::_thesis: contradiction dom (Shift (F,k)) = { (m + k) where m is Element of NAT : m in dom F } by Def12; then ex m being Element of NAT st ( 0 = m + k & m in dom F ) by A2; hence contradiction by A1; ::_thesis: verum end; registration cluster Relation-like NAT -defined Function-like non empty finite for set ; existence ex b1 being Function st ( b1 is NAT -defined & b1 is finite & not b1 is empty ) proof take f = 0 .--> 0; ::_thesis: ( f is NAT -defined & f is finite & not f is empty ) dom f = {0} by FUNCOP_1:13; hence dom f c= NAT by ZFMISC_1:31; :: according to RELAT_1:def_18 ::_thesis: ( f is finite & not f is empty ) thus ( f is finite & not f is empty ) ; ::_thesis: verum end; end; registration let F be NAT -defined Relation; cluster dom F -> natural-membered ; coherence dom F is natural-membered ; end; definition let F be NAT -defined non empty finite Function; func LastLoc F -> Element of NAT equals :: VALUED_1:def 17 max (dom F); coherence max (dom F) is Element of NAT by ORDINAL1:def_12; end; :: deftheorem defines LastLoc VALUED_1:def_17_:_ for F being NAT -defined non empty finite Function holds LastLoc F = max (dom F); definition let F be NAT -defined non empty finite Function; func CutLastLoc F -> Function equals :: VALUED_1:def 18 F \ ((LastLoc F) .--> (F . (LastLoc F))); coherence F \ ((LastLoc F) .--> (F . (LastLoc F))) is Function ; end; :: deftheorem defines CutLastLoc VALUED_1:def_18_:_ for F being NAT -defined non empty finite Function holds CutLastLoc F = F \ ((LastLoc F) .--> (F . (LastLoc F))); registration let F be NAT -defined non empty finite Function; cluster CutLastLoc F -> NAT -defined finite ; coherence ( CutLastLoc F is NAT -defined & CutLastLoc F is finite ) ; end; theorem Th30: :: VALUED_1:30 for F being NAT -defined non empty finite Function holds LastLoc F in dom F by XXREAL_2:def_8; theorem :: VALUED_1:31 for F, G being NAT -defined non empty finite Function st F c= G holds LastLoc F <= LastLoc G by RELAT_1:11, XXREAL_2:59; theorem :: VALUED_1:32 for F being NAT -defined non empty finite Function for l being Element of NAT st l in dom F holds l <= LastLoc F by XXREAL_2:def_8; definition let F be NAT -defined non empty Function; func FirstLoc F -> Element of NAT equals :: VALUED_1:def 19 min (dom F); coherence min (dom F) is Element of NAT by ORDINAL1:def_12; end; :: deftheorem defines FirstLoc VALUED_1:def_19_:_ for F being NAT -defined non empty Function holds FirstLoc F = min (dom F); theorem :: VALUED_1:33 for F being NAT -defined non empty finite Function holds FirstLoc F in dom F by XXREAL_2:def_7; theorem :: VALUED_1:34 for F, G being NAT -defined non empty finite Function st F c= G holds FirstLoc G <= FirstLoc F by RELAT_1:11, XXREAL_2:60; theorem :: VALUED_1:35 for l1 being Element of NAT for F being NAT -defined non empty finite Function st l1 in dom F holds FirstLoc F <= l1 by XXREAL_2:def_7; theorem Th36: :: VALUED_1:36 for F being NAT -defined non empty finite Function holds dom (CutLastLoc F) = (dom F) \ {(LastLoc F)} proof let F be NAT -defined non empty finite Function; ::_thesis: dom (CutLastLoc F) = (dom F) \ {(LastLoc F)} A1: dom ((LastLoc F) .--> (F . (LastLoc F))) = {(LastLoc F)} by FUNCOP_1:13; reconsider R = {[(LastLoc F),(F . (LastLoc F))]} as Relation ; A2: R = (LastLoc F) .--> (F . (LastLoc F)) by FUNCT_4:82; then A3: dom R = {(LastLoc F)} by FUNCOP_1:13; thus dom (CutLastLoc F) c= (dom F) \ {(LastLoc F)} :: according to XBOOLE_0:def_10 ::_thesis: (dom F) \ {(LastLoc F)} c= dom (CutLastLoc F) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (CutLastLoc F) or x in (dom F) \ {(LastLoc F)} ) assume x in dom (CutLastLoc F) ; ::_thesis: x in (dom F) \ {(LastLoc F)} then consider y being set such that A4: [x,y] in F \ R by A2, XTUPLE_0:def_12; A5: not [x,y] in R by A4, XBOOLE_0:def_5; A6: x in dom F by A4, XTUPLE_0:def_12; percases ( x <> LastLoc F or y <> F . (LastLoc F) ) by A5, TARSKI:def_1; suppose x <> LastLoc F ; ::_thesis: x in (dom F) \ {(LastLoc F)} then not x in dom R by A3, TARSKI:def_1; hence x in (dom F) \ {(LastLoc F)} by A1, A2, A6, XBOOLE_0:def_5; ::_thesis: verum end; supposeA7: y <> F . (LastLoc F) ; ::_thesis: x in (dom F) \ {(LastLoc F)} now__::_thesis:_not_x_in_dom_R assume x in dom R ; ::_thesis: contradiction then x = LastLoc F by A3, TARSKI:def_1; hence contradiction by A4, A7, FUNCT_1:1; ::_thesis: verum end; hence x in (dom F) \ {(LastLoc F)} by A1, A2, A6, XBOOLE_0:def_5; ::_thesis: verum end; end; end; thus (dom F) \ {(LastLoc F)} c= dom (CutLastLoc F) by A1, RELAT_1:3; ::_thesis: verum end; theorem :: VALUED_1:37 for F being NAT -defined non empty finite Function holds dom F = (dom (CutLastLoc F)) \/ {(LastLoc F)} proof let F be NAT -defined non empty finite Function; ::_thesis: dom F = (dom (CutLastLoc F)) \/ {(LastLoc F)} LastLoc F in dom F by Th30; then A1: {(LastLoc F)} c= dom F by ZFMISC_1:31; dom (CutLastLoc F) = (dom F) \ {(LastLoc F)} by Th36; hence dom F = (dom (CutLastLoc F)) \/ {(LastLoc F)} by A1, XBOOLE_1:45; ::_thesis: verum end; registration cluster Relation-like NAT -defined Function-like finite 1 -element for set ; existence ex b1 being Function st ( b1 is 1 -element & b1 is NAT -defined & b1 is finite ) proof take 0 .--> 0 ; ::_thesis: ( 0 .--> 0 is 1 -element & 0 .--> 0 is NAT -defined & 0 .--> 0 is finite ) thus ( 0 .--> 0 is 1 -element & 0 .--> 0 is NAT -defined & 0 .--> 0 is finite ) ; ::_thesis: verum end; end; registration let F be NAT -defined finite 1 -element Function; cluster CutLastLoc F -> empty ; coherence CutLastLoc F is empty proof LastLoc F in dom F by Th30; then A1: [(LastLoc F),(F . (LastLoc F))] in F by FUNCT_1:def_2; assume not CutLastLoc F is empty ; ::_thesis: contradiction then consider a being set such that A2: a in CutLastLoc F by XBOOLE_0:def_1; A3: a = [(LastLoc F),(F . (LastLoc F))] by A1, A2, ZFMISC_1:def_10; not a in (LastLoc F) .--> (F . (LastLoc F)) by A2, XBOOLE_0:def_5; then not a in {[(LastLoc F),(F . (LastLoc F))]} by FUNCT_4:82; hence contradiction by A3, TARSKI:def_1; ::_thesis: verum end; end; theorem :: VALUED_1:38 for F being NAT -defined non empty finite Function holds card (CutLastLoc F) = (card F) - 1 proof let F be NAT -defined non empty finite Function; ::_thesis: card (CutLastLoc F) = (card F) - 1 (LastLoc F) .--> (F . (LastLoc F)) c= F proof let a, b be set ; :: according to RELAT_1:def_3 ::_thesis: ( not [a,b] in (LastLoc F) .--> (F . (LastLoc F)) or [a,b] in F ) assume [a,b] in (LastLoc F) .--> (F . (LastLoc F)) ; ::_thesis: [a,b] in F then [a,b] in {[(LastLoc F),(F . (LastLoc F))]} by FUNCT_4:82; then A1: [a,b] = [(LastLoc F),(F . (LastLoc F))] by TARSKI:def_1; LastLoc F in dom F by Th30; hence [a,b] in F by A1, FUNCT_1:def_2; ::_thesis: verum end; hence card (CutLastLoc F) = (card F) - (card ((LastLoc F) .--> (F . (LastLoc F)))) by CARD_2:44 .= (card F) - (card {[(LastLoc F),(F . (LastLoc F))]}) by FUNCT_4:82 .= (card F) - 1 by CARD_1:30 ; ::_thesis: verum end; begin registration let X be set ; let f be X -defined complex-valued Function; cluster - f -> X -defined complex-valued ; coherence - f is X -defined proof A1: dom (- f) = dom f by Def5; thus dom (- f) c= X by A1; :: according to RELAT_1:def_18 ::_thesis: verum end; clusterf " -> X -defined complex-valued ; coherence f " is X -defined proof A2: dom (f ") = dom f by Def7; thus dom (f ") c= X by A2; :: according to RELAT_1:def_18 ::_thesis: verum end; clusterf ^2 -> X -defined ; coherence f ^2 is X -defined proof A3: dom (f ^2) = dom f by Th11; thus dom (f ^2) c= X by A3; :: according to RELAT_1:def_18 ::_thesis: verum end; cluster|.f.| -> X -defined real-valued ; coherence |.f.| is X -defined proof A4: dom |.f.| = dom f by Def11; thus dom |.f.| c= X by A4; :: according to RELAT_1:def_18 ::_thesis: verum end; end; registration let X be set ; cluster Relation-like X -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued for set ; existence ex b1 being X -defined natural-valued Function st b1 is total proof take the total PartFunc of X,NAT ; ::_thesis: the total PartFunc of X,NAT is total thus the total PartFunc of X,NAT is total ; ::_thesis: verum end; end; registration let X be set ; let f be X -defined total complex-valued Function; cluster - f -> total complex-valued ; coherence - f is total proof A1: dom (- f) = dom f by Def5; dom f = X by PARTFUN1:def_2; hence - f is total by A1, PARTFUN1:def_2; ::_thesis: verum end; clusterf " -> total complex-valued ; coherence f " is total proof A2: dom (f ") = dom f by Def7; dom f = X by PARTFUN1:def_2; hence f " is total by A2, PARTFUN1:def_2; ::_thesis: verum end; clusterf ^2 -> total ; coherence f ^2 is total proof A3: dom (f ^2) = dom f by Th11; dom f = X by PARTFUN1:def_2; hence f ^2 is total by A3, PARTFUN1:def_2; ::_thesis: verum end; cluster|.f.| -> total real-valued ; coherence |.f.| is total proof A4: dom |.f.| = dom f by Def11; dom f = X by PARTFUN1:def_2; hence |.f.| is total by A4, PARTFUN1:def_2; ::_thesis: verum end; end; registration let X be set ; let f be X -defined complex-valued Function; let r be complex number ; clusterr + f -> X -defined ; coherence r + f is X -defined proof A1: dom (r + f) = dom f by Def2; thus dom (r + f) c= X by A1; :: according to RELAT_1:def_18 ::_thesis: verum end; clusterf - r -> X -defined ; coherence f - r is X -defined ; clusterr (#) f -> X -defined ; coherence r (#) f is X -defined proof A2: dom (r (#) f) = dom f by Def5; thus dom (r (#) f) c= X by A2; :: according to RELAT_1:def_18 ::_thesis: verum end; end; registration let X be set ; let f be X -defined total complex-valued Function; let r be complex number ; clusterr + f -> total ; coherence r + f is total proof A1: dom (r + f) = dom f by Def2; dom f = X by PARTFUN1:def_2; hence r + f is total by A1, PARTFUN1:def_2; ::_thesis: verum end; clusterf - r -> total ; coherence f - r is total ; clusterr (#) f -> total ; coherence r (#) f is total proof A2: dom (r (#) f) = dom f by Def5; dom f = X by PARTFUN1:def_2; hence r (#) f is total by A2, PARTFUN1:def_2; ::_thesis: verum end; end; registration let X be set ; let f1 be complex-valued Function; let f2 be X -defined complex-valued Function; clusterf1 + f2 -> X -defined ; coherence f1 + f2 is X -defined proof dom (f1 + f2) = (dom f1) /\ (dom f2) by Def1; then A1: dom (f1 + f2) c= dom f2 by XBOOLE_1:17; thus dom (f1 + f2) c= X by A1, XBOOLE_1:1; :: according to RELAT_1:def_18 ::_thesis: verum end; clusterf1 - f2 -> X -defined ; coherence f1 - f2 is X -defined ; clusterf1 (#) f2 -> X -defined ; coherence f1 (#) f2 is X -defined proof dom (f1 (#) f2) = (dom f1) /\ (dom f2) by Def4; then A2: dom (f1 (#) f2) c= dom f2 by XBOOLE_1:17; thus dom (f1 (#) f2) c= X by A2, XBOOLE_1:1; :: according to RELAT_1:def_18 ::_thesis: verum end; clusterf1 /" f2 -> X -defined ; coherence f1 /" f2 is X -defined ; end; registration let X be set ; let f1, f2 be X -defined total complex-valued Function; clusterf1 + f2 -> total ; coherence f1 + f2 is total proof A1: dom (f1 + f2) = (dom f1) /\ (dom f2) by Def1; ( dom f1 = X & dom f2 = X ) by PARTFUN1:def_2; hence f1 + f2 is total by A1, PARTFUN1:def_2; ::_thesis: verum end; clusterf1 - f2 -> total ; coherence f1 - f2 is total ; clusterf1 (#) f2 -> total ; coherence f1 (#) f2 is total proof A2: dom (f1 (#) f2) = (dom f1) /\ (dom f2) by Def4; ( dom f1 = X & dom f2 = X ) by PARTFUN1:def_2; hence f1 (#) f2 is total by A2, PARTFUN1:def_2; ::_thesis: verum end; clusterf1 /" f2 -> total ; coherence f1 /" f2 is total ; end; registration let X be non empty set ; let F be NAT -defined X -valued non empty finite Function; cluster CutLastLoc F -> X -valued ; coherence CutLastLoc F is X -valued ; end; theorem :: VALUED_1:39 for f being Function for i, n being Nat st i in dom (Shift (f,n)) holds ex j being Nat st ( j in dom f & i = j + n ) proof let f be Function; ::_thesis: for i, n being Nat st i in dom (Shift (f,n)) holds ex j being Nat st ( j in dom f & i = j + n ) let i, n be Nat; ::_thesis: ( i in dom (Shift (f,n)) implies ex j being Nat st ( j in dom f & i = j + n ) ) A: dom (Shift (f,n)) = { (m + n) where m is Element of NAT : m in dom f } by Def12; assume i in dom (Shift (f,n)) ; ::_thesis: ex j being Nat st ( j in dom f & i = j + n ) then ex m being Element of NAT st ( i = m + n & m in dom f ) by A; hence ex j being Nat st ( j in dom f & i = j + n ) ; ::_thesis: verum end;