:: SIN_COS semantic presentation begin definition let m, k be Element of NAT ; func CHK (m,k) -> Element of COMPLEX equals :Def1: :: SIN_COS:def 1 1 if m <= k otherwise 0 ; correctness coherence ( ( m <= k implies 1 is Element of COMPLEX ) & ( not m <= k implies 0 is Element of COMPLEX ) ); consistency for b1 being Element of COMPLEX holds verum; proof 0c = 0 ; hence ( ( m <= k implies 1 is Element of COMPLEX ) & ( not m <= k implies 0 is Element of COMPLEX ) & ( for b1 being Element of COMPLEX holds verum ) ) by COMPLEX1:def_4; ::_thesis: verum end; end; :: deftheorem Def1 defines CHK SIN_COS:def_1_:_ for m, k being Element of NAT holds ( ( m <= k implies CHK (m,k) = 1 ) & ( not m <= k implies CHK (m,k) = 0 ) ); registration let m, k be Element of NAT ; cluster CHK (m,k) -> real ; coherence CHK (m,k) is real by Def1; end; scheme :: SIN_COS:sch 1 ExComplexCASE{ F1( Nat, Nat) -> complex number } : for k being Element of NAT ex seq being Complex_Sequence st for n being Element of NAT holds ( ( n <= k implies seq . n = F1(k,n) ) & ( n > k implies seq . n = 0 ) ) proof let k be Element of NAT ; ::_thesis: ex seq being Complex_Sequence st for n being Element of NAT holds ( ( n <= k implies seq . n = F1(k,n) ) & ( n > k implies seq . n = 0 ) ) defpred S1[ set , set ] means ex n being Element of NAT st ( n = \$1 & ( n <= k implies \$2 = F1(k,n) ) & ( n > k implies \$2 = 0c ) ); A1: now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_ ex_y_being_set_st_S1[x,y] let x be set ; ::_thesis: ( x in NAT implies ex y being set st S1[x,y] ) assume x in NAT ; ::_thesis: ex y being set st S1[x,y] then consider n being Element of NAT such that A2: n = x ; A3: now__::_thesis:_(_n_<=_k_implies_(CHK_(n,k))_*_F1(k,n)_=_F1(k,n)_) assume n <= k ; ::_thesis: (CHK (n,k)) * F1(k,n) = F1(k,n) hence (CHK (n,k)) * F1(k,n) = 1 * F1(k,n) by Def1 .= F1(k,n) ; ::_thesis: verum end; A4: now__::_thesis:_(_n_>_k_implies_(CHK_(n,k))_*_F1(k,n)_=_0_) assume n > k ; ::_thesis: (CHK (n,k)) * F1(k,n) = 0 hence (CHK (n,k)) * F1(k,n) = 0 * F1(k,n) by Def1 .= 0 ; ::_thesis: verum end; reconsider y = (CHK (n,k)) * F1(k,n) as set ; take y = y; ::_thesis: S1[x,y] thus S1[x,y] by A2, A3, A4; ::_thesis: verum end; consider f being Function such that A5: dom f = NAT and A6: for x being set st x in NAT holds S1[x,f . x] from CLASSES1:sch_1(A1); now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_ f_._x_is_Element_of_COMPLEX let x be set ; ::_thesis: ( x in NAT implies f . x is Element of COMPLEX ) assume x in NAT ; ::_thesis: f . x is Element of COMPLEX then ex n being Element of NAT st ( n = x & ( n <= k implies f . x = F1(k,n) ) & ( n > k implies f . x = 0c ) ) by A6; hence f . x is Element of COMPLEX by XCMPLX_0:def_2; ::_thesis: verum end; then reconsider f = f as Complex_Sequence by A5, COMSEQ_1:1; take seq = f; ::_thesis: for n being Element of NAT holds ( ( n <= k implies seq . n = F1(k,n) ) & ( n > k implies seq . n = 0 ) ) let n be Element of NAT ; ::_thesis: ( ( n <= k implies seq . n = F1(k,n) ) & ( n > k implies seq . n = 0 ) ) ex l being Element of NAT st ( l = n & ( l <= k implies seq . n = F1(k,l) ) & ( l > k implies seq . n = 0c ) ) by A6; hence ( ( n <= k implies seq . n = F1(k,n) ) & ( n > k implies seq . n = 0 ) ) ; ::_thesis: verum end; scheme :: SIN_COS:sch 2 ExRealCASE{ F1( Nat, Nat) -> real number } : for k being Element of NAT ex rseq being Real_Sequence st for n being Element of NAT holds ( ( n <= k implies rseq . n = F1(k,n) ) & ( n > k implies rseq . n = 0 ) ) proof let k be Element of NAT ; ::_thesis: ex rseq being Real_Sequence st for n being Element of NAT holds ( ( n <= k implies rseq . n = F1(k,n) ) & ( n > k implies rseq . n = 0 ) ) defpred S1[ set , set ] means ex n being Element of NAT st ( n = \$1 & ( n <= k implies \$2 = F1(k,n) ) & ( n > k implies \$2 = 0 ) ); A1: now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_ ex_y_being_set_st_S1[x,y] let x be set ; ::_thesis: ( x in NAT implies ex y being set st S1[x,y] ) assume x in NAT ; ::_thesis: ex y being set st S1[x,y] then consider n being Element of NAT such that A2: n = x ; A3: now__::_thesis:_(_n_<=_k_implies_(CHK_(n,k))_*_F1(k,n)_=_F1(k,n)_) assume n <= k ; ::_thesis: (CHK (n,k)) * F1(k,n) = F1(k,n) hence (CHK (n,k)) * F1(k,n) = 1 * F1(k,n) by Def1 .= F1(k,n) ; ::_thesis: verum end; A4: now__::_thesis:_(_n_>_k_implies_(CHK_(n,k))_*_F1(k,n)_=_0_) assume n > k ; ::_thesis: (CHK (n,k)) * F1(k,n) = 0 hence (CHK (n,k)) * F1(k,n) = 0 * F1(k,n) by Def1 .= 0 ; ::_thesis: verum end; reconsider y = (CHK (n,k)) * F1(k,n) as set ; take y = y; ::_thesis: S1[x,y] thus S1[x,y] by A2, A3, A4; ::_thesis: verum end; consider f being Function such that A5: dom f = NAT and A6: for x being set st x in NAT holds S1[x,f . x] from CLASSES1:sch_1(A1); now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_ f_._x_is_real let x be set ; ::_thesis: ( x in NAT implies f . x is real ) assume x in NAT ; ::_thesis: f . x is real then ex n being Element of NAT st ( n = x & ( n <= k implies f . x = F1(k,n) ) & ( n > k implies f . x = 0 ) ) by A6; hence f . x is real ; ::_thesis: verum end; then reconsider f = f as Real_Sequence by A5, SEQ_1:1; take rseq = f; ::_thesis: for n being Element of NAT holds ( ( n <= k implies rseq . n = F1(k,n) ) & ( n > k implies rseq . n = 0 ) ) let n be Element of NAT ; ::_thesis: ( ( n <= k implies rseq . n = F1(k,n) ) & ( n > k implies rseq . n = 0 ) ) ex l being Element of NAT st ( l = n & ( l <= k implies rseq . n = F1(k,l) ) & ( l > k implies rseq . n = 0 ) ) by A6; hence ( ( n <= k implies rseq . n = F1(k,n) ) & ( n > k implies rseq . n = 0 ) ) ; ::_thesis: verum end; definition func Prod_real_n -> Real_Sequence means :Def2: :: SIN_COS:def 2 ( it . 0 = 1 & ( for n being Element of NAT holds it . (n + 1) = (it . n) * (n + 1) ) ); existence ex b1 being Real_Sequence st ( b1 . 0 = 1 & ( for n being Element of NAT holds b1 . (n + 1) = (b1 . n) * (n + 1) ) ) proof deffunc H1( Nat, Element of REAL ) -> Element of REAL = \$2 * (\$1 + 1); consider f being Function of NAT,REAL such that A1: ( f . 0 = 1 & ( for n being Nat holds f . (n + 1) = H1(n,f . n) ) ) from NAT_1:sch_12(); take f ; ::_thesis: ( f . 0 = 1 & ( for n being Element of NAT holds f . (n + 1) = (f . n) * (n + 1) ) ) thus ( f . 0 = 1 & ( for n being Element of NAT holds f . (n + 1) = (f . n) * (n + 1) ) ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Real_Sequence st b1 . 0 = 1 & ( for n being Element of NAT holds b1 . (n + 1) = (b1 . n) * (n + 1) ) & b2 . 0 = 1 & ( for n being Element of NAT holds b2 . (n + 1) = (b2 . n) * (n + 1) ) holds b1 = b2 proof let rseq1, rseq2 be Real_Sequence; ::_thesis: ( rseq1 . 0 = 1 & ( for n being Element of NAT holds rseq1 . (n + 1) = (rseq1 . n) * (n + 1) ) & rseq2 . 0 = 1 & ( for n being Element of NAT holds rseq2 . (n + 1) = (rseq2 . n) * (n + 1) ) implies rseq1 = rseq2 ) assume that A2: rseq1 . 0 = 1 and A3: for n being Element of NAT holds rseq1 . (n + 1) = (rseq1 . n) * (n + 1) and A4: rseq2 . 0 = 1 and A5: for n being Element of NAT holds rseq2 . (n + 1) = (rseq2 . n) * (n + 1) ; ::_thesis: rseq1 = rseq2 defpred S1[ Element of NAT ] means rseq1 . \$1 = rseq2 . \$1; A6: S1[ 0 ] by A2, A4; A7: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume rseq1 . k = rseq2 . k ; ::_thesis: S1[k + 1] hence rseq1 . (k + 1) = (rseq2 . k) * (k + 1) by A3 .= rseq2 . (k + 1) by A5 ; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A6, A7); hence rseq1 = rseq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def2 defines Prod_real_n SIN_COS:def_2_:_ for b1 being Real_Sequence holds ( b1 = Prod_real_n iff ( b1 . 0 = 1 & ( for n being Element of NAT holds b1 . (n + 1) = (b1 . n) * (n + 1) ) ) ); definition let n be Nat; redefine func n ! equals :: SIN_COS:def 3 Prod_real_n . n; compatibility for b1 being Element of REAL holds ( b1 = n ! iff b1 = Prod_real_n . n ) proof defpred S1[ Nat] means Prod_real_n . \$1 = \$1 ! ; A1: S1[ 0 ] by Def2, NEWTON:12; A2: now__::_thesis:_for_l_being_Nat_st_S1[l]_holds_ S1[l_+_1] let l be Nat; ::_thesis: ( S1[l] implies S1[l + 1] ) assume A3: S1[l] ; ::_thesis: S1[l + 1] l in NAT by ORDINAL1:def_12; then Prod_real_n . (l + 1) = (Prod_real_n . l) * (l + 1) by Def2 .= (l + 1) ! by A3, NEWTON:15 ; hence S1[l + 1] ; ::_thesis: verum end; for k being Nat holds S1[k] from NAT_1:sch_2(A1, A2); hence for b1 being Element of REAL holds ( b1 = n ! iff b1 = Prod_real_n . n ) ; ::_thesis: verum end; end; :: deftheorem defines ! SIN_COS:def_3_:_ for n being Nat holds n ! = Prod_real_n . n; definition let z be complex number ; deffunc H1( Element of NAT ) -> Element of COMPLEX = (z #N \$1) / (\$1 !); funcz ExpSeq -> Complex_Sequence means :Def4: :: SIN_COS:def 4 for n being Element of NAT holds it . n = (z #N n) / (n !); existence ex b1 being Complex_Sequence st for n being Element of NAT holds b1 . n = (z #N n) / (n !) proof thus ex s being Complex_Sequence st for n being Element of NAT holds s . n = H1(n) from COMSEQ_1:sch_1(); ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st ( for n being Element of NAT holds b1 . n = (z #N n) / (n !) ) & ( for n being Element of NAT holds b2 . n = (z #N n) / (n !) ) holds b1 = b2 proof thus for s1, s2 being Complex_Sequence st ( for x being Element of NAT holds s1 . x = H1(x) ) & ( for x being Element of NAT holds s2 . x = H1(x) ) holds s1 = s2 from BINOP_2:sch_1(); ::_thesis: verum end; end; :: deftheorem Def4 defines ExpSeq SIN_COS:def_4_:_ for z being complex number for b2 being Complex_Sequence holds ( b2 = z ExpSeq iff for n being Element of NAT holds b2 . n = (z #N n) / (n !) ); definition let a be real number ; deffunc H1( Element of NAT ) -> Element of REAL = (a |^ \$1) / (\$1 !); funca rExpSeq -> Real_Sequence means :Def5: :: SIN_COS:def 5 for n being Element of NAT holds it . n = (a |^ n) / (n !); existence ex b1 being Real_Sequence st for n being Element of NAT holds b1 . n = (a |^ n) / (n !) proof thus ex s being Real_Sequence st for n being Element of NAT holds s . n = H1(n) from SEQ_1:sch_1(); ::_thesis: verum end; uniqueness for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = (a |^ n) / (n !) ) & ( for n being Element of NAT holds b2 . n = (a |^ n) / (n !) ) holds b1 = b2 proof thus for s1, s2 being Real_Sequence st ( for x being Element of NAT holds s1 . x = H1(x) ) & ( for x being Element of NAT holds s2 . x = H1(x) ) holds s1 = s2 from BINOP_2:sch_1(); ::_thesis: verum end; end; :: deftheorem Def5 defines rExpSeq SIN_COS:def_5_:_ for a being real number for b2 being Real_Sequence holds ( b2 = a rExpSeq iff for n being Element of NAT holds b2 . n = (a |^ n) / (n !) ); theorem Th1: :: SIN_COS:1 for n being Element of NAT holds ( 0 ! = 1 & n ! <> 0 & (n + 1) ! = (n !) * (n + 1) ) by Def2; theorem Th2: :: SIN_COS:2 for k, m being Element of NAT holds ( ( 0 < k implies ((k -' 1) !) * k = k ! ) & ( k <= m implies ((m -' k) !) * ((m + 1) - k) = ((m + 1) -' k) ! ) ) proof let k, m be Element of NAT ; ::_thesis: ( ( 0 < k implies ((k -' 1) !) * k = k ! ) & ( k <= m implies ((m -' k) !) * ((m + 1) - k) = ((m + 1) -' k) ! ) ) A1: now__::_thesis:_for_k_being_Element_of_NAT_st_0_<_k_holds_ k_!_=_((k_-'_1)_!)_*_k let k be Element of NAT ; ::_thesis: ( 0 < k implies k ! = ((k -' 1) !) * k ) assume 0 < k ; ::_thesis: k ! = ((k -' 1) !) * k then 0 + 1 <= k by INT_1:7; then (k -' 1) + 1 = (k - 1) + 1 by XREAL_1:233 .= k ; hence k ! = ((k -' 1) !) * k by Th1; ::_thesis: verum end; now__::_thesis:_for_m,_k_being_Element_of_NAT_st_k_<=_m_holds_ ((m_+_1)_-'_k)_!_=_((m_-'_k)_!)_*_((m_+_1)_-_k) let m, k be Element of NAT ; ::_thesis: ( k <= m implies ((m + 1) -' k) ! = ((m -' k) !) * ((m + 1) - k) ) assume A2: k <= m ; ::_thesis: ((m + 1) -' k) ! = ((m -' k) !) * ((m + 1) - k) m <= m + 1 by XREAL_1:29; then (m + 1) -' k = (m + 1) - k by A2, XREAL_1:233, XXREAL_0:2 .= (m - k) + 1 .= (m -' k) + 1 by A2, XREAL_1:233 ; hence ((m + 1) -' k) ! = ((m -' k) !) * (((m -' k) + 1) + (0 * )) by Th1 .= ((m -' k) !) * (((m - k) + 1) + (0 * )) by A2, XREAL_1:233 .= ((m -' k) !) * ((m + 1) - k) ; ::_thesis: verum end; hence ( ( 0 < k implies ((k -' 1) !) * k = k ! ) & ( k <= m implies ((m -' k) !) * ((m + 1) - k) = ((m + 1) -' k) ! ) ) by A1; ::_thesis: verum end; definition let n be Element of NAT ; func Coef n -> Complex_Sequence means :Def6: :: SIN_COS:def 6 for k being Element of NAT holds ( ( k <= n implies it . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies it . k = 0 ) ); existence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies b1 . k = 0 ) ) proof deffunc H1( Element of NAT , Element of NAT ) -> Element of REAL = (\$1 !) / ((\$2 !) * ((\$1 -' \$2) !)); for n being Element of NAT ex seq being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies seq . k = H1(n,k) ) & ( k > n implies seq . k = 0 ) ) from SIN_COS:sch_1(); hence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies b1 . k = 0 ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st ( for k being Element of NAT holds ( ( k <= n implies b1 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies b1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies b2 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies b2 . k = 0 ) ) ) holds b1 = b2 proof let seq1, seq2 be Complex_Sequence; ::_thesis: ( ( for k being Element of NAT holds ( ( k <= n implies seq1 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies seq1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies seq2 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies seq2 . k = 0 ) ) ) implies seq1 = seq2 ) assume that A1: for k being Element of NAT holds ( ( k <= n implies seq1 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies seq1 . k = 0 ) ) and A2: for k being Element of NAT holds ( ( k <= n implies seq2 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies seq2 . k = 0 ) ) ; ::_thesis: seq1 = seq2 now__::_thesis:_for_k_being_Element_of_NAT_holds_seq1_._k_=_seq2_._k let k be Element of NAT ; ::_thesis: seq1 . b1 = seq2 . b1 percases ( k <= n or k > n ) ; supposeA3: k <= n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = (n !) / ((k !) * ((n -' k) !)) by A1 .= seq2 . k by A2, A3 ; ::_thesis: verum end; supposeA4: k > n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = 0c by A1 .= seq2 . k by A2, A4 ; ::_thesis: verum end; end; end; hence seq1 = seq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def6 defines Coef SIN_COS:def_6_:_ for n being Element of NAT for b2 being Complex_Sequence holds ( b2 = Coef n iff for k being Element of NAT holds ( ( k <= n implies b2 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies b2 . k = 0 ) ) ); definition let n be Element of NAT ; func Coef_e n -> Complex_Sequence means :Def7: :: SIN_COS:def 7 for k being Element of NAT holds ( ( k <= n implies it . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies it . k = 0 ) ); existence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies b1 . k = 0 ) ) proof deffunc H1( Element of NAT , Element of NAT ) -> Element of COMPLEX = 1r / ((\$2 !) * ((\$1 -' \$2) !)); for n being Element of NAT ex seq being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies seq . k = H1(n,k) ) & ( k > n implies seq . k = 0 ) ) from SIN_COS:sch_1(); hence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies b1 . k = 0 ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st ( for k being Element of NAT holds ( ( k <= n implies b1 . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies b1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies b2 . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies b2 . k = 0 ) ) ) holds b1 = b2 proof let seq1, seq2 be Complex_Sequence; ::_thesis: ( ( for k being Element of NAT holds ( ( k <= n implies seq1 . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies seq1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies seq2 . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies seq2 . k = 0 ) ) ) implies seq1 = seq2 ) assume that A1: for k being Element of NAT holds ( ( k <= n implies seq1 . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies seq1 . k = 0 ) ) and A2: for k being Element of NAT holds ( ( k <= n implies seq2 . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies seq2 . k = 0 ) ) ; ::_thesis: seq1 = seq2 now__::_thesis:_for_k_being_Element_of_NAT_holds_seq1_._k_=_seq2_._k let k be Element of NAT ; ::_thesis: seq1 . b1 = seq2 . b1 percases ( k <= n or k > n ) ; supposeA3: k <= n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = 1r / ((k !) * ((n -' k) !)) by A1 .= seq2 . k by A2, A3 ; ::_thesis: verum end; supposeA4: k > n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = 0c by A1 .= seq2 . k by A2, A4 ; ::_thesis: verum end; end; end; hence seq1 = seq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def7 defines Coef_e SIN_COS:def_7_:_ for n being Element of NAT for b2 being Complex_Sequence holds ( b2 = Coef_e n iff for k being Element of NAT holds ( ( k <= n implies b2 . k = 1r / ((k !) * ((n -' k) !)) ) & ( k > n implies b2 . k = 0 ) ) ); definition let seq be Complex_Sequence; func Shift seq -> Complex_Sequence means :Def8: :: SIN_COS:def 8 ( it . 0 = 0 & ( for k being Element of NAT holds it . (k + 1) = seq . k ) ); existence ex b1 being Complex_Sequence st ( b1 . 0 = 0 & ( for k being Element of NAT holds b1 . (k + 1) = seq . k ) ) proof deffunc H1( Nat, Element of COMPLEX ) -> Element of COMPLEX = seq . \$1; consider f being Function of NAT,COMPLEX such that A1: ( f . 0 = 0c & ( for n being Nat holds f . (n + 1) = H1(n,f . n) ) ) from NAT_1:sch_12(); take f ; ::_thesis: ( f . 0 = 0 & ( for k being Element of NAT holds f . (k + 1) = seq . k ) ) thus ( f . 0 = 0 & ( for k being Element of NAT holds f . (k + 1) = seq . k ) ) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st b1 . 0 = 0 & ( for k being Element of NAT holds b1 . (k + 1) = seq . k ) & b2 . 0 = 0 & ( for k being Element of NAT holds b2 . (k + 1) = seq . k ) holds b1 = b2 proof let seq1, seq2 be Complex_Sequence; ::_thesis: ( seq1 . 0 = 0 & ( for k being Element of NAT holds seq1 . (k + 1) = seq . k ) & seq2 . 0 = 0 & ( for k being Element of NAT holds seq2 . (k + 1) = seq . k ) implies seq1 = seq2 ) assume that A2: seq1 . 0 = 0 and A3: for n being Element of NAT holds seq1 . (n + 1) = seq . n and A4: seq2 . 0 = 0 and A5: for n being Element of NAT holds seq2 . (n + 1) = seq . n ; ::_thesis: seq1 = seq2 defpred S1[ Element of NAT ] means seq1 . \$1 = seq2 . \$1; A6: S1[ 0 ] by A2, A4; A7: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume seq1 . k = seq2 . k ; ::_thesis: S1[k + 1] thus seq1 . (k + 1) = seq . k by A3 .= seq2 . (k + 1) by A5 ; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A6, A7); hence seq1 = seq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def8 defines Shift SIN_COS:def_8_:_ for seq, b2 being Complex_Sequence holds ( b2 = Shift seq iff ( b2 . 0 = 0 & ( for k being Element of NAT holds b2 . (k + 1) = seq . k ) ) ); definition let n be Element of NAT ; let z, w be Element of COMPLEX ; func Expan (n,z,w) -> Complex_Sequence means :Def9: :: SIN_COS:def 9 for k being Element of NAT holds ( ( k <= n implies it . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies it . k = 0 ) ); existence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b1 . k = 0 ) ) proof deffunc H1( Element of NAT , Element of NAT ) -> Element of COMPLEX = (((Coef \$1) . \$2) * (z #N \$2)) * (w #N (\$1 -' \$2)); for n being Element of NAT ex seq being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies seq . k = H1(n,k) ) & ( k > n implies seq . k = 0 ) ) from SIN_COS:sch_1(); hence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b1 . k = 0 ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st ( for k being Element of NAT holds ( ( k <= n implies b1 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies b2 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b2 . k = 0 ) ) ) holds b1 = b2 proof let seq1, seq2 be Complex_Sequence; ::_thesis: ( ( for k being Element of NAT holds ( ( k <= n implies seq1 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies seq1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies seq2 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies seq2 . k = 0 ) ) ) implies seq1 = seq2 ) assume that A1: for k being Element of NAT holds ( ( k <= n implies seq1 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( k > n implies seq1 . k = 0 ) ) and A2: for k being Element of NAT holds ( ( k <= n implies seq2 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( k > n implies seq2 . k = 0 ) ) ; ::_thesis: seq1 = seq2 now__::_thesis:_for_k_being_Element_of_NAT_holds_seq1_._k_=_seq2_._k let k be Element of NAT ; ::_thesis: seq1 . b1 = seq2 . b1 percases ( k <= n or k > n ) ; supposeA3: k <= n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) by A1 .= seq2 . k by A2, A3 ; ::_thesis: verum end; supposeA4: k > n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = 0c by A1 .= seq2 . k by A2, A4 ; ::_thesis: verum end; end; end; hence seq1 = seq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def9 defines Expan SIN_COS:def_9_:_ for n being Element of NAT for z, w being Element of COMPLEX for b4 being Complex_Sequence holds ( b4 = Expan (n,z,w) iff for k being Element of NAT holds ( ( k <= n implies b4 . k = (((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b4 . k = 0 ) ) ); definition let n be Element of NAT ; let z, w be Element of COMPLEX ; func Expan_e (n,z,w) -> Complex_Sequence means :Def10: :: SIN_COS:def 10 for k being Element of NAT holds ( ( k <= n implies it . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies it . k = 0 ) ); existence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b1 . k = 0 ) ) proof deffunc H1( Element of NAT , Element of NAT ) -> Element of COMPLEX = (((Coef_e \$1) . \$2) * (z #N \$2)) * (w #N (\$1 -' \$2)); for n being Element of NAT ex seq being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies seq . k = H1(n,k) ) & ( k > n implies seq . k = 0 ) ) from SIN_COS:sch_1(); hence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b1 . k = 0 ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st ( for k being Element of NAT holds ( ( k <= n implies b1 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies b2 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b2 . k = 0 ) ) ) holds b1 = b2 proof let seq1, seq2 be Complex_Sequence; ::_thesis: ( ( for k being Element of NAT holds ( ( k <= n implies seq1 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies seq1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies seq2 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies seq2 . k = 0 ) ) ) implies seq1 = seq2 ) assume that A1: for k being Element of NAT holds ( ( k <= n implies seq1 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( k > n implies seq1 . k = 0 ) ) and A2: for k being Element of NAT holds ( ( k <= n implies seq2 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( k > n implies seq2 . k = 0 ) ) ; ::_thesis: seq1 = seq2 now__::_thesis:_for_k_being_Element_of_NAT_holds_seq1_._k_=_seq2_._k let k be Element of NAT ; ::_thesis: seq1 . b1 = seq2 . b1 percases ( k <= n or k > n ) ; supposeA3: k <= n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) by A1 .= seq2 . k by A2, A3 ; ::_thesis: verum end; supposeA4: k > n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = 0c by A1 .= seq2 . k by A2, A4 ; ::_thesis: verum end; end; end; hence seq1 = seq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def10 defines Expan_e SIN_COS:def_10_:_ for n being Element of NAT for z, w being Element of COMPLEX for b4 being Complex_Sequence holds ( b4 = Expan_e (n,z,w) iff for k being Element of NAT holds ( ( k <= n implies b4 . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) ) & ( n < k implies b4 . k = 0 ) ) ); definition let n be Element of NAT ; let z, w be Element of COMPLEX ; func Alfa (n,z,w) -> Complex_Sequence means :Def11: :: SIN_COS:def 11 for k being Element of NAT holds ( ( k <= n implies it . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies it . k = 0 ) ); existence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies b1 . k = 0 ) ) proof deffunc H1( Element of NAT , Element of NAT ) -> Element of COMPLEX = ((z ExpSeq) . \$2) * ((Partial_Sums (w ExpSeq)) . (\$1 -' \$2)); for n being Element of NAT ex seq being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies seq . k = H1(n,k) ) & ( k > n implies seq . k = 0 ) ) from SIN_COS:sch_1(); hence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies b1 . k = 0 ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st ( for k being Element of NAT holds ( ( k <= n implies b1 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies b1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies b2 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies b2 . k = 0 ) ) ) holds b1 = b2 proof let seq1, seq2 be Complex_Sequence; ::_thesis: ( ( for k being Element of NAT holds ( ( k <= n implies seq1 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies seq1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies seq2 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies seq2 . k = 0 ) ) ) implies seq1 = seq2 ) assume that A1: for k being Element of NAT holds ( ( k <= n implies seq1 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( k > n implies seq1 . k = 0 ) ) and A2: for k being Element of NAT holds ( ( k <= n implies seq2 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( k > n implies seq2 . k = 0 ) ) ; ::_thesis: seq1 = seq2 now__::_thesis:_for_k_being_Element_of_NAT_holds_seq1_._k_=_seq2_._k let k be Element of NAT ; ::_thesis: seq1 . b1 = seq2 . b1 percases ( k <= n or k > n ) ; supposeA3: k <= n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) by A1 .= seq2 . k by A2, A3 ; ::_thesis: verum end; supposeA4: k > n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = 0c by A1 .= seq2 . k by A2, A4 ; ::_thesis: verum end; end; end; hence seq1 = seq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def11 defines Alfa SIN_COS:def_11_:_ for n being Element of NAT for z, w being Element of COMPLEX for b4 being Complex_Sequence holds ( b4 = Alfa (n,z,w) iff for k being Element of NAT holds ( ( k <= n implies b4 . k = ((z ExpSeq) . k) * ((Partial_Sums (w ExpSeq)) . (n -' k)) ) & ( n < k implies b4 . k = 0 ) ) ); definition let a, b be real number ; let n be Element of NAT ; func Conj (n,a,b) -> Real_Sequence means :: SIN_COS:def 12 for k being Element of NAT holds ( ( k <= n implies it . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( n < k implies it . k = 0 ) ); existence ex b1 being Real_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( n < k implies b1 . k = 0 ) ) proof deffunc H1( Element of NAT , Element of NAT ) -> Element of REAL = ((a rExpSeq) . \$2) * (((Partial_Sums (b rExpSeq)) . \$1) - ((Partial_Sums (b rExpSeq)) . (\$1 -' \$2))); for n being Element of NAT ex rseq being Real_Sequence st for k being Element of NAT holds ( ( k <= n implies rseq . k = H1(n,k) ) & ( k > n implies rseq . k = 0 ) ) from SIN_COS:sch_2(); hence ex b1 being Real_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( n < k implies b1 . k = 0 ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Real_Sequence st ( for k being Element of NAT holds ( ( k <= n implies b1 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( n < k implies b1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies b2 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( n < k implies b2 . k = 0 ) ) ) holds b1 = b2 proof let rseq1, rseq2 be Real_Sequence; ::_thesis: ( ( for k being Element of NAT holds ( ( k <= n implies rseq1 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( n < k implies rseq1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies rseq2 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( n < k implies rseq2 . k = 0 ) ) ) implies rseq1 = rseq2 ) assume that A1: for k being Element of NAT holds ( ( k <= n implies rseq1 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( k > n implies rseq1 . k = 0 ) ) and A2: for k being Element of NAT holds ( ( k <= n implies rseq2 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( k > n implies rseq2 . k = 0 ) ) ; ::_thesis: rseq1 = rseq2 now__::_thesis:_for_k_being_Element_of_NAT_holds_rseq1_._k_=_rseq2_._k let k be Element of NAT ; ::_thesis: rseq1 . b1 = rseq2 . b1 percases ( k <= n or k > n ) ; supposeA3: k <= n ; ::_thesis: rseq1 . b1 = rseq2 . b1 hence rseq1 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) by A1 .= rseq2 . k by A2, A3 ; ::_thesis: verum end; supposeA4: k > n ; ::_thesis: rseq1 . b1 = rseq2 . b1 hence rseq1 . k = 0 by A1 .= rseq2 . k by A2, A4 ; ::_thesis: verum end; end; end; hence rseq1 = rseq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem defines Conj SIN_COS:def_12_:_ for a, b being real number for n being Element of NAT for b4 being Real_Sequence holds ( b4 = Conj (n,a,b) iff for k being Element of NAT holds ( ( k <= n implies b4 . k = ((a rExpSeq) . k) * (((Partial_Sums (b rExpSeq)) . n) - ((Partial_Sums (b rExpSeq)) . (n -' k))) ) & ( n < k implies b4 . k = 0 ) ) ); definition let z, w be Element of COMPLEX ; let n be Element of NAT ; func Conj (n,z,w) -> Complex_Sequence means :Def13: :: SIN_COS:def 13 for k being Element of NAT holds ( ( k <= n implies it . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies it . k = 0 ) ); existence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies b1 . k = 0 ) ) proof deffunc H1( Element of NAT , Element of NAT ) -> Element of COMPLEX = ((z ExpSeq) . \$2) * (((Partial_Sums (w ExpSeq)) . \$1) - ((Partial_Sums (w ExpSeq)) . (\$1 -' \$2))); for n being Element of NAT ex seq being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies seq . k = H1(n,k) ) & ( k > n implies seq . k = 0 ) ) from SIN_COS:sch_1(); hence ex b1 being Complex_Sequence st for k being Element of NAT holds ( ( k <= n implies b1 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies b1 . k = 0 ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st ( for k being Element of NAT holds ( ( k <= n implies b1 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies b1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies b2 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies b2 . k = 0 ) ) ) holds b1 = b2 proof let seq1, seq2 be Complex_Sequence; ::_thesis: ( ( for k being Element of NAT holds ( ( k <= n implies seq1 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies seq1 . k = 0 ) ) ) & ( for k being Element of NAT holds ( ( k <= n implies seq2 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies seq2 . k = 0 ) ) ) implies seq1 = seq2 ) assume that A1: for k being Element of NAT holds ( ( k <= n implies seq1 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( k > n implies seq1 . k = 0 ) ) and A2: for k being Element of NAT holds ( ( k <= n implies seq2 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( k > n implies seq2 . k = 0 ) ) ; ::_thesis: seq1 = seq2 now__::_thesis:_for_k_being_Element_of_NAT_holds_seq1_._k_=_seq2_._k let k be Element of NAT ; ::_thesis: seq1 . b1 = seq2 . b1 percases ( k <= n or k > n ) ; supposeA3: k <= n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) by A1 .= seq2 . k by A2, A3 ; ::_thesis: verum end; supposeA4: k > n ; ::_thesis: seq1 . b1 = seq2 . b1 hence seq1 . k = 0c by A1 .= seq2 . k by A2, A4 ; ::_thesis: verum end; end; end; hence seq1 = seq2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def13 defines Conj SIN_COS:def_13_:_ for z, w being Element of COMPLEX for n being Element of NAT for b4 being Complex_Sequence holds ( b4 = Conj (n,z,w) iff for k being Element of NAT holds ( ( k <= n implies b4 . k = ((z ExpSeq) . k) * (((Partial_Sums (w ExpSeq)) . n) - ((Partial_Sums (w ExpSeq)) . (n -' k))) ) & ( n < k implies b4 . k = 0 ) ) ); Lm1: for p1, p2, g1, g2 being Element of REAL holds ( (p1 + (g1 * )) * (p2 + (g2 * )) = ((p1 * p2) - (g1 * g2)) + (((p1 * g2) + (p2 * g1)) * ) & (p2 + (g2 * )) *' = p2 + ((- g2) * ) ) proof let p1, p2, g1, g2 be Element of REAL ; ::_thesis: ( (p1 + (g1 * )) * (p2 + (g2 * )) = ((p1 * p2) - (g1 * g2)) + (((p1 * g2) + (p2 * g1)) * ) & (p2 + (g2 * )) *' = p2 + ((- g2) * ) ) thus (p1 + (g1 * )) * (p2 + (g2 * )) = ((p1 * p2) - (g1 * g2)) + (((p1 * g2) + (p2 * g1)) * ) ; ::_thesis: (p2 + (g2 * )) *' = p2 + ((- g2) * ) thus (p2 + (g2 * )) *' = p2 - ((Im (p2 + (g2 * ))) * ) by COMPLEX1:12 .= p2 - (g2 * ) by COMPLEX1:12 .= p2 + ((- g2) * ) ; ::_thesis: verum end; theorem Th3: :: SIN_COS:3 for n being Element of NAT for z being complex number holds ( (z ExpSeq) . (n + 1) = (((z ExpSeq) . n) * z) / ((n + 1) + (0 * )) & (z ExpSeq) . 0 = 1 & |.((z ExpSeq) . n).| = (|.z.| rExpSeq) . n ) proof let n be Element of NAT ; ::_thesis: for z being complex number holds ( (z ExpSeq) . (n + 1) = (((z ExpSeq) . n) * z) / ((n + 1) + (0 * )) & (z ExpSeq) . 0 = 1 & |.((z ExpSeq) . n).| = (|.z.| rExpSeq) . n ) let z be complex number ; ::_thesis: ( (z ExpSeq) . (n + 1) = (((z ExpSeq) . n) * z) / ((n + 1) + (0 * )) & (z ExpSeq) . 0 = 1 & |.((z ExpSeq) . n).| = (|.z.| rExpSeq) . n ) A1: now__::_thesis:_for_n_being_Element_of_NAT_holds_(z_ExpSeq)_._(n_+_1)_=_(((z_ExpSeq)_._n)_*_z)_/_((n_+_1)_+_(0_*_)) let n be Element of NAT ; ::_thesis: (z ExpSeq) . (n + 1) = (((z ExpSeq) . n) * z) / ((n + 1) + (0 * )) thus (z ExpSeq) . (n + 1) = (z |^ (n + 1)) / ((n + 1) !) by Def4 .= (((z GeoSeq) . n) * z) / ((n + 1) !) by COMSEQ_3:def_1 .= ((z |^ n) * z) / ((n !) * ((n + 1) + (0 * ))) by Th1 .= ((z |^ n) / (n !)) * (z / ((n + 1) + (0 * ))) by XCMPLX_1:76 .= (((z |^ n) / (n !)) * z) / ((n + 1) + (0 * )) .= (((z ExpSeq) . n) * z) / ((n + 1) + (0 * )) by Def4 ; ::_thesis: verum end; A2: (z ExpSeq) . 0 = (z |^ 0) / (0 !) by Def4 .= 1 by Th1, COMSEQ_3:def_1 ; defpred S1[ Element of NAT ] means |.((z ExpSeq) . \$1).| = (|.z.| rExpSeq) . \$1; (|.z.| rExpSeq) . 0 = (|.z.| |^ 0) / (0 !) by Def5 .= 1 / (Prod_real_n . 0) by NEWTON:4 .= 1 / 1 by Def2 .= 1 ; then A3: S1[ 0 ] by A2, COMPLEX1:48; A4: now__::_thesis:_for_n_being_Element_of_NAT_st_S1[n]_holds_ S1[n_+_1] let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A5: S1[n] ; ::_thesis: S1[n + 1] A6: |.((n + 1) + (0 * )).| = n + 1 by ABSVALUE:def_1; |.((z ExpSeq) . (n + 1)).| = |.((((z ExpSeq) . n) * z) / ((n + 1) + (0 * ))).| by A1 .= |.(((z ExpSeq) . n) * z).| / |.((n + 1) + (0 * )).| by COMPLEX1:67 .= (((|.z.| rExpSeq) . n) * |.z.|) / |.((n + 1) + (0 * )).| by A5, COMPLEX1:65 .= (((|.z.| |^ n) / (n !)) * |.z.|) / |.((n + 1) + (0 * )).| by Def5 .= ((|.z.| |^ n) * |.z.|) / ((n !) * |.((n + 1) + (0 * )).|) by XCMPLX_1:83 .= ((|.z.| |^ n) * |.z.|) / ((n + 1) !) by A6, NEWTON:15 .= (|.z.| |^ (n + 1)) / ((n + 1) !) by NEWTON:6 .= (|.z.| rExpSeq) . (n + 1) by Def5 ; hence S1[n + 1] ; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A3, A4); hence ( (z ExpSeq) . (n + 1) = (((z ExpSeq) . n) * z) / ((n + 1) + (0 * )) & (z ExpSeq) . 0 = 1 & |.((z ExpSeq) . n).| = (|.z.| rExpSeq) . n ) by A1, A2; ::_thesis: verum end; theorem Th4: :: SIN_COS:4 for k being Element of NAT for seq being Complex_Sequence st 0 < k holds (Shift seq) . k = seq . (k -' 1) proof let k be Element of NAT ; ::_thesis: for seq being Complex_Sequence st 0 < k holds (Shift seq) . k = seq . (k -' 1) let seq be Complex_Sequence; ::_thesis: ( 0 < k implies (Shift seq) . k = seq . (k -' 1) ) assume A1: 0 < k ; ::_thesis: (Shift seq) . k = seq . (k -' 1) then A2: 0 + 1 <= k by INT_1:7; consider m being Nat such that A3: m + 1 = k by A1, NAT_1:6; A4: m = k - 1 by A3; m in NAT by ORDINAL1:def_12; hence (Shift seq) . k = seq . m by A3, Def8 .= seq . (k -' 1) by A2, A4, XREAL_1:233 ; ::_thesis: verum end; theorem Th5: :: SIN_COS:5 for k being Element of NAT for seq being Complex_Sequence holds (Partial_Sums seq) . k = ((Partial_Sums (Shift seq)) . k) + (seq . k) proof let k be Element of NAT ; ::_thesis: for seq being Complex_Sequence holds (Partial_Sums seq) . k = ((Partial_Sums (Shift seq)) . k) + (seq . k) let seq be Complex_Sequence; ::_thesis: (Partial_Sums seq) . k = ((Partial_Sums (Shift seq)) . k) + (seq . k) defpred S1[ Element of NAT ] means (Partial_Sums seq) . \$1 = ((Partial_Sums (Shift seq)) . \$1) + (seq . \$1); (Partial_Sums seq) . 0 = 0c + (seq . 0) by SERIES_1:def_1 .= ((Shift seq) . 0) + (seq . 0) by Def8 .= ((Partial_Sums (Shift seq)) . 0) + (seq . 0) by SERIES_1:def_1 ; then A1: S1[ 0 ] ; A2: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: (Partial_Sums seq) . k = ((Partial_Sums (Shift seq)) . k) + (seq . k) ; ::_thesis: S1[k + 1] thus (Partial_Sums seq) . (k + 1) = (((Partial_Sums (Shift seq)) . k) + (seq . k)) + (seq . (k + 1)) by A3, SERIES_1:def_1 .= (((Partial_Sums (Shift seq)) . k) + ((Shift seq) . (k + 1))) + (seq . (k + 1)) by Def8 .= ((Partial_Sums (Shift seq)) . (k + 1)) + (seq . (k + 1)) by SERIES_1:def_1 ; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A1, A2); hence (Partial_Sums seq) . k = ((Partial_Sums (Shift seq)) . k) + (seq . k) ; ::_thesis: verum end; theorem Th6: :: SIN_COS:6 for z, w being Element of COMPLEX for n being Element of NAT holds (z + w) |^ n = (Partial_Sums (Expan (n,z,w))) . n proof let z, w be Element of COMPLEX ; ::_thesis: for n being Element of NAT holds (z + w) |^ n = (Partial_Sums (Expan (n,z,w))) . n let n be Element of NAT ; ::_thesis: (z + w) |^ n = (Partial_Sums (Expan (n,z,w))) . n A1: 0 -' 0 = 0 by XREAL_1:232; defpred S1[ Element of NAT ] means (z + w) |^ \$1 = (Partial_Sums (Expan (\$1,z,w))) . \$1; (Partial_Sums (Expan (0,z,w))) . 0 = (Expan (0,z,w)) . 0 by SERIES_1:def_1 .= (((Coef 0) . 0) * (z |^ 0)) * (w |^ 0) by A1, Def9 .= ((1 / (1 * 1)) * (z |^ 0)) * (w |^ 0) by A1, Def6, Th1 .= 1r * ((w GeoSeq) . 0) by COMSEQ_3:def_1 .= 1r by COMSEQ_3:def_1 ; then A2: S1[ 0 ] by COMSEQ_3:def_1; A3: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A4: (z + w) |^ n = (Partial_Sums (Expan (n,z,w))) . n ; ::_thesis: S1[n + 1] A5: (z + w) |^ (n + 1) = (((z + w) GeoSeq) . n) * (z + w) by COMSEQ_3:def_1 .= ((z + w) (#) (Partial_Sums (Expan (n,z,w)))) . n by A4, VALUED_1:6 .= (Partial_Sums ((z + w) (#) (Expan (n,z,w)))) . n by COMSEQ_3:29 ; now__::_thesis:_for_k_being_Element_of_NAT_holds_((z_+_w)_(#)_(Expan_(n,z,w)))_._k_=_((z_(#)_(Expan_(n,z,w)))_+_(w_(#)_(Expan_(n,z,w))))_._k let k be Element of NAT ; ::_thesis: ((z + w) (#) (Expan (n,z,w))) . k = ((z (#) (Expan (n,z,w))) + (w (#) (Expan (n,z,w)))) . k thus ((z + w) (#) (Expan (n,z,w))) . k = (z + w) * ((Expan (n,z,w)) . k) by VALUED_1:6 .= (z * ((Expan (n,z,w)) . k)) + (w * ((Expan (n,z,w)) . k)) .= ((z (#) (Expan (n,z,w))) . k) + (w * ((Expan (n,z,w)) . k)) by VALUED_1:6 .= ((z (#) (Expan (n,z,w))) . k) + ((w (#) (Expan (n,z,w))) . k) by VALUED_1:6 .= ((z (#) (Expan (n,z,w))) + (w (#) (Expan (n,z,w)))) . k by VALUED_1:1 ; ::_thesis: verum end; then (z + w) (#) (Expan (n,z,w)) = (z (#) (Expan (n,z,w))) + (w (#) (Expan (n,z,w))) by FUNCT_2:63; then A6: (z + w) |^ (n + 1) = ((Partial_Sums (z (#) (Expan (n,z,w)))) + (Partial_Sums (w (#) (Expan (n,z,w))))) . n by A5, COMSEQ_3:27 .= ((Partial_Sums (z (#) (Expan (n,z,w)))) . n) + ((Partial_Sums (w (#) (Expan (n,z,w)))) . n) by VALUED_1:1 ; A7: (Partial_Sums (z (#) (Expan (n,z,w)))) . (n + 1) = ((Partial_Sums (z (#) (Expan (n,z,w)))) . n) + ((z (#) (Expan (n,z,w))) . (n + 1)) by SERIES_1:def_1 .= ((Partial_Sums (z (#) (Expan (n,z,w)))) . n) + (z * ((Expan (n,z,w)) . (n + 1))) by VALUED_1:6 ; n < n + 1 by XREAL_1:29; then A8: (Expan (n,z,w)) . (n + 1) = 0c by Def9; A9: (Partial_Sums (w (#) (Expan (n,z,w)))) . (n + 1) = ((Partial_Sums (w (#) (Expan (n,z,w)))) . n) + ((w (#) (Expan (n,z,w))) . (n + 1)) by SERIES_1:def_1 .= ((Partial_Sums (w (#) (Expan (n,z,w)))) . n) + (w * ((Expan (n,z,w)) . (n + 1))) by VALUED_1:6 ; A10: (Partial_Sums (z (#) (Expan (n,z,w)))) . (n + 1) = ((Partial_Sums (Shift (z (#) (Expan (n,z,w))))) . (n + 1)) + ((z (#) (Expan (n,z,w))) . (n + 1)) by Th5; 0 + n < n + 1 by XREAL_1:29; then (Expan (n,z,w)) . (n + 1) = 0c by Def9; then z * ((Expan (n,z,w)) . (n + 1)) = 0c ; then A11: (Partial_Sums (z (#) (Expan (n,z,w)))) . (n + 1) = ((Partial_Sums (Shift (z (#) (Expan (n,z,w))))) . (n + 1)) + 0c by A10, VALUED_1:6 .= (Partial_Sums (Shift (z (#) (Expan (n,z,w))))) . (n + 1) ; now__::_thesis:_for_k_being_Element_of_NAT_holds_((w_(#)_(Expan_(n,z,w)))_+_(Shift_(z_(#)_(Expan_(n,z,w)))))_._k_=_(Expan_((n_+_1),z,w))_._k let k be Element of NAT ; ::_thesis: ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k A12: now__::_thesis:_(_n_+_1_<_k_implies_((w_(#)_(Expan_(n,z,w)))_+_(Shift_(z_(#)_(Expan_(n,z,w)))))_._k_=_(Expan_((n_+_1),z,w))_._k_) assume A13: n + 1 < k ; ::_thesis: ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k A14: 0 + 1 <= n + 1 by XREAL_1:6; A15: (n + 1) - 1 < k - 1 by A13, XREAL_1:9; then A16: n + 0 < (k - 1) + 1 by XREAL_1:8; A17: k - 1 = k -' 1 by A13, A14, XREAL_1:233, XXREAL_0:2; ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = ((w (#) (Expan (n,z,w))) . k) + ((Shift (z (#) (Expan (n,z,w)))) . k) by VALUED_1:1 .= (w * ((Expan (n,z,w)) . k)) + ((Shift (z (#) (Expan (n,z,w)))) . k) by VALUED_1:6 .= (w * ((Expan (n,z,w)) . k)) + ((z (#) (Expan (n,z,w))) . (k -' 1)) by A16, Th4 .= (w * ((Expan (n,z,w)) . k)) + (z * ((Expan (n,z,w)) . (k -' 1))) by VALUED_1:6 .= (w * 0c) + (z * ((Expan (n,z,w)) . (k -' 1))) by A16, Def9 .= 0c + (z * 0c) by A15, A17, Def9 .= 0c ; hence ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k by A13, Def9; ::_thesis: verum end; now__::_thesis:_(_k_<=_n_+_1_implies_((w_(#)_(Expan_(n,z,w)))_+_(Shift_(z_(#)_(Expan_(n,z,w)))))_._k_=_(Expan_((n_+_1),z,w))_._k_) assume A18: k <= n + 1 ; ::_thesis: ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k A19: now__::_thesis:_(_k_=_n_+_1_implies_((w_(#)_(Expan_(n,z,w)))_+_(Shift_(z_(#)_(Expan_(n,z,w)))))_._k_=_(Expan_((n_+_1),z,w))_._k_) assume A20: k = n + 1 ; ::_thesis: ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k A21: n < n + 1 by XREAL_1:29; A22: n -' n = n - n by XREAL_1:233 .= 0 ; A23: (n + 1) -' (n + 1) = (n + 1) - (n + 1) by XREAL_1:233 .= 0 ; A24: (Coef n) . n = (n !) / ((n !) * (0 !)) by A22, Def6 .= 1 by Th1, XCMPLX_1:60 ; A25: (Coef (n + 1)) . (n + 1) = ((n + 1) !) / (((n + 1) !) * (0 !)) by A23, Def6 .= 1 by Th1, XCMPLX_1:60 ; ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = ((w (#) (Expan (n,z,w))) . (n + 1)) + ((Shift (z (#) (Expan (n,z,w)))) . (n + 1)) by A20, VALUED_1:1 .= (w * ((Expan (n,z,w)) . (n + 1))) + ((Shift (z (#) (Expan (n,z,w)))) . (n + 1)) by VALUED_1:6 .= (w * 0c) + ((Shift (z (#) (Expan (n,z,w)))) . (n + 1)) by A21, Def9 .= (z (#) (Expan (n,z,w))) . n by Def8 .= z * ((Expan (n,z,w)) . n) by VALUED_1:6 .= z * ((((Coef n) . n) * (z |^ n)) * (w |^ (n -' n))) by Def9 .= (((Coef n) . n) * (((z GeoSeq) . n) * z)) * (w |^ (n -' n)) .= (((Coef (n + 1)) . (n + 1)) * (z |^ (n + 1))) * (w |^ (n -' n)) by A24, A25, COMSEQ_3:def_1 .= (Expan ((n + 1),z,w)) . k by A20, A22, A23, Def9 ; hence ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k ; ::_thesis: verum end; now__::_thesis:_(_k_<_n_+_1_implies_((w_(#)_(Expan_(n,z,w)))_+_(Shift_(z_(#)_(Expan_(n,z,w)))))_._k_=_(Expan_((n_+_1),z,w))_._k_) assume A26: k < n + 1 ; ::_thesis: ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k A27: now__::_thesis:_(_k_=_0_implies_((w_(#)_(Expan_(n,z,w)))_+_(Shift_(z_(#)_(Expan_(n,z,w)))))_._k_=_(Expan_((n_+_1),z,w))_._k_) assume A28: k = 0 ; ::_thesis: ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k A29: n -' 0 = n - 0 by XREAL_1:233; A30: (n + 1) -' 0 = (n + 1) - 0 by XREAL_1:233; A31: (Coef n) . 0 = (n !) / ((0 !) * (n !)) by A29, Def6 .= 1 by Th1, XCMPLX_1:60 ; A32: (Coef (n + 1)) . 0 = ((n + 1) !) / ((0 !) * ((n + 1) !)) by A30, Def6 .= 1 by Th1, XCMPLX_1:60 ; ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = ((w (#) (Expan (n,z,w))) . 0) + ((Shift (z (#) (Expan (n,z,w)))) . 0) by A28, VALUED_1:1 .= (w * ((Expan (n,z,w)) . 0)) + ((Shift (z (#) (Expan (n,z,w)))) . 0) by VALUED_1:6 .= (w * ((Expan (n,z,w)) . 0)) + 0c by Def8 .= w * ((((Coef n) . 0) * (z |^ 0)) * (w |^ (n -' 0))) by Def9 .= (((Coef n) . 0) * (z |^ 0)) * (((w GeoSeq) . n) * w) by A29 .= (((Coef (n + 1)) . 0) * (z |^ 0)) * (w |^ ((n + 1) -' 0)) by A30, A31, A32, COMSEQ_3:def_1 .= (Expan ((n + 1),z,w)) . k by A28, Def9 ; hence ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k ; ::_thesis: verum end; now__::_thesis:_(_k_<>_0_implies_((w_(#)_(Expan_(n,z,w)))_+_(Shift_(z_(#)_(Expan_(n,z,w)))))_._k_=_(Expan_((n_+_1),z,w))_._k_) assume A33: k <> 0 ; ::_thesis: ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k then A34: 0 + 1 <= k by INT_1:7; A35: (k + 1) - 1 <= (n + 1) - 1 by A26, INT_1:7; k < k + 1 by XREAL_1:29; then k - 1 <= (k + 1) - 1 by XREAL_1:9; then k - 1 <= n by A35, XXREAL_0:2; then A36: k -' 1 <= n by A34, XREAL_1:233; A37: ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = ((w (#) (Expan (n,z,w))) . k) + ((Shift (z (#) (Expan (n,z,w)))) . k) by VALUED_1:1 .= (w * ((Expan (n,z,w)) . k)) + ((Shift (z (#) (Expan (n,z,w)))) . k) by VALUED_1:6 .= (w * ((Expan (n,z,w)) . k)) + ((z (#) (Expan (n,z,w))) . (k -' 1)) by A33, Th4 .= (w * ((Expan (n,z,w)) . k)) + (z * ((Expan (n,z,w)) . (k -' 1))) by VALUED_1:6 .= (w * ((((Coef n) . k) * (z |^ k)) * (w |^ (n -' k)))) + (z * ((Expan (n,z,w)) . (k -' 1))) by A35, Def9 .= ((w * (((Coef n) . k) * (z |^ k))) * (w |^ (n -' k))) + (z * ((((Coef n) . (k -' 1)) * (z |^ (k -' 1))) * (w |^ (n -' (k -' 1))))) by A36, Def9 .= (((Coef n) . k) * ((w * (z |^ k)) * (w |^ (n -' k)))) + (((Coef n) . (k -' 1)) * (((z |^ (k -' 1)) * z) * (w |^ (n -' (k -' 1))))) ; A38: (n -' k) + 1 = (n - k) + 1 by A35, XREAL_1:233 .= (n + 1) - k .= (n + 1) -' k by A26, XREAL_1:233 ; A39: n -' (k -' 1) = n - (k -' 1) by A36, XREAL_1:233 .= n - (k - 1) by A34, XREAL_1:233 .= (n + 1) - k .= (n + 1) -' k by A26, XREAL_1:233 ; (k -' 1) + 1 = (k - 1) + 1 by A34, XREAL_1:233 .= k ; then A40: ( (w |^ (n -' k)) * w = w |^ ((n -' k) + 1) & (z |^ (k -' 1)) * z = z |^ k ) by COMSEQ_3:def_1; A41: ((Coef n) . k) + ((Coef n) . (k -' 1)) = ((n !) / ((k !) * ((n -' k) !))) + ((Coef n) . (k -' 1)) by A35, Def6 .= ((n !) / ((k !) * ((n -' k) !))) + ((n !) / (((k -' 1) !) * ((n -' (k -' 1)) !))) by A36, Def6 ; A42: ((k !) * ((n -' k) !)) * (((n + 1) - k) + (0 * )) = (k !) * (((n -' k) !) * (((n + 1) - k) + (0 * ))) ; A43: (((k -' 1) !) * ((n -' (k -' 1)) !)) * (k + (0 * )) = ((k + (0 * )) * ((k -' 1) !)) * ((n -' (k -' 1)) !) .= (k !) * (((n + 1) -' k) !) by A33, A39, Th2 ; ((n + 1) - k) + (0 * ) <> 0c by A26; then A44: (n !) / ((k !) * ((n -' k) !)) = ((n !) * (((n + 1) - k) + (0 * ))) / (((k !) * ((n -' k) !)) * (((n + 1) - k) + (0 * ))) by XCMPLX_1:91 .= ((n !) * (((n + 1) - k) + (0 * ))) / ((k !) * (((n + 1) -' k) !)) by A35, A42, Th2 ; (n !) / (((k -' 1) !) * ((n -' (k -' 1)) !)) = ((n !) * (k + (0 * ))) / ((k !) * (((n + 1) -' k) !)) by A33, A43, XCMPLX_1:91; then ((Coef n) . k) + ((Coef n) . (k -' 1)) = ((n !) * ((((n + 1) - k) + k) + ((0 + 0) * ))) / ((k !) * (((n + 1) -' k) !)) by A41, A44 .= ((n + 1) !) / ((k !) * (((n + 1) -' k) !)) by Th1 .= (Coef (n + 1)) . k by A26, Def6 ; then ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (((Coef (n + 1)) . k) * (z |^ k)) * (w |^ ((n + 1) -' k)) by A37, A38, A39, A40 .= (Expan ((n + 1),z,w)) . k by A26, Def9 ; hence ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k ; ::_thesis: verum end; hence ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k by A27; ::_thesis: verum end; hence ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k by A18, A19, XXREAL_0:1; ::_thesis: verum end; hence ((w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w))))) . k = (Expan ((n + 1),z,w)) . k by A12; ::_thesis: verum end; then A45: (w (#) (Expan (n,z,w))) + (Shift (z (#) (Expan (n,z,w)))) = Expan ((n + 1),z,w) by FUNCT_2:63; thus (z + w) |^ (n + 1) = ((Partial_Sums (w (#) (Expan (n,z,w)))) + (Partial_Sums (Shift (z (#) (Expan (n,z,w)))))) . (n + 1) by A6, A7, A8, A9, A11, VALUED_1:1 .= (Partial_Sums (Expan ((n + 1),z,w))) . (n + 1) by A45, COMSEQ_3:27 ; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A2, A3); hence (z + w) |^ n = (Partial_Sums (Expan (n,z,w))) . n ; ::_thesis: verum end; theorem Th7: :: SIN_COS:7 for z, w being Element of COMPLEX for n being Element of NAT holds Expan_e (n,z,w) = (1r / (n !)) (#) (Expan (n,z,w)) proof let z, w be Element of COMPLEX ; ::_thesis: for n being Element of NAT holds Expan_e (n,z,w) = (1r / (n !)) (#) (Expan (n,z,w)) let n be Element of NAT ; ::_thesis: Expan_e (n,z,w) = (1r / (n !)) (#) (Expan (n,z,w)) now__::_thesis:_for_k_being_Element_of_NAT_holds_(Expan_e_(n,z,w))_._k_=_((1r_/_(n_!))_(#)_(Expan_(n,z,w)))_._k let k be Element of NAT ; ::_thesis: (Expan_e (n,z,w)) . k = ((1r / (n !)) (#) (Expan (n,z,w))) . k A1: now__::_thesis:_(_n_<_k_implies_(Expan_e_(n,z,w))_._k_=_((1r_/_(n_!))_(#)_(Expan_(n,z,w)))_._k_) assume A2: n < k ; ::_thesis: (Expan_e (n,z,w)) . k = ((1r / (n !)) (#) (Expan (n,z,w))) . k hence (Expan_e (n,z,w)) . k = (1r / (n !)) * 0c by Def10 .= (1r / (n !)) * ((Expan (n,z,w)) . k) by A2, Def9 .= ((1r / (n !)) (#) (Expan (n,z,w))) . k by VALUED_1:6 ; ::_thesis: verum end; now__::_thesis:_(_k_<=_n_implies_(Expan_e_(n,z,w))_._k_=_((1r_/_(n_!))_(#)_(Expan_(n,z,w)))_._k_) assume A3: k <= n ; ::_thesis: (Expan_e (n,z,w)) . k = ((1r / (n !)) (#) (Expan (n,z,w))) . k then A4: (Expan_e (n,z,w)) . k = (((Coef_e n) . k) * (z |^ k)) * (w |^ (n -' k)) by Def10 .= ((1r / ((k !) * ((n -' k) !))) * (z |^ k)) * (w |^ (n -' k)) by A3, Def7 ; 1r / ((k !) * ((n -' k) !)) = (((n !) * 1r) / (n !)) / ((k !) * ((n -' k) !)) by XCMPLX_1:60 .= ((1r / (n !)) * (n !)) / ((k !) * ((n -' k) !)) ; hence (Expan_e (n,z,w)) . k = (((1r / (n !)) * (n !)) / ((k !) * ((n -' k) !))) * ((z |^ k) * (w |^ (n -' k))) by A4 .= (1r / (n !)) * ((((n !) / ((k !) * ((n -' k) !))) * (z |^ k)) * (w |^ (n -' k))) .= (1r / (n !)) * ((((Coef n) . k) * (z |^ k)) * (w |^ (n -' k))) by A3, Def6 .= (1r / (n !)) * ((Expan (n,z,w)) . k) by A3, Def9 .= ((1r / (n !)) (#) (Expan (n,z,w))) . k by VALUED_1:6 ; ::_thesis: verum end; hence (Expan_e (n,z,w)) . k = ((1r / (n !)) (#) (Expan (n,z,w))) . k by A1; ::_thesis: verum end; hence Expan_e (n,z,w) = (1r / (n !)) (#) (Expan (n,z,w)) by FUNCT_2:63; ::_thesis: verum end; theorem Th8: :: SIN_COS:8 for z, w being Element of COMPLEX for n being Element of NAT holds ((z + w) |^ n) / (n !) = (Partial_Sums (Expan_e (n,z,w))) . n proof let z, w be Element of COMPLEX ; ::_thesis: for n being Element of NAT holds ((z + w) |^ n) / (n !) = (Partial_Sums (Expan_e (n,z,w))) . n let n be Element of NAT ; ::_thesis: ((z + w) |^ n) / (n !) = (Partial_Sums (Expan_e (n,z,w))) . n thus ((z + w) |^ n) / (n !) = ((Partial_Sums (Expan (n,z,w))) . n) * (1r / (n !)) by Th6 .= ((1r / (n !)) (#) (Partial_Sums (Expan (n,z,w)))) . n by VALUED_1:6 .= (Partial_Sums ((1r / (n !)) (#) (Expan (n,z,w)))) . n by COMSEQ_3:29 .= (Partial_Sums (Expan_e (n,z,w))) . n by Th7 ; ::_thesis: verum end; theorem Th9: :: SIN_COS:9 ( 0c ExpSeq is absolutely_summable & Sum (0c ExpSeq) = 1r ) proof defpred S1[ set ] means (Partial_Sums |.(0c ExpSeq).|) . \$1 = 1; (Partial_Sums |.(0c ExpSeq).|) . 0 = |.(0c ExpSeq).| . 0 by SERIES_1:def_1 .= |.((0c ExpSeq) . 0).| by VALUED_1:18 .= 1 by Th3, COMPLEX1:48 ; then A1: S1[ 0 ] ; A2: for n being Nat st S1[n] holds S1[n + 1] proof let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A3: (Partial_Sums |.(0c ExpSeq).|) . n = 1 ; ::_thesis: S1[n + 1] A4: n in NAT by ORDINAL1:def_12; hence (Partial_Sums |.(0c ExpSeq).|) . (n + 1) = 1 + (|.(0c ExpSeq).| . (n + 1)) by A3, SERIES_1:def_1 .= 1 + |.((0c ExpSeq) . (n + 1)).| by VALUED_1:18 .= 1 + |.((((0c ExpSeq) . n) * 0c) / ((n + 1) + (0 * ))).| by A4, Th3 .= 1 by COMPLEX1:44 ; ::_thesis: verum end; for n being Nat holds S1[n] from NAT_1:sch_2(A1, A2); then Partial_Sums |.(0c ExpSeq).| is constant by VALUED_0:def_18; then A5: |.(0c ExpSeq).| is summable by SERIES_1:def_2; defpred S2[ set ] means (Partial_Sums (0c ExpSeq)) . \$1 = 1; (Partial_Sums (0c ExpSeq)) . 0 = (0c ExpSeq) . 0 by SERIES_1:def_1 .= 1 by Th3 ; then A6: S2[ 0 ] ; A7: for n being Element of NAT st S2[n] holds S2[n + 1] proof let n be Element of NAT ; ::_thesis: ( S2[n] implies S2[n + 1] ) assume A8: (Partial_Sums (0c ExpSeq)) . n = 1 ; ::_thesis: S2[n + 1] thus (Partial_Sums (0c ExpSeq)) . (n + 1) = 1r + ((0c ExpSeq) . (n + 1)) by A8, SERIES_1:def_1 .= 1r + ((((0c ExpSeq) . n) * 0c) / ((n + 1) + (0 * ))) by Th3 .= 1 ; ::_thesis: verum end; for n being Element of NAT holds S2[n] from NAT_1:sch_1(A6, A7); hence ( 0c ExpSeq is absolutely_summable & Sum (0c ExpSeq) = 1r ) by A5, COMSEQ_2:10, COMSEQ_3:def_9; ::_thesis: verum end; registration let z be complex number ; clusterz ExpSeq -> absolutely_summable ; coherence z ExpSeq is absolutely_summable proof now__::_thesis:_(_z_<>_0c_implies_(_(_for_n_being_Element_of_NAT_holds_S1[n]_)_&_z_ExpSeq_is_absolutely_summable_)_) assume A1: z <> 0c ; ::_thesis: ( ( for n being Element of NAT holds S1[n] ) & z ExpSeq is absolutely_summable ) defpred S1[ set ] means (z ExpSeq) . z <> 0c ; A2: S1[ 0 ] by Th3; A3: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A4: (z ExpSeq) . n <> 0c ; ::_thesis: S1[n + 1] thus (z ExpSeq) . (n + 1) <> 0c ::_thesis: verum proof assume (z ExpSeq) . (n + 1) = 0c ; ::_thesis: contradiction then A5: (((z ExpSeq) . n) * z) / ((n + 1) + (0 * )) = 0c by Th3; 0c = 0c / z .= (((z ExpSeq) . n) * z) / z by A5, XCMPLX_1:50 .= ((z ExpSeq) . n) * (z / z) .= ((z ExpSeq) . n) * 1 by A1, XCMPLX_1:60 .= (z ExpSeq) . n ; hence contradiction by A4; ::_thesis: verum end; end; deffunc H1( Element of NAT ) -> Element of REAL = (|.(z ExpSeq).| . (z + 1)) / (|.(z ExpSeq).| . z); thus A6: for n being Element of NAT holds S1[n] from NAT_1:sch_1(A2, A3); ::_thesis: z ExpSeq is absolutely_summable ex rseq being Real_Sequence st for n being Element of NAT holds rseq . n = H1(n) from SEQ_1:sch_1(); then consider rseq being Real_Sequence such that A7: for n being Element of NAT holds rseq . n = (|.(z ExpSeq).| . (n + 1)) / (|.(z ExpSeq).| . n) ; now__::_thesis:_for_n_being_Element_of_NAT_holds_rseq_._n_=_|.z.|_/_(n_+_1) let n be Element of NAT ; ::_thesis: rseq . n = |.z.| / (n + 1) thus rseq . n = (|.(z ExpSeq).| . (n + 1)) / (|.(z ExpSeq).| . n) by A7 .= |.((z ExpSeq) . (n + 1)).| / (|.(z ExpSeq).| . n) by VALUED_1:18 .= |.((z ExpSeq) . (n + 1)).| / |.((z ExpSeq) . n).| by VALUED_1:18 .= |.(((z ExpSeq) . (n + 1)) / ((z ExpSeq) . n)).| by COMPLEX1:67 .= |.(((((z ExpSeq) . n) * z) / ((n + 1) + (0 * ))) / ((z ExpSeq) . n)).| by Th3 .= |.((((z ExpSeq) . n) * (z / ((n + 1) + (0 * )))) / ((z ExpSeq) . n)).| .= |.(z / ((n + 1) + (0 * ))).| by A6, XCMPLX_1:89 .= |.z.| / (abs (n + 1)) by COMPLEX1:67 .= |.z.| / (n + 1) by ABSVALUE:def_1 ; ::_thesis: verum end; then ( rseq is convergent & lim rseq < 1 ) by SEQ_4:31; hence z ExpSeq is absolutely_summable by A6, A7, COMSEQ_3:75; ::_thesis: verum end; hence z ExpSeq is absolutely_summable by Th9; ::_thesis: verum end; end; theorem Th10: :: SIN_COS:10 for z, w being Element of COMPLEX holds ( (z ExpSeq) . 0 = 1 & (Expan (0,z,w)) . 0 = 1 ) proof let z, w be Element of COMPLEX ; ::_thesis: ( (z ExpSeq) . 0 = 1 & (Expan (0,z,w)) . 0 = 1 ) thus (z ExpSeq) . 0 = (z |^ 0) / (0 !) by Def4 .= 1 by Th1, COMSEQ_3:def_1 ; ::_thesis: (Expan (0,z,w)) . 0 = 1 A1: 0 -' 0 = 0 by XREAL_1:232; hence (Expan (0,z,w)) . 0 = (((Coef 0) . 0) * (z |^ 0)) * (w |^ 0) by Def9 .= ((1 / (1 * 1)) * (z |^ 0)) * (w |^ 0) by A1, Def6, Th1 .= 1r * ((w GeoSeq) . 0) by COMSEQ_3:def_1 .= 1 by COMSEQ_3:def_1 ; ::_thesis: verum end; theorem Th11: :: SIN_COS:11 for z, w being Element of COMPLEX for l, k being Element of NAT st l <= k holds (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l) proof let z, w be Element of COMPLEX ; ::_thesis: for l, k being Element of NAT st l <= k holds (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l) let l, k be Element of NAT ; ::_thesis: ( l <= k implies (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l) ) assume A1: l <= k ; ::_thesis: (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l) A2: k < k + 1 by XREAL_1:29; then A3: l <= k + 1 by A1, XXREAL_0:2; (k + 1) -' l = (k + 1) - l by A1, A2, XREAL_1:233, XXREAL_0:2; then A4: (k + 1) -' l = (k - l) + 1 .= (k -' l) + 1 by A1, XREAL_1:233 ; then A5: (Alfa ((k + 1),z,w)) . l = ((z ExpSeq) . l) * ((Partial_Sums (w ExpSeq)) . ((k -' l) + 1)) by A3, Def11 .= ((z ExpSeq) . l) * (((Partial_Sums (w ExpSeq)) . (k -' l)) + ((w ExpSeq) . ((k + 1) -' l))) by A4, SERIES_1:def_1 .= (((z ExpSeq) . l) * ((Partial_Sums (w ExpSeq)) . (k -' l))) + (((z ExpSeq) . l) * ((w ExpSeq) . ((k + 1) -' l))) .= ((Alfa (k,z,w)) . l) + (((z ExpSeq) . l) * ((w ExpSeq) . ((k + 1) -' l))) by A1, Def11 ; ((z ExpSeq) . l) * ((w ExpSeq) . ((k + 1) -' l)) = ((z |^ l) / (l !)) * ((w ExpSeq) . ((k + 1) -' l)) by Def4 .= ((z |^ l) / (l !)) * ((w |^ ((k + 1) -' l)) / (((k + 1) -' l) !)) by Def4 .= (((z |^ l) * (w |^ ((k + 1) -' l))) * 1r) / ((l !) * (((k + 1) -' l) !)) by XCMPLX_1:76 .= ((z |^ l) * (w |^ ((k + 1) -' l))) * (1r / ((l !) * (((k + 1) -' l) !))) .= ((Coef_e (k + 1)) . l) * ((z |^ l) * (w |^ ((k + 1) -' l))) by A3, Def7 .= (((Coef_e (k + 1)) . l) * (z |^ l)) * (w |^ ((k + 1) -' l)) .= (Expan_e ((k + 1),z,w)) . l by A3, Def10 ; hence (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l) by A5; ::_thesis: verum end; theorem Th12: :: SIN_COS:12 for z, w being Element of COMPLEX for k being Element of NAT holds (Partial_Sums (Alfa ((k + 1),z,w))) . k = ((Partial_Sums (Alfa (k,z,w))) . k) + ((Partial_Sums (Expan_e ((k + 1),z,w))) . k) proof let z, w be Element of COMPLEX ; ::_thesis: for k being Element of NAT holds (Partial_Sums (Alfa ((k + 1),z,w))) . k = ((Partial_Sums (Alfa (k,z,w))) . k) + ((Partial_Sums (Expan_e ((k + 1),z,w))) . k) let k be Element of NAT ; ::_thesis: (Partial_Sums (Alfa ((k + 1),z,w))) . k = ((Partial_Sums (Alfa (k,z,w))) . k) + ((Partial_Sums (Expan_e ((k + 1),z,w))) . k) now__::_thesis:_for_l_being_Element_of_NAT_st_l_<=_k_holds_ (Alfa_((k_+_1),z,w))_._l_=_((Alfa_(k,z,w))_+_(Expan_e_((k_+_1),z,w)))_._l let l be Element of NAT ; ::_thesis: ( l <= k implies (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) + (Expan_e ((k + 1),z,w))) . l ) assume l <= k ; ::_thesis: (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) + (Expan_e ((k + 1),z,w))) . l hence (Alfa ((k + 1),z,w)) . l = ((Alfa (k,z,w)) . l) + ((Expan_e ((k + 1),z,w)) . l) by Th11 .= ((Alfa (k,z,w)) + (Expan_e ((k + 1),z,w))) . l by VALUED_1:1 ; ::_thesis: verum end; hence (Partial_Sums (Alfa ((k + 1),z,w))) . k = (Partial_Sums ((Alfa (k,z,w)) + (Expan_e ((k + 1),z,w)))) . k by COMSEQ_3:35 .= ((Partial_Sums (Alfa (k,z,w))) + (Partial_Sums (Expan_e ((k + 1),z,w)))) . k by COMSEQ_3:27 .= ((Partial_Sums (Alfa (k,z,w))) . k) + ((Partial_Sums (Expan_e ((k + 1),z,w))) . k) by VALUED_1:1 ; ::_thesis: verum end; theorem Th13: :: SIN_COS:13 for z, w being Element of COMPLEX for k being Element of NAT holds (z ExpSeq) . k = (Expan_e (k,z,w)) . k proof let z, w be Element of COMPLEX ; ::_thesis: for k being Element of NAT holds (z ExpSeq) . k = (Expan_e (k,z,w)) . k let k be Element of NAT ; ::_thesis: (z ExpSeq) . k = (Expan_e (k,z,w)) . k A1: 0 = k - k ; then A2: k -' k = 0 by XREAL_1:233; A3: (k -' k) ! = 1 by A1, Th1, XREAL_1:233; thus (Expan_e (k,z,w)) . k = (((Coef_e k) . k) * (z |^ k)) * (w |^ 0) by A2, Def10 .= (((Coef_e k) . k) * (z |^ k)) * 1r by COMSEQ_3:11 .= (1r / ((k !) * 1r)) * (z |^ k) by A3, Def7 .= ((z |^ k) * 1r) / (k !) .= (z ExpSeq) . k by Def4 ; ::_thesis: verum end; theorem Th14: :: SIN_COS:14 for z, w being Element of COMPLEX for n being Element of NAT holds (Partial_Sums ((z + w) ExpSeq)) . n = (Partial_Sums (Alfa (n,z,w))) . n proof let z, w be Element of COMPLEX ; ::_thesis: for n being Element of NAT holds (Partial_Sums ((z + w) ExpSeq)) . n = (Partial_Sums (Alfa (n,z,w))) . n let n be Element of NAT ; ::_thesis: (Partial_Sums ((z + w) ExpSeq)) . n = (Partial_Sums (Alfa (n,z,w))) . n A1: (Partial_Sums ((z + w) ExpSeq)) . 0 = ((z + w) ExpSeq) . 0 by SERIES_1:def_1 .= 1 by Th10 ; defpred S1[ Element of NAT ] means (Partial_Sums ((z + w) ExpSeq)) . \$1 = (Partial_Sums (Alfa (\$1,z,w))) . \$1; A2: 0 -' 0 = 0 by XREAL_1:232; (Partial_Sums (Alfa (0,z,w))) . 0 = (Alfa (0,z,w)) . 0 by SERIES_1:def_1 .= ((z ExpSeq) . 0) * ((Partial_Sums (w ExpSeq)) . 0) by A2, Def11 .= ((z ExpSeq) . 0) * ((w ExpSeq) . 0) by SERIES_1:def_1 .= 1r * ((w ExpSeq) . 0) by Th10 .= 1 by Th10 ; then A3: S1[ 0 ] by A1; A4: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A5: (Partial_Sums ((z + w) ExpSeq)) . k = (Partial_Sums (Alfa (k,z,w))) . k ; ::_thesis: S1[k + 1] A6: (Partial_Sums (Alfa ((k + 1),z,w))) . (k + 1) = ((Partial_Sums (Alfa ((k + 1),z,w))) . k) + ((Alfa ((k + 1),z,w)) . (k + 1)) by SERIES_1:def_1 .= (((Partial_Sums (Alfa (k,z,w))) . k) + ((Partial_Sums (Expan_e ((k + 1),z,w))) . k)) + ((Alfa ((k + 1),z,w)) . (k + 1)) by Th12 .= ((Partial_Sums ((z + w) ExpSeq)) . k) + (((Partial_Sums (Expan_e ((k + 1),z,w))) . k) + ((Alfa ((k + 1),z,w)) . (k + 1))) by A5 ; (k + 1) -' (k + 1) = 0 by XREAL_1:232; then (Alfa ((k + 1),z,w)) . (k + 1) = ((z ExpSeq) . (k + 1)) * ((Partial_Sums (w ExpSeq)) . 0) by Def11 .= ((z ExpSeq) . (k + 1)) * ((w ExpSeq) . 0) by SERIES_1:def_1 .= ((z ExpSeq) . (k + 1)) * 1 by Th10 .= (Expan_e ((k + 1),z,w)) . (k + 1) by Th13 ; then ((Partial_Sums (Expan_e ((k + 1),z,w))) . k) + ((Alfa ((k + 1),z,w)) . (k + 1)) = (Partial_Sums (Expan_e ((k + 1),z,w))) . (k + 1) by SERIES_1:def_1 .= ((z + w) |^ (k + 1)) / ((k + 1) !) by Th8 ; then (Partial_Sums (Alfa ((k + 1),z,w))) . (k + 1) = ((Partial_Sums ((z + w) ExpSeq)) . k) + (((z + w) ExpSeq) . (k + 1)) by A6, Def4 .= (Partial_Sums ((z + w) ExpSeq)) . (k + 1) by SERIES_1:def_1 ; hence (Partial_Sums ((z + w) ExpSeq)) . (k + 1) = (Partial_Sums (Alfa ((k + 1),z,w))) . (k + 1) ; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A3, A4); hence (Partial_Sums ((z + w) ExpSeq)) . n = (Partial_Sums (Alfa (n,z,w))) . n ; ::_thesis: verum end; theorem Th15: :: SIN_COS:15 for z, w being Element of COMPLEX for k being Element of NAT holds (((Partial_Sums (z ExpSeq)) . k) * ((Partial_Sums (w ExpSeq)) . k)) - ((Partial_Sums ((z + w) ExpSeq)) . k) = (Partial_Sums (Conj (k,z,w))) . k proof let z, w be Element of COMPLEX ; ::_thesis: for k being Element of NAT holds (((Partial_Sums (z ExpSeq)) . k) * ((Partial_Sums (w ExpSeq)) . k)) - ((Partial_Sums ((z + w) ExpSeq)) . k) = (Partial_Sums (Conj (k,z,w))) . k let k be Element of NAT ; ::_thesis: (((Partial_Sums (z ExpSeq)) . k) * ((Partial_Sums (w ExpSeq)) . k)) - ((Partial_Sums ((z + w) ExpSeq)) . k) = (Partial_Sums (Conj (k,z,w))) . k A1: (((Partial_Sums (z ExpSeq)) . k) * ((Partial_Sums (w ExpSeq)) . k)) - ((Partial_Sums ((z + w) ExpSeq)) . k) = (((Partial_Sums (z ExpSeq)) . k) * ((Partial_Sums (w ExpSeq)) . k)) - ((Partial_Sums (Alfa (k,z,w))) . k) by Th14 .= ((((Partial_Sums (w ExpSeq)) . k) (#) (Partial_Sums (z ExpSeq))) . k) - ((Partial_Sums (Alfa (k,z,w))) . k) by VALUED_1:6 .= ((Partial_Sums (((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq))) . k) - ((Partial_Sums (Alfa (k,z,w))) . k) by COMSEQ_3:29 .= ((Partial_Sums (((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq))) . k) + ((- (Partial_Sums (Alfa (k,z,w)))) . k) by VALUED_1:8 .= ((Partial_Sums (((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq))) - (Partial_Sums (Alfa (k,z,w)))) . k by VALUED_1:1 .= (Partial_Sums ((((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq)) - (Alfa (k,z,w)))) . k by COMSEQ_3:28 ; for l being Element of NAT st l <= k holds ((((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq)) - (Alfa (k,z,w))) . l = (Conj (k,z,w)) . l proof let l be Element of NAT ; ::_thesis: ( l <= k implies ((((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq)) - (Alfa (k,z,w))) . l = (Conj (k,z,w)) . l ) assume A2: l <= k ; ::_thesis: ((((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq)) - (Alfa (k,z,w))) . l = (Conj (k,z,w)) . l thus ((((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq)) - (Alfa (k,z,w))) . l = ((((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq)) . l) + ((- (Alfa (k,z,w))) . l) by VALUED_1:1 .= ((((Partial_Sums (w ExpSeq)) . k) (#) (z ExpSeq)) . l) - ((Alfa (k,z,w)) . l) by VALUED_1:8 .= (((Partial_Sums (w ExpSeq)) . k) * ((z ExpSeq) . l)) - ((Alfa (k,z,w)) . l) by VALUED_1:6 .= (((z ExpSeq) . l) * ((Partial_Sums (w ExpSeq)) . k)) - (((z ExpSeq) . l) * ((Partial_Sums (w ExpSeq)) . (k -' l))) by A2, Def11 .= ((z ExpSeq) . l) * (((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))) .= (Conj (k,z,w)) . l by A2, Def13 ; ::_thesis: verum end; hence (((Partial_Sums (z ExpSeq)) . k) * ((Partial_Sums (w ExpSeq)) . k)) - ((Partial_Sums ((z + w) ExpSeq)) . k) = (Partial_Sums (Conj (k,z,w))) . k by A1, COMSEQ_3:35; ::_thesis: verum end; theorem Th16: :: SIN_COS:16 for z being Element of COMPLEX for k being Element of NAT holds ( |.((Partial_Sums (z ExpSeq)) . k).| <= (Partial_Sums (|.z.| rExpSeq)) . k & (Partial_Sums (|.z.| rExpSeq)) . k <= Sum (|.z.| rExpSeq) & |.((Partial_Sums (z ExpSeq)) . k).| <= Sum (|.z.| rExpSeq) ) proof let z be Element of COMPLEX ; ::_thesis: for k being Element of NAT holds ( |.((Partial_Sums (z ExpSeq)) . k).| <= (Partial_Sums (|.z.| rExpSeq)) . k & (Partial_Sums (|.z.| rExpSeq)) . k <= Sum (|.z.| rExpSeq) & |.((Partial_Sums (z ExpSeq)) . k).| <= Sum (|.z.| rExpSeq) ) let k be Element of NAT ; ::_thesis: ( |.((Partial_Sums (z ExpSeq)) . k).| <= (Partial_Sums (|.z.| rExpSeq)) . k & (Partial_Sums (|.z.| rExpSeq)) . k <= Sum (|.z.| rExpSeq) & |.((Partial_Sums (z ExpSeq)) . k).| <= Sum (|.z.| rExpSeq) ) defpred S1[ Element of NAT ] means |.((Partial_Sums (z ExpSeq)) . \$1).| <= (Partial_Sums (|.z.| rExpSeq)) . \$1; A1: |.((Partial_Sums (z ExpSeq)) . 0).| = |.((z ExpSeq) . 0).| by SERIES_1:def_1 .= |.((z |^ 0) / (0 !)).| by Def4 .= 1 by Th1, COMPLEX1:48, COMSEQ_3:def_1 ; (Partial_Sums (|.z.| rExpSeq)) . 0 = (|.z.| rExpSeq) . 0 by SERIES_1:def_1 .= (|.z.| |^ 0) / (0 !) by Def5 .= 1 by NEWTON:4, NEWTON:12 ; then A2: S1[ 0 ] by A1; A3: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A4: |.((Partial_Sums (z ExpSeq)) . k).| <= (Partial_Sums (|.z.| rExpSeq)) . k ; ::_thesis: S1[k + 1] ( |.((Partial_Sums (z ExpSeq)) . (k + 1)).| = |.(((Partial_Sums (z ExpSeq)) . k) + ((z ExpSeq) . (k + 1))).| & |.(((Partial_Sums (z ExpSeq)) . k) + ((z ExpSeq) . (k + 1))).| <= |.((Partial_Sums (z ExpSeq)) . k).| + |.((z ExpSeq) . (k + 1)).| ) by COMPLEX1:56, SERIES_1:def_1; then A5: |.((Partial_Sums (z ExpSeq)) . (k + 1)).| <= |.((Partial_Sums (z ExpSeq)) . k).| + ((|.z.| rExpSeq) . (k + 1)) by Th3; A6: |.((Partial_Sums (z ExpSeq)) . k).| + ((|.z.| rExpSeq) . (k + 1)) <= ((Partial_Sums (|.z.| rExpSeq)) . k) + ((|.z.| rExpSeq) . (k + 1)) by A4, XREAL_1:6; ((Partial_Sums (|.z.| rExpSeq)) . k) + ((|.z.| rExpSeq) . (k + 1)) = (Partial_Sums (|.z.| rExpSeq)) . (k + 1) by SERIES_1:def_1; hence S1[k + 1] by A5, A6, XXREAL_0:2; ::_thesis: verum end; A7: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A3); hence |.((Partial_Sums (z ExpSeq)) . k).| <= (Partial_Sums (|.z.| rExpSeq)) . k ; ::_thesis: ( (Partial_Sums (|.z.| rExpSeq)) . k <= Sum (|.z.| rExpSeq) & |.((Partial_Sums (z ExpSeq)) . k).| <= Sum (|.z.| rExpSeq) ) now__::_thesis:_for_k_being_set_st_k_in_NAT_holds_ |.(z_ExpSeq).|_._k_=_(|.z.|_rExpSeq)_._k let k be set ; ::_thesis: ( k in NAT implies |.(z ExpSeq).| . k = (|.z.| rExpSeq) . k ) assume A8: k in NAT ; ::_thesis: |.(z ExpSeq).| . k = (|.z.| rExpSeq) . k thus |.(z ExpSeq).| . k = |.((z ExpSeq) . k).| by VALUED_1:18 .= (|.z.| rExpSeq) . k by A8, Th3 ; ::_thesis: verum end; then A9: |.(z ExpSeq).| = |.z.| rExpSeq by FUNCT_2:12; then (Partial_Sums (|.z.| rExpSeq)) . k <= lim (Partial_Sums (|.z.| rExpSeq)) by SEQ_4:37; hence (Partial_Sums (|.z.| rExpSeq)) . k <= Sum (|.z.| rExpSeq) by SERIES_1:def_3; ::_thesis: |.((Partial_Sums (z ExpSeq)) . k).| <= Sum (|.z.| rExpSeq) A10: now__::_thesis:_for_k_being_Element_of_NAT_holds_(Partial_Sums_(|.z.|_rExpSeq))_._k_<=_Sum_(|.z.|_rExpSeq) let k be Element of NAT ; ::_thesis: (Partial_Sums (|.z.| rExpSeq)) . k <= Sum (|.z.| rExpSeq) lim (Partial_Sums (|.z.| rExpSeq)) = Sum (|.z.| rExpSeq) by SERIES_1:def_3; hence (Partial_Sums (|.z.| rExpSeq)) . k <= Sum (|.z.| rExpSeq) by A9, SEQ_4:37; ::_thesis: verum end; A11: |.((Partial_Sums (z ExpSeq)) . k).| <= (Partial_Sums (|.z.| rExpSeq)) . k by A7; (Partial_Sums (|.z.| rExpSeq)) . k <= Sum (|.z.| rExpSeq) by A10; hence |.((Partial_Sums (z ExpSeq)) . k).| <= Sum (|.z.| rExpSeq) by A11, XXREAL_0:2; ::_thesis: verum end; theorem Th17: :: SIN_COS:17 for z being Element of COMPLEX holds 1 <= Sum (|.z.| rExpSeq) proof let z be Element of COMPLEX ; ::_thesis: 1 <= Sum (|.z.| rExpSeq) |.((Partial_Sums (z ExpSeq)) . 0).| = |.((z ExpSeq) . 0).| by SERIES_1:def_1 .= 1 by Th3, COMPLEX1:48 ; hence 1 <= Sum (|.z.| rExpSeq) by Th16; ::_thesis: verum end; theorem Th18: :: SIN_COS:18 for z being Element of COMPLEX for n being Element of NAT holds 0 <= (|.z.| rExpSeq) . n proof let z be Element of COMPLEX ; ::_thesis: for n being Element of NAT holds 0 <= (|.z.| rExpSeq) . n let n be Element of NAT ; ::_thesis: 0 <= (|.z.| rExpSeq) . n (|.z.| rExpSeq) . n = |.((z ExpSeq) . n).| by Th3; hence 0 <= (|.z.| rExpSeq) . n by COMPLEX1:46; ::_thesis: verum end; theorem Th19: :: SIN_COS:19 for z being Element of COMPLEX for n, m being Element of NAT holds ( abs ((Partial_Sums (|.z.| rExpSeq)) . n) = (Partial_Sums (|.z.| rExpSeq)) . n & ( n <= m implies abs (((Partial_Sums (|.z.| rExpSeq)) . m) - ((Partial_Sums (|.z.| rExpSeq)) . n)) = ((Partial_Sums (|.z.| rExpSeq)) . m) - ((Partial_Sums (|.z.| rExpSeq)) . n) ) ) proof let z be Element of COMPLEX ; ::_thesis: for n, m being Element of NAT holds ( abs ((Partial_Sums (|.z.| rExpSeq)) . n) = (Partial_Sums (|.z.| rExpSeq)) . n & ( n <= m implies abs (((Partial_Sums (|.z.| rExpSeq)) . m) - ((Partial_Sums (|.z.| rExpSeq)) . n)) = ((Partial_Sums (|.z.| rExpSeq)) . m) - ((Partial_Sums (|.z.| rExpSeq)) . n) ) ) let n, m be Element of NAT ; ::_thesis: ( abs ((Partial_Sums (|.z.| rExpSeq)) . n) = (Partial_Sums (|.z.| rExpSeq)) . n & ( n <= m implies abs (((Partial_Sums (|.z.| rExpSeq)) . m) - ((Partial_Sums (|.z.| rExpSeq)) . n)) = ((Partial_Sums (|.z.| rExpSeq)) . m) - ((Partial_Sums (|.z.| rExpSeq)) . n) ) ) for n being Element of NAT holds 0 <= (|.z.| rExpSeq) . n by Th18; hence ( abs ((Partial_Sums (|.z.| rExpSeq)) . n) = (Partial_Sums (|.z.| rExpSeq)) . n & ( n <= m implies abs (((Partial_Sums (|.z.| rExpSeq)) . m) - ((Partial_Sums (|.z.| rExpSeq)) . n)) = ((Partial_Sums (|.z.| rExpSeq)) . m) - ((Partial_Sums (|.z.| rExpSeq)) . n) ) ) by COMSEQ_3:9; ::_thesis: verum end; theorem Th20: :: SIN_COS:20 for z, w being Element of COMPLEX for k, n being Element of NAT holds abs ((Partial_Sums |.(Conj (k,z,w)).|) . n) = (Partial_Sums |.(Conj (k,z,w)).|) . n proof let z, w be Element of COMPLEX ; ::_thesis: for k, n being Element of NAT holds abs ((Partial_Sums |.(Conj (k,z,w)).|) . n) = (Partial_Sums |.(Conj (k,z,w)).|) . n let k, n be Element of NAT ; ::_thesis: abs ((Partial_Sums |.(Conj (k,z,w)).|) . n) = (Partial_Sums |.(Conj (k,z,w)).|) . n A1: now__::_thesis:_for_n_being_Element_of_NAT_holds_0_<=_|.(Conj_(k,z,w)).|_._n let n be Element of NAT ; ::_thesis: 0 <= |.(Conj (k,z,w)).| . n |.(Conj (k,z,w)).| . n = |.((Conj (k,z,w)) . n).| by VALUED_1:18; hence 0 <= |.(Conj (k,z,w)).| . n by COMPLEX1:46; ::_thesis: verum end; A2: ( (Partial_Sums |.(Conj (k,z,w)).|) . 0 <= (Partial_Sums |.(Conj (k,z,w)).|) . n & (Partial_Sums |.(Conj (k,z,w)).|) . 0 = |.(Conj (k,z,w)).| . 0 ) by SEQM_3:6, SERIES_1:def_1; 0 <= |.(Conj (k,z,w)).| . 0 by A1; hence abs ((Partial_Sums |.(Conj (k,z,w)).|) . n) = (Partial_Sums |.(Conj (k,z,w)).|) . n by A2, ABSVALUE:def_1; ::_thesis: verum end; theorem Th21: :: SIN_COS:21 for z, w being Element of COMPLEX for p being real number st p > 0 holds ex n being Element of NAT st for k being Element of NAT st n <= k holds abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p proof let z, w be Element of COMPLEX ; ::_thesis: for p being real number st p > 0 holds ex n being Element of NAT st for k being Element of NAT st n <= k holds abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p let p be real number ; ::_thesis: ( p > 0 implies ex n being Element of NAT st for k being Element of NAT st n <= k holds abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p ) assume A1: p > 0 ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p reconsider pp = p as Real by XREAL_0:def_1; A2: 1 <= Sum (|.z.| rExpSeq) by Th17; A3: 0 < Sum (|.w.| rExpSeq) by Th17; set p1 = min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq))))); A4: min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq))))) > 0 by A1, A2, A3, XXREAL_0:15; now__::_thesis:_for_k_being_set_st_k_in_NAT_holds_ |.(z_ExpSeq).|_._k_=_(|.z.|_rExpSeq)_._k let k be set ; ::_thesis: ( k in NAT implies |.(z ExpSeq).| . k = (|.z.| rExpSeq) . k ) assume A5: k in NAT ; ::_thesis: |.(z ExpSeq).| . k = (|.z.| rExpSeq) . k thus |.(z ExpSeq).| . k = |.((z ExpSeq) . k).| by VALUED_1:18 .= (|.z.| rExpSeq) . k by A5, Th3 ; ::_thesis: verum end; then |.(z ExpSeq).| = |.z.| rExpSeq by FUNCT_2:12; then consider n1 being Element of NAT such that A6: for k, l being Element of NAT st n1 <= k & n1 <= l holds abs (((Partial_Sums (|.z.| rExpSeq)) . k) - ((Partial_Sums (|.z.| rExpSeq)) . l)) < min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq))))) by A4, COMSEQ_3:4; consider n2 being Element of NAT such that A7: for k, l being Element of NAT st n2 <= k & n2 <= l holds |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . l)).| < min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq))))) by A4, COMSEQ_3:47; set n3 = n1 + n2; take n4 = (n1 + n2) + (n1 + n2); ::_thesis: for k being Element of NAT st n4 <= k holds abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p A8: now__::_thesis:_for_n,_k,_l_being_Element_of_NAT_st_l_<=_k_holds_ |.(Conj_(k,z,w)).|_._l_=_((|.z.|_rExpSeq)_._l)_*_|.(((Partial_Sums_(w_ExpSeq))_._k)_-_((Partial_Sums_(w_ExpSeq))_._(k_-'_l))).| let n, k be Element of NAT ; ::_thesis: for l being Element of NAT st l <= k holds |.(Conj (k,z,w)).| . l = ((|.z.| rExpSeq) . l) * |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| now__::_thesis:_for_l_being_Element_of_NAT_st_l_<=_k_holds_ |.(Conj_(k,z,w)).|_._l_=_((|.z.|_rExpSeq)_._l)_*_|.(((Partial_Sums_(w_ExpSeq))_._k)_-_((Partial_Sums_(w_ExpSeq))_._(k_-'_l))).| let l be Element of NAT ; ::_thesis: ( l <= k implies |.(Conj (k,z,w)).| . l = ((|.z.| rExpSeq) . l) * |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| ) assume A9: l <= k ; ::_thesis: |.(Conj (k,z,w)).| . l = ((|.z.| rExpSeq) . l) * |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| thus |.(Conj (k,z,w)).| . l = |.((Conj (k,z,w)) . l).| by VALUED_1:18 .= |.(((z ExpSeq) . l) * (((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l)))).| by A9, Def13 .= |.((z ExpSeq) . l).| * |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| by COMPLEX1:65 .= ((|.z.| rExpSeq) . l) * |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| by Th3 ; ::_thesis: verum end; hence for l being Element of NAT st l <= k holds |.(Conj (k,z,w)).| . l = ((|.z.| rExpSeq) . l) * |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| ; ::_thesis: verum end; A10: now__::_thesis:_for_k,_l_being_Element_of_NAT_st_l_<=_k_holds_ |.(Conj_(k,z,w)).|_._l_<=_((|.z.|_rExpSeq)_._l)_*_(2_*_(Sum_(|.w.|_rExpSeq))) let k be Element of NAT ; ::_thesis: for l being Element of NAT st l <= k holds |.(Conj (k,z,w)).| . l <= ((|.z.| rExpSeq) . l) * (2 * (Sum (|.w.| rExpSeq))) now__::_thesis:_for_l_being_Element_of_NAT_st_l_<=_k_holds_ |.(Conj_(k,z,w)).|_._l_<=_((|.z.|_rExpSeq)_._l)_*_(2_*_(Sum_(|.w.|_rExpSeq))) let l be Element of NAT ; ::_thesis: ( l <= k implies |.(Conj (k,z,w)).| . l <= ((|.z.| rExpSeq) . l) * (2 * (Sum (|.w.| rExpSeq))) ) assume l <= k ; ::_thesis: |.(Conj (k,z,w)).| . l <= ((|.z.| rExpSeq) . l) * (2 * (Sum (|.w.| rExpSeq))) then A11: |.(Conj (k,z,w)).| . l = ((|.z.| rExpSeq) . l) * |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| by A8; |.((Partial_Sums (w ExpSeq)) . k).| <= Sum (|.w.| rExpSeq) by Th16; then A12: |.((Partial_Sums (w ExpSeq)) . k).| + |.((Partial_Sums (w ExpSeq)) . (k -' l)).| <= (Sum (|.w.| rExpSeq)) + |.((Partial_Sums (w ExpSeq)) . (k -' l)).| by XREAL_1:6; |.((Partial_Sums (w ExpSeq)) . (k -' l)).| <= Sum (|.w.| rExpSeq) by Th16; then (Sum (|.w.| rExpSeq)) + |.((Partial_Sums (w ExpSeq)) . (k -' l)).| <= (Sum (|.w.| rExpSeq)) + (Sum (|.w.| rExpSeq)) by XREAL_1:6; then ( |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| <= |.((Partial_Sums (w ExpSeq)) . k).| + |.((Partial_Sums (w ExpSeq)) . (k -' l)).| & |.((Partial_Sums (w ExpSeq)) . k).| + |.((Partial_Sums (w ExpSeq)) . (k -' l)).| <= 2 * (Sum (|.w.| rExpSeq)) ) by A12, COMPLEX1:57, XXREAL_0:2; then A13: |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| <= 2 * (Sum (|.w.| rExpSeq)) by XXREAL_0:2; 0 <= (|.z.| rExpSeq) . l by Th18; hence |.(Conj (k,z,w)).| . l <= ((|.z.| rExpSeq) . l) * (2 * (Sum (|.w.| rExpSeq))) by A11, A13, XREAL_1:64; ::_thesis: verum end; hence for l being Element of NAT st l <= k holds |.(Conj (k,z,w)).| . l <= ((|.z.| rExpSeq) . l) * (2 * (Sum (|.w.| rExpSeq))) ; ::_thesis: verum end; now__::_thesis:_for_k_being_Element_of_NAT_st_n4_<=_k_holds_ abs_((Partial_Sums_|.(Conj_(k,z,w)).|)_._k)_<_p let k be Element of NAT ; ::_thesis: ( n4 <= k implies abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p ) assume A14: n4 <= k ; ::_thesis: abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p A15: 0 + (n1 + n2) <= (n1 + n2) + (n1 + n2) by XREAL_1:6; then A16: n1 + n2 <= k by A14, XXREAL_0:2; A17: n1 + 0 <= n1 + n2 by XREAL_1:6; then A18: n1 <= k by A16, XXREAL_0:2; now__::_thesis:_for_l_being_Element_of_NAT_st_l_<=_n1_+_n2_holds_ |.(Conj_(k,z,w)).|_._l_<=_(min_((pp_/_(4_*_(Sum_(|.z.|_rExpSeq)))),(pp_/_(4_*_(Sum_(|.w.|_rExpSeq))))))_*_((|.z.|_rExpSeq)_._l) let l be Element of NAT ; ::_thesis: ( l <= n1 + n2 implies |.(Conj (k,z,w)).| . l <= (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) * ((|.z.| rExpSeq) . l) ) assume A19: l <= n1 + n2 ; ::_thesis: |.(Conj (k,z,w)).| . l <= (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) * ((|.z.| rExpSeq) . l) then A20: k -' l = k - l by A16, XREAL_1:233, XXREAL_0:2; A21: 0 + n2 <= n1 + n2 by XREAL_1:6; A22: n4 - l <= k - l by A14, XREAL_1:9; ((n1 + n2) + (n1 + n2)) - (n1 + n2) <= ((n1 + n2) + (n1 + n2)) - l by A19, XREAL_1:10; then n1 + n2 <= k - l by A22, XXREAL_0:2; then A23: n2 <= k -' l by A20, A21, XXREAL_0:2; 0 + (n1 + n2) <= (n1 + n2) + (n1 + n2) by XREAL_1:6; then n2 <= n4 by A21, XXREAL_0:2; then n2 <= k by A14, XXREAL_0:2; then A24: |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| < min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq))))) by A7, A23; 0 <= (|.z.| rExpSeq) . l by Th18; then ((|.z.| rExpSeq) . l) * |.(((Partial_Sums (w ExpSeq)) . k) - ((Partial_Sums (w ExpSeq)) . (k -' l))).| <= ((|.z.| rExpSeq) . l) * (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) by A24, XREAL_1:64; hence |.(Conj (k,z,w)).| . l <= (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) * ((|.z.| rExpSeq) . l) by A8, A16, A19, XXREAL_0:2; ::_thesis: verum end; then A25: (Partial_Sums |.(Conj (k,z,w)).|) . (n1 + n2) <= ((Partial_Sums (|.z.| rExpSeq)) . (n1 + n2)) * (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) by COMSEQ_3:7; ((Partial_Sums (|.z.| rExpSeq)) . (n1 + n2)) * (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) <= (Sum (|.z.| rExpSeq)) * (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) by A4, Th16, XREAL_1:64; then A26: (Partial_Sums |.(Conj (k,z,w)).|) . (n1 + n2) <= (Sum (|.z.| rExpSeq)) * (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) by A25, XXREAL_0:2; A27: (Sum (|.z.| rExpSeq)) * (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) <= (Sum (|.z.| rExpSeq)) * (p / (4 * (Sum (|.z.| rExpSeq)))) by A2, XREAL_1:64, XXREAL_0:17; A28: 0 <> Sum (|.z.| rExpSeq) by Th17; (Sum (|.z.| rExpSeq)) * (p / (4 * (Sum (|.z.| rExpSeq)))) = ((Sum (|.z.| rExpSeq)) * p) / (4 * (Sum (|.z.| rExpSeq))) .= p / 4 by A28, XCMPLX_1:91 ; then A29: (Partial_Sums |.(Conj (k,z,w)).|) . (n1 + n2) <= p / 4 by A26, A27, XXREAL_0:2; 0 + (p / 4) < (p / 4) + (p / 4) by A1, XREAL_1:6; then A30: (Partial_Sums |.(Conj (k,z,w)).|) . (n1 + n2) < p / 2 by A29, XXREAL_0:2; k -' (n1 + n2) = k - (n1 + n2) by A14, A15, XREAL_1:233, XXREAL_0:2; then A31: k = (k -' (n1 + n2)) + (n1 + n2) ; for l being Element of NAT st l <= k holds |.(Conj (k,z,w)).| . l <= (2 * (Sum (|.w.| rExpSeq))) * ((|.z.| rExpSeq) . l) by A10; then A32: ((Partial_Sums |.(Conj (k,z,w)).|) . k) - ((Partial_Sums |.(Conj (k,z,w)).|) . (n1 + n2)) <= (((Partial_Sums (|.z.| rExpSeq)) . k) - ((Partial_Sums (|.z.| rExpSeq)) . (n1 + n2))) * (2 * (Sum (|.w.| rExpSeq))) by A31, COMSEQ_3:8; abs (((Partial_Sums (|.z.| rExpSeq)) . k) - ((Partial_Sums (|.z.| rExpSeq)) . (n1 + n2))) <= min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq))))) by A6, A17, A18; then ((Partial_Sums (|.z.| rExpSeq)) . k) - ((Partial_Sums (|.z.| rExpSeq)) . (n1 + n2)) <= min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq))))) by A14, A15, Th19, XXREAL_0:2; then (((Partial_Sums (|.z.| rExpSeq)) . k) - ((Partial_Sums (|.z.| rExpSeq)) . (n1 + n2))) * (2 * (Sum (|.w.| rExpSeq))) <= (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) * (2 * (Sum (|.w.| rExpSeq))) by A3, XREAL_1:64; then A33: ((Partial_Sums |.(Conj (k,z,w)).|) . k) - ((Partial_Sums |.(Conj (k,z,w)).|) . (n1 + n2)) <= (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) * (2 * (Sum (|.w.| rExpSeq))) by A32, XXREAL_0:2; A34: (2 * (Sum (|.w.| rExpSeq))) * (min ((pp / (4 * (Sum (|.z.| rExpSeq)))),(pp / (4 * (Sum (|.w.| rExpSeq)))))) <= (2 * (Sum (|.w.| rExpSeq))) * (p / (4 * (Sum (|.w.| rExpSeq)))) by A3, XREAL_1:64, XXREAL_0:17; A35: 0 <> Sum (|.w.| rExpSeq) by Th17; (2 * (Sum (|.w.| rExpSeq))) * (p / (4 * (Sum (|.w.| rExpSeq)))) = ((2 * p) * (Sum (|.w.| rExpSeq))) / (4 * (Sum (|.w.| rExpSeq))) .= (p + p) / 4 by A35, XCMPLX_1:91 .= p / 2 ; then ((Partial_Sums |.(Conj (k,z,w)).|) . k) - ((Partial_Sums |.(Conj (k,z,w)).|) . (n1 + n2)) <= p / 2 by A33, A34, XXREAL_0:2; then (((Partial_Sums |.(Conj (k,z,w)).|) . k) - ((Partial_Sums |.(Conj (k,z,w)).|) . (n1 + n2))) + ((Partial_Sums |.(Conj (k,z,w)).|) . (n1 + n2)) < (p / 2) + (p / 2) by A30, XREAL_1:8; hence abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p by Th20; ::_thesis: verum end; hence for k being Element of NAT st n4 <= k holds abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p ; ::_thesis: verum end; theorem Th22: :: SIN_COS:22 for z, w being Element of COMPLEX for seq being Complex_Sequence st ( for k being Element of NAT holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ) holds ( seq is convergent & lim seq = 0 ) proof let z, w be Element of COMPLEX ; ::_thesis: for seq being Complex_Sequence st ( for k being Element of NAT holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ) holds ( seq is convergent & lim seq = 0 ) let seq be Complex_Sequence; ::_thesis: ( ( for k being Element of NAT holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ) implies ( seq is convergent & lim seq = 0 ) ) now__::_thesis:_for_seq_being_Complex_Sequence_st_(_for_k_being_Element_of_NAT_holds_seq_._k_=_(Partial_Sums_(Conj_(k,z,w)))_._k_)_holds_ (_seq_is_convergent_&_lim_seq_=_0c_) let seq be Complex_Sequence; ::_thesis: ( ( for k being Element of NAT holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ) implies ( seq is convergent & lim seq = 0c ) ) assume A1: for k being Element of NAT holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ; ::_thesis: ( seq is convergent & lim seq = 0c ) A2: now__::_thesis:_for_k_being_Element_of_NAT_holds_|.(seq_._k).|_<=_(Partial_Sums_|.(Conj_(k,z,w)).|)_._k let k be Element of NAT ; ::_thesis: |.(seq . k).| <= (Partial_Sums |.(Conj (k,z,w)).|) . k |.(seq . k).| = |.((Partial_Sums (Conj (k,z,w))) . k).| by A1; hence |.(seq . k).| <= (Partial_Sums |.(Conj (k,z,w)).|) . k by COMSEQ_3:30; ::_thesis: verum end; deffunc H1( Element of NAT ) -> Element of REAL = (Partial_Sums |.(Conj (\$1,z,w)).|) . \$1; ex rseq being Real_Sequence st for k being Element of NAT holds rseq . k = H1(k) from SEQ_1:sch_1(); then consider rseq being Real_Sequence such that A3: for k being Element of NAT holds rseq . k = (Partial_Sums |.(Conj (k,z,w)).|) . k ; A4: now__::_thesis:_for_k_being_Element_of_NAT_holds_|.(seq_._k).|_<=_rseq_._k let k be Element of NAT ; ::_thesis: |.(seq . k).| <= rseq . k |.(seq . k).| <= (Partial_Sums |.(Conj (k,z,w)).|) . k by A2; hence |.(seq . k).| <= rseq . k by A3; ::_thesis: verum end; A5: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_ ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_((rseq_._k)_-_0)_<_p let p be real number ; ::_thesis: ( p > 0 implies ex n being Element of NAT st for k being Element of NAT st n <= k holds abs ((rseq . k) - 0) < p ) assume p > 0 ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds abs ((rseq . k) - 0) < p then consider n being Element of NAT such that A6: for k being Element of NAT st n <= k holds abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) < p by Th21; take n = n; ::_thesis: for k being Element of NAT st n <= k holds abs ((rseq . k) - 0) < p now__::_thesis:_for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_((rseq_._k)_-_0)_<_p let k be Element of NAT ; ::_thesis: ( n <= k implies abs ((rseq . k) - 0) < p ) assume A7: n <= k ; ::_thesis: abs ((rseq . k) - 0) < p abs ((rseq . k) - 0) = abs ((Partial_Sums |.(Conj (k,z,w)).|) . k) by A3; hence abs ((rseq . k) - 0) < p by A6, A7; ::_thesis: verum end; hence for k being Element of NAT st n <= k holds abs ((rseq . k) - 0) < p ; ::_thesis: verum end; then A8: rseq is convergent by SEQ_2:def_6; then lim rseq = 0 by A5, SEQ_2:def_7; hence ( seq is convergent & lim seq = 0c ) by A4, A8, COMSEQ_3:48; ::_thesis: verum end; hence ( ( for k being Element of NAT holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ) implies ( seq is convergent & lim seq = 0 ) ) ; ::_thesis: verum end; Lm2: for z, w being complex number holds (Sum (z ExpSeq)) * (Sum (w ExpSeq)) = Sum ((z + w) ExpSeq) proof let z0, w0 be complex number ; ::_thesis: (Sum (z0 ExpSeq)) * (Sum (w0 ExpSeq)) = Sum ((z0 + w0) ExpSeq) reconsider z = z0, w = w0 as Element of COMPLEX by XCMPLX_0:def_2; deffunc H1( Element of NAT ) -> Element of COMPLEX = (Partial_Sums (Conj (\$1,z,w))) . \$1; ex seq being Complex_Sequence st for k being Element of NAT holds seq . k = H1(k) from COMSEQ_1:sch_1(); then consider seq being Complex_Sequence such that A1: for k being Element of NAT holds seq . k = (Partial_Sums (Conj (k,z,w))) . k ; now__::_thesis:_for_k_being_Element_of_NAT_holds_seq_._k_=_(((Partial_Sums_(z_ExpSeq))_(#)_(Partial_Sums_(w_ExpSeq)))_-_(Partial_Sums_((z_+_w)_ExpSeq)))_._k let k be Element of NAT ; ::_thesis: seq . k = (((Partial_Sums (z ExpSeq)) (#) (Partial_Sums (w ExpSeq))) - (Partial_Sums ((z + w) ExpSeq))) . k thus seq . k = (Partial_Sums (Conj (k,z,w))) . k by A1 .= (((Partial_Sums (z ExpSeq)) . k) * ((Partial_Sums (w ExpSeq)) . k)) - ((Partial_Sums ((z + w) ExpSeq)) . k) by Th15 .= (((Partial_Sums (z ExpSeq)) (#) (Partial_Sums (w ExpSeq))) . k) - ((Partial_Sums ((z + w) ExpSeq)) . k) by VALUED_1:5 .= (((Partial_Sums (z ExpSeq)) (#) (Partial_Sums (w ExpSeq))) . k) + ((- (Partial_Sums ((z + w) ExpSeq))) . k) by VALUED_1:8 .= (((Partial_Sums (z ExpSeq)) (#) (Partial_Sums (w ExpSeq))) - (Partial_Sums ((z + w) ExpSeq))) . k by VALUED_1:1 ; ::_thesis: verum end; then A2: seq = ((Partial_Sums (z ExpSeq)) (#) (Partial_Sums (w ExpSeq))) - (Partial_Sums ((z + w) ExpSeq)) by FUNCT_2:63; A3: ( (Partial_Sums (z ExpSeq)) (#) (Partial_Sums (w ExpSeq)) is convergent & lim ((Partial_Sums (z ExpSeq)) (#) (Partial_Sums (w ExpSeq))) = (lim (Partial_Sums (z ExpSeq))) * (lim (Partial_Sums (w ExpSeq))) ) by COMSEQ_2:30; lim seq = 0c by A1, Th22; hence (Sum (z0 ExpSeq)) * (Sum (w0 ExpSeq)) = Sum ((z0 + w0) ExpSeq) by A2, A3, COMSEQ_3:10; ::_thesis: verum end; begin definition func exp -> Function of COMPLEX,COMPLEX means :Def14: :: SIN_COS:def 14 for z being complex number holds it . z = Sum (z ExpSeq); existence ex b1 being Function of COMPLEX,COMPLEX st for z being complex number holds b1 . z = Sum (z ExpSeq) proof deffunc H1( Element of COMPLEX ) -> Element of COMPLEX = Sum (\$1 ExpSeq); consider f being Function of COMPLEX,COMPLEX such that A1: for x being Element of COMPLEX holds f . x = H1(x) from FUNCT_2:sch_4(); take f ; ::_thesis: for z being complex number holds f . z = Sum (z ExpSeq) let z be complex number ; ::_thesis: f . z = Sum (z ExpSeq) z in COMPLEX by XCMPLX_0:def_2; hence f . z = Sum (z ExpSeq) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Function of COMPLEX,COMPLEX st ( for z being complex number holds b1 . z = Sum (z ExpSeq) ) & ( for z being complex number holds b2 . z = Sum (z ExpSeq) ) holds b1 = b2 proof let f1, f2 be Function of COMPLEX,COMPLEX; ::_thesis: ( ( for z being complex number holds f1 . z = Sum (z ExpSeq) ) & ( for z being complex number holds f2 . z = Sum (z ExpSeq) ) implies f1 = f2 ) assume A2: for z being complex number holds f1 . z = Sum (z ExpSeq) ; ::_thesis: ( ex z being complex number st not f2 . z = Sum (z ExpSeq) or f1 = f2 ) assume A3: for z being complex number holds f2 . z = Sum (z ExpSeq) ; ::_thesis: f1 = f2 for z being Element of COMPLEX holds f1 . z = f2 . z proof let z be Element of COMPLEX ; ::_thesis: f1 . z = f2 . z thus f1 . z = Sum (z ExpSeq) by A2 .= f2 . z by A3 ; ::_thesis: verum end; hence f1 = f2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def14 defines exp SIN_COS:def_14_:_ for b1 being Function of COMPLEX,COMPLEX holds ( b1 = exp iff for z being complex number holds b1 . z = Sum (z ExpSeq) ); definition let z be complex number ; func exp z -> complex number equals :: SIN_COS:def 15 exp . z; coherence exp . z is complex number ; end; :: deftheorem defines exp SIN_COS:def_15_:_ for z being complex number holds exp z = exp . z; definition let z be complex number ; :: original: exp redefine func exp z -> Element of COMPLEX ; coherence exp z is Element of COMPLEX by XCMPLX_0:def_2; end; theorem :: SIN_COS:23 for z1, z2 being complex number holds exp (z1 + z2) = (exp z1) * (exp z2) proof let z1, z2 be complex number ; ::_thesis: exp (z1 + z2) = (exp z1) * (exp z2) exp (z1 + z2) = Sum ((z1 + z2) ExpSeq) by Def14 .= (Sum (z1 ExpSeq)) * (Sum (z2 ExpSeq)) by Lm2 .= (exp z1) * (Sum (z2 ExpSeq)) by Def14 .= (exp z1) * (exp z2) by Def14 ; hence exp (z1 + z2) = (exp z1) * (exp z2) ; ::_thesis: verum end; begin definition func sin -> Function of REAL,REAL means :Def16: :: SIN_COS:def 16 for d being Element of REAL holds it . d = Im (Sum ((d * ) ExpSeq)); existence ex b1 being Function of REAL,REAL st for d being Element of REAL holds b1 . d = Im (Sum ((d * ) ExpSeq)) proof deffunc H1( Element of REAL ) -> Element of REAL = Im (Sum ((\$1 * ) ExpSeq)); consider f being Function of REAL,REAL such that A1: for x being Element of REAL holds f . x = H1(x) from FUNCT_2:sch_4(); take f ; ::_thesis: for d being Element of REAL holds f . d = Im (Sum ((d * ) ExpSeq)) thus for d being Element of REAL holds f . d = Im (Sum ((d * ) ExpSeq)) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Function of REAL,REAL st ( for d being Element of REAL holds b1 . d = Im (Sum ((d * ) ExpSeq)) ) & ( for d being Element of REAL holds b2 . d = Im (Sum ((d * ) ExpSeq)) ) holds b1 = b2 proof let f1, f2 be Function of REAL,REAL; ::_thesis: ( ( for d being Element of REAL holds f1 . d = Im (Sum ((d * ) ExpSeq)) ) & ( for d being Element of REAL holds f2 . d = Im (Sum ((d * ) ExpSeq)) ) implies f1 = f2 ) assume A2: for d being Element of REAL holds f1 . d = Im (Sum ((d * ) ExpSeq)) ; ::_thesis: ( ex d being Element of REAL st not f2 . d = Im (Sum ((d * ) ExpSeq)) or f1 = f2 ) assume A3: for d being Element of REAL holds f2 . d = Im (Sum ((d * ) ExpSeq)) ; ::_thesis: f1 = f2 for d being Element of REAL holds f1 . d = f2 . d proof let d be Element of REAL ; ::_thesis: f1 . d = f2 . d thus f1 . d = Im (Sum ((d * ) ExpSeq)) by A2 .= f2 . d by A3 ; ::_thesis: verum end; hence f1 = f2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def16 defines sin SIN_COS:def_16_:_ for b1 being Function of REAL,REAL holds ( b1 = sin iff for d being Element of REAL holds b1 . d = Im (Sum ((d * ) ExpSeq)) ); definition let th be real number ; func sin th -> set equals :: SIN_COS:def 17 sin . th; coherence sin . th is set ; end; :: deftheorem defines sin SIN_COS:def_17_:_ for th being real number holds sin th = sin . th; registration let th be real number ; cluster sin th -> real ; coherence sin th is real ; end; definition let th be Real; :: original: sin redefine func sin th -> Real; coherence sin th is Real ; end; definition func cos -> Function of REAL,REAL means :Def18: :: SIN_COS:def 18 for d being Real holds it . d = Re (Sum ((d * ) ExpSeq)); existence ex b1 being Function of REAL,REAL st for d being Real holds b1 . d = Re (Sum ((d * ) ExpSeq)) proof deffunc H1( Element of REAL ) -> Element of REAL = Re (Sum ((\$1 * ) ExpSeq)); consider f being Function of REAL,REAL such that A1: for x being Element of REAL holds f . x = H1(x) from FUNCT_2:sch_4(); take f ; ::_thesis: for d being Real holds f . d = Re (Sum ((d * ) ExpSeq)) thus for d being Real holds f . d = Re (Sum ((d * ) ExpSeq)) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Function of REAL,REAL st ( for d being Real holds b1 . d = Re (Sum ((d * ) ExpSeq)) ) & ( for d being Real holds b2 . d = Re (Sum ((d * ) ExpSeq)) ) holds b1 = b2 proof let f1, f2 be Function of REAL,REAL; ::_thesis: ( ( for d being Real holds f1 . d = Re (Sum ((d * ) ExpSeq)) ) & ( for d being Real holds f2 . d = Re (Sum ((d * ) ExpSeq)) ) implies f1 = f2 ) assume A2: for d being Real holds f1 . d = Re (Sum ((d * ) ExpSeq)) ; ::_thesis: ( ex d being Real st not f2 . d = Re (Sum ((d * ) ExpSeq)) or f1 = f2 ) assume A3: for d being Real holds f2 . d = Re (Sum ((d * ) ExpSeq)) ; ::_thesis: f1 = f2 for d being Real holds f1 . d = f2 . d proof let d be Real; ::_thesis: f1 . d = f2 . d thus f1 . d = Re (Sum ((d * ) ExpSeq)) by A2 .= f2 . d by A3 ; ::_thesis: verum end; hence f1 = f2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def18 defines cos SIN_COS:def_18_:_ for b1 being Function of REAL,REAL holds ( b1 = cos iff for d being Real holds b1 . d = Re (Sum ((d * ) ExpSeq)) ); definition let th be real number ; func cos th -> set equals :: SIN_COS:def 19 cos . th; coherence cos . th is set ; end; :: deftheorem defines cos SIN_COS:def_19_:_ for th being real number holds cos th = cos . th; registration let th be real number ; cluster cos th -> real ; coherence cos th is real ; end; definition let th be Real; :: original: cos redefine func cos th -> Real; coherence cos th is Real ; end; theorem Th24: :: SIN_COS:24 ( dom sin = REAL & dom cos = REAL ) by FUNCT_2:def_1; Lm3: for th being Real holds Sum ((th * ) ExpSeq) = (cos . th) + ((sin . th) * ) proof let th be Real; ::_thesis: Sum ((th * ) ExpSeq) = (cos . th) + ((sin . th) * ) ( Im (Sum ((th * ) ExpSeq)) = sin . th & Re (Sum ((th * ) ExpSeq)) = cos . th ) by Def16, Def18; hence Sum ((th * ) ExpSeq) = (cos . th) + ((sin . th) * ) by COMPLEX1:13; ::_thesis: verum end; theorem :: SIN_COS:25 for th being Real holds exp (th * ) = (cos th) + ((sin th) * ) proof let th be Real; ::_thesis: exp (th * ) = (cos th) + ((sin th) * ) exp (th * ) = Sum ((th * ) ExpSeq) by Def14 .= (cos th) + ((sin th) * ) by Lm3 ; hence exp (th * ) = (cos th) + ((sin th) * ) ; ::_thesis: verum end; Lm4: for th being Real holds (Sum ((th * ) ExpSeq)) *' = Sum ((- (th * )) ExpSeq) proof let th be Real; ::_thesis: (Sum ((th * ) ExpSeq)) *' = Sum ((- (th * )) ExpSeq) (Partial_Sums ((th * ) ExpSeq)) *' = Partial_Sums ((- (th * )) ExpSeq) proof A1: for n being Element of NAT holds (((th * ) ExpSeq) . n) *' = ((- (th * )) ExpSeq) . n proof defpred S1[ Element of NAT ] means (((th * ) ExpSeq) . \$1) *' = ((- (th * )) ExpSeq) . \$1; (((th * ) ExpSeq) . 0) *' = 1r by Th3, COMPLEX1:30 .= ((- (th * )) ExpSeq) . 0 by Th3 ; then A2: S1[ 0 ] ; A3: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A4: (((th * ) ExpSeq) . n) *' = ((- (th * )) ExpSeq) . n ; ::_thesis: S1[n + 1] thus (((th * ) ExpSeq) . (n + 1)) *' = (((((th * ) ExpSeq) . n) * (th * )) / ((n + 1) + (0 * ))) *' by Th3 .= ((((th * ) ExpSeq) . n) * ((th * ) / ((n + 1) + (0 * )))) *' .= (((- (th * )) ExpSeq) . n) * (((th * ) / (n + 1)) *') by A4, COMPLEX1:35 .= (((- (th * )) ExpSeq) . n) * (((0 + (th * )) *') / ((n + 1) *')) by COMPLEX1:37 .= (((- (th * )) ExpSeq) . n) * ((0 + ((- th) * )) / (((n + 1) + (0 * )) *')) by Lm1 .= (((- (th * )) ExpSeq) . n) * ((0 + ((- th) * )) / ((n + 1) + ((- 0) * ))) by Lm1 .= ((((- (th * )) ExpSeq) . n) * (- (th * ))) / ((n + 1) + (0 * )) .= ((- (th * )) ExpSeq) . (n + 1) by Th3 ; ::_thesis: verum end; thus for n being Element of NAT holds S1[n] from NAT_1:sch_1(A2, A3); ::_thesis: verum end; defpred S1[ Element of NAT ] means ((Partial_Sums ((th * ) ExpSeq)) *') . \$1 = (Partial_Sums ((- (th * )) ExpSeq)) . \$1; ((Partial_Sums ((th * ) ExpSeq)) *') . 0 = ((Partial_Sums ((th * ) ExpSeq)) . 0) *' by COMSEQ_2:def_2 .= (((th * ) ExpSeq) . 0) *' by SERIES_1:def_1 .= 1 by Th3, COMPLEX1:30 .= ((- (th * )) ExpSeq) . 0 by Th3 .= (Partial_Sums ((- (th * )) ExpSeq)) . 0 by SERIES_1:def_1 ; then A5: S1[ 0 ] ; A6: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A7: ((Partial_Sums ((th * ) ExpSeq)) *') . n = (Partial_Sums ((- (th * )) ExpSeq)) . n ; ::_thesis: S1[n + 1] thus ((Partial_Sums ((th * ) ExpSeq)) *') . (n + 1) = ((Partial_Sums ((th * ) ExpSeq)) . (n + 1)) *' by COMSEQ_2:def_2 .= (((Partial_Sums ((th * ) ExpSeq)) . n) + (((th * ) ExpSeq) . (n + 1))) *' by SERIES_1:def_1 .= (((Partial_Sums ((th * ) ExpSeq)) . n) *') + ((((th * ) ExpSeq) . (n + 1)) *') by COMPLEX1:32 .= ((Partial_Sums ((- (th * )) ExpSeq)) . n) + ((((th * ) ExpSeq) . (n + 1)) *') by A7, COMSEQ_2:def_2 .= ((Partial_Sums ((- (th * )) ExpSeq)) . n) + (((- (th * )) ExpSeq) . (n + 1)) by A1 .= (Partial_Sums ((- (th * )) ExpSeq)) . (n + 1) by SERIES_1:def_1 ; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A5, A6); hence (Partial_Sums ((th * ) ExpSeq)) *' = Partial_Sums ((- (th * )) ExpSeq) by FUNCT_2:63; ::_thesis: verum end; hence (Sum ((th * ) ExpSeq)) *' = Sum ((- (th * )) ExpSeq) by COMSEQ_2:12; ::_thesis: verum end; theorem :: SIN_COS:26 for th being Real holds (exp (th * )) *' = exp (- (th * )) proof let th be Real; ::_thesis: (exp (th * )) *' = exp (- (th * )) (exp (th * )) *' = (Sum ((th * ) ExpSeq)) *' by Def14 .= Sum ((- (th * )) ExpSeq) by Lm4 .= exp (- (th * )) by Def14 ; hence (exp (th * )) *' = exp (- (th * )) ; ::_thesis: verum end; Lm5: for th being Real for th1 being real number st th = th1 holds ( |.(Sum ((th * ) ExpSeq)).| = 1 & abs (sin . th1) <= 1 & abs (cos . th1) <= 1 ) proof let th be Real; ::_thesis: for th1 being real number st th = th1 holds ( |.(Sum ((th * ) ExpSeq)).| = 1 & abs (sin . th1) <= 1 & abs (cos . th1) <= 1 ) let th1 be real number ; ::_thesis: ( th = th1 implies ( |.(Sum ((th * ) ExpSeq)).| = 1 & abs (sin . th1) <= 1 & abs (cos . th1) <= 1 ) ) assume A1: th = th1 ; ::_thesis: ( |.(Sum ((th * ) ExpSeq)).| = 1 & abs (sin . th1) <= 1 & abs (cos . th1) <= 1 ) A2: (Sum ((th * ) ExpSeq)) * (Sum ((- (th * )) ExpSeq)) = Sum (((th * ) + (- (th * ))) ExpSeq) by Lm2 .= 1r by Th9 ; |.(Sum ((th * ) ExpSeq)).| * |.(Sum ((th * ) ExpSeq)).| = |.((Sum ((th * ) ExpSeq)) * (Sum ((th * ) ExpSeq))).| by COMPLEX1:65 .= |.((Sum ((th * ) ExpSeq)) * ((Sum ((th * ) ExpSeq)) *')).| by COMPLEX1:69 .= 1 by A2, Lm4, COMPLEX1:48 ; then A3: ( |.(Sum ((th * ) ExpSeq)).| = 1 / |.(Sum ((th * ) ExpSeq)).| & |.(Sum ((th * ) ExpSeq)).| <> 0 ) by XCMPLX_1:73; |.(Sum ((th * ) ExpSeq)).| <> - 1 by COMPLEX1:46; hence A4: |.(Sum ((th * ) ExpSeq)).| = 1 by A3, XCMPLX_1:199; ::_thesis: ( abs (sin . th1) <= 1 & abs (cos . th1) <= 1 ) ( abs (sin . th) = abs (Im (Sum ((th * ) ExpSeq))) & abs (cos . th) = abs (Re (Sum ((th * ) ExpSeq))) ) by Def16, Def18; hence ( abs (sin . th1) <= 1 & abs (cos . th1) <= 1 ) by A1, A4, COMSEQ_3:13; ::_thesis: verum end; theorem :: SIN_COS:27 for th being Real holds ( |.(exp (th * )).| = 1 & ( for th being real number holds ( abs (sin th) <= 1 & abs (cos th) <= 1 ) ) ) proof let th be Real; ::_thesis: ( |.(exp (th * )).| = 1 & ( for th being real number holds ( abs (sin th) <= 1 & abs (cos th) <= 1 ) ) ) thus |.(exp (th * )).| = |.(Sum ((th * ) ExpSeq)).| by Def14 .= 1 by Lm5 ; ::_thesis: for th being real number holds ( abs (sin th) <= 1 & abs (cos th) <= 1 ) let th be real number ; ::_thesis: ( abs (sin th) <= 1 & abs (cos th) <= 1 ) A1: th is Real by XREAL_0:def_1; hence abs (sin th) <= 1 by Lm5; ::_thesis: abs (cos th) <= 1 thus abs (cos th) <= 1 by A1, Lm5; ::_thesis: verum end; theorem Th28: :: SIN_COS:28 for th being real number holds ( ((cos . th) ^2) + ((sin . th) ^2) = 1 & ((cos . th) * (cos . th)) + ((sin . th) * (sin . th)) = 1 ) proof let th be real number ; ::_thesis: ( ((cos . th) ^2) + ((sin . th) ^2) = 1 & ((cos . th) * (cos . th)) + ((sin . th) * (sin . th)) = 1 ) reconsider th1 = th as Real by XREAL_0:def_1; A1: (Sum ((th1 * ) ExpSeq)) * (Sum ((- (th1 * )) ExpSeq)) = Sum (((th1 * ) + (- (th1 * ))) ExpSeq) by Lm2 .= 1r by Th9 ; thus ((cos . th) ^2) + ((sin . th) ^2) = ((Re (Sum ((th1 * ) ExpSeq))) ^2) + ((sin . th) ^2) by Def18 .= ((Re (Sum ((th1 * ) ExpSeq))) ^2) + ((Im (Sum ((th1 * ) ExpSeq))) ^2) by Def16 .= |.((Sum ((th1 * ) ExpSeq)) * (Sum ((th1 * ) ExpSeq))).| by COMPLEX1:68 .= |.((Sum ((th1 * ) ExpSeq)) * ((Sum ((th1 * ) ExpSeq)) *')).| by COMPLEX1:69 .= 1 by A1, Lm4, COMPLEX1:48 ; ::_thesis: ((cos . th) * (cos . th)) + ((sin . th) * (sin . th)) = 1 hence ((cos . th) * (cos . th)) + ((sin . th) * (sin . th)) = 1 ; ::_thesis: verum end; theorem :: SIN_COS:29 for th being real number holds ( ((cos th) ^2) + ((sin th) ^2) = 1 & ((cos th) * (cos th)) + ((sin th) * (sin th)) = 1 ) by Th28; theorem Th30: :: SIN_COS:30 for th being real number holds ( cos . 0 = 1 & sin . 0 = 0 & cos . (- th) = cos . th & sin . (- th) = - (sin . th) ) proof let th be real number ; ::_thesis: ( cos . 0 = 1 & sin . 0 = 0 & cos . (- th) = cos . th & sin . (- th) = - (sin . th) ) thus cos . 0 = Re (Sum ((0 * ) ExpSeq)) by Def18 .= 1 by Th9, COMPLEX1:6 ; ::_thesis: ( sin . 0 = 0 & cos . (- th) = cos . th & sin . (- th) = - (sin . th) ) thus sin . 0 = Im (Sum ((0 * ) ExpSeq)) by Def16 .= 0 by Th9, COMPLEX1:6 ; ::_thesis: ( cos . (- th) = cos . th & sin . (- th) = - (sin . th) ) reconsider th1 = th as Real by XREAL_0:def_1; thus cos . (- th) = Re (Sum (((- 0) + ((- th1) * )) ExpSeq)) by Def18 .= Re (Sum ((- (th1 * )) ExpSeq)) .= Re ((Sum ((th1 * ) ExpSeq)) *') by Lm4 .= Re (((cos . th) + ((sin . th) * )) *') by Lm3 .= Re ((cos . th) + ((- (sin . th)) * )) by Lm1 .= cos . th by COMPLEX1:12 ; ::_thesis: sin . (- th) = - (sin . th) thus sin . (- th) = Im (Sum (((- 0) + ((- th1) * )) ExpSeq)) by Def16 .= Im (Sum ((- (th1 * )) ExpSeq)) .= Im ((Sum ((th1 * ) ExpSeq)) *') by Lm4 .= Im (((cos . th) + ((sin . th) * )) *') by Lm3 .= Im ((cos . th) + ((- (sin . th)) * )) by Lm1 .= - (sin . th) by COMPLEX1:12 ; ::_thesis: verum end; theorem :: SIN_COS:31 for th being real number holds ( cos 0 = 1 & sin 0 = 0 & cos (- th) = cos th & sin (- th) = - (sin th) ) by Th30; definition let th be real number ; deffunc H1( Element of NAT ) -> Element of REAL = (((- 1) |^ \$1) * (th |^ ((2 * \$1) + 1))) / (((2 * \$1) + 1) !); functh P_sin -> Real_Sequence means :Def20: :: SIN_COS:def 20 for n being Element of NAT holds it . n = (((- 1) |^ n) * (th |^ ((2 * n) + 1))) / (((2 * n) + 1) !); existence ex b1 being Real_Sequence st for n being Element of NAT holds b1 . n = (((- 1) |^ n) * (th |^ ((2 * n) + 1))) / (((2 * n) + 1) !) proof thus ex s being Real_Sequence st for n being Element of NAT holds s . n = H1(n) from SEQ_1:sch_1(); ::_thesis: verum end; uniqueness for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = (((- 1) |^ n) * (th |^ ((2 * n) + 1))) / (((2 * n) + 1) !) ) & ( for n being Element of NAT holds b2 . n = (((- 1) |^ n) * (th |^ ((2 * n) + 1))) / (((2 * n) + 1) !) ) holds b1 = b2 proof thus for s1, s2 being Real_Sequence st ( for x being Element of NAT holds s1 . x = H1(x) ) & ( for x being Element of NAT holds s2 . x = H1(x) ) holds s1 = s2 from BINOP_2:sch_1(); ::_thesis: verum end; deffunc H2( Element of NAT ) -> Element of REAL = (((- 1) |^ \$1) * (th |^ (2 * \$1))) / ((2 * \$1) !); functh P_cos -> Real_Sequence means :Def21: :: SIN_COS:def 21 for n being Element of NAT holds it . n = (((- 1) |^ n) * (th |^ (2 * n))) / ((2 * n) !); existence ex b1 being Real_Sequence st for n being Element of NAT holds b1 . n = (((- 1) |^ n) * (th |^ (2 * n))) / ((2 * n) !) proof thus ex s being Real_Sequence st for n being Element of NAT holds s . n = H2(n) from SEQ_1:sch_1(); ::_thesis: verum end; uniqueness for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = (((- 1) |^ n) * (th |^ (2 * n))) / ((2 * n) !) ) & ( for n being Element of NAT holds b2 . n = (((- 1) |^ n) * (th |^ (2 * n))) / ((2 * n) !) ) holds b1 = b2 proof thus for s1, s2 being Real_Sequence st ( for x being Element of NAT holds s1 . x = H2(x) ) & ( for x being Element of NAT holds s2 . x = H2(x) ) holds s1 = s2 from BINOP_2:sch_1(); ::_thesis: verum end; end; :: deftheorem Def20 defines P_sin SIN_COS:def_20_:_ for th being real number for b2 being Real_Sequence holds ( b2 = th P_sin iff for n being Element of NAT holds b2 . n = (((- 1) |^ n) * (th |^ ((2 * n) + 1))) / (((2 * n) + 1) !) ); :: deftheorem Def21 defines P_cos SIN_COS:def_21_:_ for th being real number for b2 being Real_Sequence holds ( b2 = th P_cos iff for n being Element of NAT holds b2 . n = (((- 1) |^ n) * (th |^ (2 * n))) / ((2 * n) !) ); theorem Th32: :: SIN_COS:32 for z being complex number for k being Element of NAT holds ( z |^ (2 * k) = (z |^ k) |^ 2 & z |^ (2 * k) = (z |^ 2) |^ k ) proof let z be complex number ; ::_thesis: for k being Element of NAT holds ( z |^ (2 * k) = (z |^ k) |^ 2 & z |^ (2 * k) = (z |^ 2) |^ k ) let k be Element of NAT ; ::_thesis: ( z |^ (2 * k) = (z |^ k) |^ 2 & z |^ (2 * k) = (z |^ 2) |^ k ) defpred S1[ Element of NAT ] means ( z |^ (2 * \$1) = (z |^ \$1) |^ 2 & z |^ (2 * \$1) = (z |^ 2) |^ \$1 ); A1: z |^ (2 * 0) = 1 * 1 by NEWTON:4 .= (1 |^ 1) * 1 by NEWTON:5 .= 1 |^ (1 + 1) by NEWTON:6 .= (z |^ 0) |^ 2 by NEWTON:4 ; z |^ (2 * 0) = 1 by NEWTON:4 .= (z |^ 2) |^ 0 by NEWTON:4 ; then A2: S1[ 0 ] by A1; A3: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume that A4: z |^ (2 * k) = (z |^ k) |^ 2 and A5: z |^ (2 * k) = (z |^ 2) |^ k ; ::_thesis: S1[k + 1] A6: z |^ (2 * (k + 1)) = (z GeoSeq) . (((2 * k) + 1) + 1) .= ((z GeoSeq) . ((2 * k) + 1)) * z by COMSEQ_3:def_1 .= ((z |^ (2 * k)) * z) * z by COMSEQ_3:def_1 ; then A7: z |^ (2 * (k + 1)) = ((((z |^ k) GeoSeq) . (1 + 1)) * z) * z by A4 .= (((((z |^ k) GeoSeq) . (0 + 1)) * (z |^ k)) * z) * z by COMSEQ_3:def_1 .= ((((((z |^ k) GeoSeq) . 0) * (z |^ k)) * (z |^ k)) * z) * z by COMSEQ_3:def_1 .= (((1r * (z |^ k)) * (z |^ k)) * z) * z by COMSEQ_3:def_1 .= (1r * (((z GeoSeq) . k) * z)) * ((z |^ k) * z) .= (1r * ((z GeoSeq) . (k + 1))) * (((z GeoSeq) . k) * z) by COMSEQ_3:def_1 .= (1r * (z |^ (k + 1))) * ((z GeoSeq) . (k + 1)) by COMSEQ_3:def_1 .= ((((z |^ (k + 1)) GeoSeq) . 0) * (z |^ (k + 1))) * (z |^ (k + 1)) by COMSEQ_3:def_1 .= (((z |^ (k + 1)) GeoSeq) . (0 + 1)) * (z |^ (k + 1)) by COMSEQ_3:def_1 .= ((z |^ (k + 1)) GeoSeq) . ((0 + 1) + 1) by COMSEQ_3:def_1 .= (z |^ (k + 1)) |^ 2 ; z |^ (2 * (k + 1)) = ((z |^ 2) |^ k) * ((1r * z) * z) by A5, A6 .= ((z |^ 2) |^ k) * ((((z GeoSeq) . 0) * z) * z) by COMSEQ_3:def_1 .= ((z |^ 2) |^ k) * (((z GeoSeq) . (0 + 1)) * z) by COMSEQ_3:def_1 .= ((z |^ 2) |^ k) * ((z GeoSeq) . ((0 + 1) + 1)) by COMSEQ_3:def_1 .= (z |^ 2) |^ (k + 1) by COMSEQ_3:def_1 ; hence S1[k + 1] by A7; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A3); hence ( z |^ (2 * k) = (z |^ k) |^ 2 & z |^ (2 * k) = (z |^ 2) |^ k ) ; ::_thesis: verum end; theorem Th33: :: SIN_COS:33 for k being Element of NAT for th being Real holds ( (th * ) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) & (th * ) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * ) proof let k be Element of NAT ; ::_thesis: for th being Real holds ( (th * ) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) & (th * ) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * ) let th be Real; ::_thesis: ( (th * ) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) & (th * ) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * ) A1: (((- 1) |^ 0) * (th |^ (2 * 0))) * = (((- 1) |^ 0) * ((th GeoSeq) . 0)) * by PREPOWER:def_1 .= (((- 1) |^ 0) * 1) * by PREPOWER:3 .= (((- 1) GeoSeq) . 0) * by PREPOWER:def_1 .= 1 * by PREPOWER:3 ; A2: (th * ) |^ ((2 * 0) + 1) = (((th * ) GeoSeq) . 0) * (th * ) by COMSEQ_3:def_1 .= 1 * (th * ) by COMSEQ_3:def_1 .= th * ; defpred S1[ Element of NAT ] means ( (th * ) |^ (2 * \$1) = ((- 1) |^ \$1) * (th |^ (2 * \$1)) & (th * ) |^ ((2 * \$1) + 1) = (((- 1) |^ \$1) * (th |^ ((2 * \$1) + 1))) * ); (((- 1) |^ 0) * (th |^ ((2 * 0) + 1))) * = (((- 1) |^ 0) * ((th GeoSeq) . ((2 * 0) + 1))) * by PREPOWER:def_1 .= (((- 1) |^ 0) * (((th GeoSeq) . 0) * th)) * by PREPOWER:3 .= (((- 1) |^ 0) * (1 * th)) * by PREPOWER:3 .= ((((- 1) GeoSeq) . 0) * th) * by PREPOWER:def_1 .= (1 * th) * by PREPOWER:3 .= th * ; then A3: S1[ 0 ] by A1, A2, COMSEQ_3:def_1; A4: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume that A5: (th * ) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) and A6: (th * ) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * ; ::_thesis: S1[k + 1] A7: (th * ) |^ (2 * (k + 1)) = ((th * ) |^ 2) |^ (k + 1) by Th32 .= (((th * ) |^ 2) |^ k) * ((th * ) |^ 2) by COMSEQ_3:def_1 .= (((- 1) |^ k) * (th |^ (2 * k))) * (((th * ) GeoSeq) . (1 + 1)) by A5, Th32 .= (((- 1) |^ k) * (th |^ (2 * k))) * ((((th * ) GeoSeq) . (0 + 1)) * (th * )) by COMSEQ_3:def_1 .= (((- 1) |^ k) * (th |^ (2 * k))) * (((((th * ) GeoSeq) . 0) * (th * )) * (th * )) by COMSEQ_3:def_1 .= (((- 1) |^ k) * (th |^ (2 * k))) * ((1r * (th * )) * (th * )) by COMSEQ_3:def_1 .= (((((- 1) |^ k) * (- 1)) * (th |^ (2 * k))) * th) * th .= ((((((- 1) GeoSeq) . k) * (- 1)) * (th |^ (2 * k))) * th) * th by PREPOWER:def_1 .= (((((- 1) GeoSeq) . (k + 1)) * (th |^ (2 * k))) * th) * th by PREPOWER:3 .= ((((- 1) |^ (k + 1)) * (th |^ (2 * k))) * th) * th by PREPOWER:def_1 .= ((((- 1) |^ (k + 1)) * ((th GeoSeq) . (2 * k))) * th) * th by PREPOWER:def_1 .= (((- 1) |^ (k + 1)) * (((th GeoSeq) . (2 * k)) * th)) * th .= (((- 1) |^ (k + 1)) * ((th GeoSeq) . ((2 * k) + 1))) * th by PREPOWER:3 .= ((- 1) |^ (k + 1)) * (((th GeoSeq) . ((2 * k) + 1)) * th) .= ((- 1) |^ (k + 1)) * ((th GeoSeq) . (((2 * k) + 1) + 1)) by PREPOWER:3 .= ((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1))) by PREPOWER:def_1 ; (th * ) |^ ((2 * (k + 1)) + 1) = (((th * ) GeoSeq) . (((2 * k) + 1) + 1)) * (th * ) by COMSEQ_3:def_1 .= ((((th * ) GeoSeq) . ((2 * k) + 1)) * (th * )) * (th * ) by COMSEQ_3:def_1 .= ((((((- 1) |^ k) * (- 1)) * (th |^ ((2 * k) + 1))) * th) * th) * by A6 .= (((((((- 1) GeoSeq) . k) * (- 1)) * (th |^ ((2 * k) + 1))) * th) * th) * by PREPOWER:def_1 .= ((((((- 1) GeoSeq) . (k + 1)) * (th |^ ((2 * k) + 1))) * th) * th) * by PREPOWER:3 .= (((((- 1) |^ (k + 1)) * (th |^ ((2 * k) + 1))) * th) * th) * by PREPOWER:def_1 .= (((((- 1) |^ (k + 1)) * ((th GeoSeq) . ((2 * k) + 1))) * th) * th) * by PREPOWER:def_1 .= ((((- 1) |^ (k + 1)) * (((th GeoSeq) . ((2 * k) + 1)) * th)) * th) * .= ((((- 1) |^ (k + 1)) * ((th GeoSeq) . (((2 * k) + 1) + 1))) * th) * by PREPOWER:3 .= (((- 1) |^ (k + 1)) * (((th GeoSeq) . (((2 * k) + 1) + 1)) * th)) * .= (((- 1) |^ (k + 1)) * ((th GeoSeq) . ((2 * (k + 1)) + 1))) * by PREPOWER:3 .= (((- 1) |^ (k + 1)) * (th |^ ((2 * (k + 1)) + 1))) * by PREPOWER:def_1 ; hence S1[k + 1] by A7; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A3, A4); hence ( (th * ) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) & (th * ) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * ) ; ::_thesis: verum end; theorem :: SIN_COS:34 for n being Element of NAT holds n ! = n ! ; theorem Th35: :: SIN_COS:35 for n being Element of NAT for th being Real holds ( (Partial_Sums (th P_sin)) . n = (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * n) + 1) & (Partial_Sums (th P_cos)) . n = (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * n) ) proof let n be Element of NAT ; ::_thesis: for th being Real holds ( (Partial_Sums (th P_sin)) . n = (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * n) + 1) & (Partial_Sums (th P_cos)) . n = (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * n) ) let th be Real; ::_thesis: ( (Partial_Sums (th P_sin)) . n = (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * n) + 1) & (Partial_Sums (th P_cos)) . n = (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * n) ) now__::_thesis:_for_n_being_Element_of_NAT_holds_S1[n] A1: (Partial_Sums (th P_sin)) . 0 = (th P_sin) . 0 by SERIES_1:def_1 .= (((- 1) |^ 0) * (th |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !) by Def20 ; A2: th |^ ((2 * 0) + 1) = (th GeoSeq) . (0 + 1) by PREPOWER:def_1 .= ((th GeoSeq) . 0) * th by PREPOWER:3 .= 1 * th by PREPOWER:3 .= th ; A3: ((2 * 0) + 1) ! = (0 !) * 1 by NEWTON:15 .= 1 by NEWTON:12 ; A4: (- 1) |^ 0 = ((- 1) GeoSeq) . 0 by PREPOWER:def_1 .= 1 by PREPOWER:3 ; A5: (Partial_Sums (th P_cos)) . 0 = (th P_cos) . 0 by SERIES_1:def_1 .= (((- 1) |^ 0) * (th |^ (2 * 0))) / ((2 * 0) !) by Def21 .= (1 * ((th GeoSeq) . 0)) / 1 by A4, NEWTON:12, PREPOWER:def_1 .= 1 by PREPOWER:3 ; A6: ( (Im ((th * ) ExpSeq)) . 0 = Im (((th * ) ExpSeq) . 0) & (Im ((th * ) ExpSeq)) . 1 = Im (((th * ) ExpSeq) . 1) ) by COMSEQ_3:def_6; A7: th * = 0 + (th * ) ; A8: (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * 0) + 1) = ((Partial_Sums (Im ((th * ) ExpSeq))) . 0) + ((Im ((th * ) ExpSeq)) . 1) by SERIES_1:def_1 .= ((Im ((th * ) ExpSeq)) . 0) + ((Im ((th * ) ExpSeq)) . 1) by SERIES_1:def_1 .= 0 + (Im (((th * ) ExpSeq) . (0 + 1))) by A6, Th3, COMPLEX1:6 .= 0 + (Im (((((th * ) ExpSeq) . 0) * (th * )) / ((0 + 1) + (0 * )))) by Th3 .= Im ((1 * (th * )) / 1) by Th3 .= th by A7, COMPLEX1:12 ; defpred S1[ Element of NAT ] means ( (Partial_Sums (th P_sin)) . \$1 = (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * \$1) + 1) & (Partial_Sums (th P_cos)) . \$1 = (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * \$1) ); (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * 0) = (Re ((th * ) ExpSeq)) . 0 by SERIES_1:def_1 .= Re (((th * ) ExpSeq) . 0) by COMSEQ_3:def_5 .= 1 by Th3, COMPLEX1:6 ; then A9: S1[ 0 ] by A1, A2, A3, A4, A5, A8; A10: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume that A11: (Partial_Sums (th P_sin)) . k = (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * k) + 1) and A12: (Partial_Sums (th P_cos)) . k = (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * k) ; ::_thesis: S1[k + 1] (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * (k + 1)) + 1) = ((Partial_Sums (Im ((th * ) ExpSeq))) . (((2 * k) + 1) + 1)) + ((Im ((th * ) ExpSeq)) . ((2 * (k + 1)) + 1)) by SERIES_1:def_1 .= (((Partial_Sums (th P_sin)) . k) + ((Im ((th * ) ExpSeq)) . (2 * (k + 1)))) + ((Im ((th * ) ExpSeq)) . ((2 * (k + 1)) + 1)) by A11, SERIES_1:def_1 .= (((Partial_Sums (th P_sin)) . k) + (Im (((th * ) ExpSeq) . (2 * (k + 1))))) + ((Im ((th * ) ExpSeq)) . ((2 * (k + 1)) + 1)) by COMSEQ_3:def_6 .= (((Partial_Sums (th P_sin)) . k) + (Im (((th * ) ExpSeq) . (2 * (k + 1))))) + (Im (((th * ) ExpSeq) . ((2 * (k + 1)) + 1))) by COMSEQ_3:def_6 .= (((Partial_Sums (th P_sin)) . k) + (Im (((th * ) |^ (2 * (k + 1))) / ((2 * (k + 1)) !)))) + (Im (((th * ) ExpSeq) . ((2 * (k + 1)) + 1))) by Def4 .= (((Partial_Sums (th P_sin)) . k) + (Im (((th * ) |^ (2 * (k + 1))) / ((2 * (k + 1)) !)))) + (Im (((th * ) |^ ((2 * (k + 1)) + 1)) / (((2 * (k + 1)) + 1) !))) by Def4 .= (((Partial_Sums (th P_sin)) . k) + (Im ((((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1)))) / ((2 * (k + 1)) !)))) + (Im (((th * ) |^ ((2 * (k + 1)) + 1)) / (((2 * (k + 1)) + 1) !))) by Th33 .= (((Partial_Sums (th P_sin)) . k) + (Im ((((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1)))) / ((2 * (k + 1)) !)))) + (Im (((((- 1) |^ (k + 1)) * (th |^ ((2 * (k + 1)) + 1))) * ) / (((2 * (k + 1)) + 1) !))) by Th33 ; then (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * (k + 1)) + 1) = (((Partial_Sums (th P_sin)) . k) + (Im (((((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1)))) / ((2 * (k + 1)) !)) + ((0 / ((2 * (k + 1)) !)) * )))) + (Im (((((- 1) |^ (k + 1)) * (th |^ ((2 * (k + 1)) + 1))) * ) / (((2 * (k + 1)) + 1) !))) ; then (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * (k + 1)) + 1) = (((Partial_Sums (th P_sin)) . k) + 0) + (Im ((0 / (((2 * (k + 1)) + 1) !)) + (((((- 1) |^ (k + 1)) * (th |^ ((2 * (k + 1)) + 1))) / (((2 * (k + 1)) + 1) !)) * ))) by COMPLEX1:12 .= (((Partial_Sums (th P_sin)) . k) + (0 / ((2 * (k + 1)) !))) + ((((- 1) |^ (k + 1)) * (th |^ ((2 * (k + 1)) + 1))) / (((2 * (k + 1)) + 1) !)) by COMPLEX1:12 ; then A13: (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * (k + 1)) + 1) = ((Partial_Sums (th P_sin)) . k) + ((th P_sin) . (k + 1)) by Def20 .= (Partial_Sums (th P_sin)) . (k + 1) by SERIES_1:def_1 ; (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * (k + 1)) = ((Partial_Sums (Re ((th * ) ExpSeq))) . ((2 * k) + 1)) + ((Re ((th * ) ExpSeq)) . (((2 * k) + 1) + 1)) by SERIES_1:def_1 .= (((Partial_Sums (th P_cos)) . k) + ((Re ((th * ) ExpSeq)) . ((2 * k) + 1))) + ((Re ((th * ) ExpSeq)) . (((2 * k) + 1) + 1)) by A12, SERIES_1:def_1 .= (((Partial_Sums (th P_cos)) . k) + (Re (((th * ) ExpSeq) . ((2 * k) + 1)))) + ((Re ((th * ) ExpSeq)) . (((2 * k) + 1) + 1)) by COMSEQ_3:def_5 .= (((Partial_Sums (th P_cos)) . k) + (Re (((th * ) ExpSeq) . ((2 * k) + 1)))) + (Re (((th * ) ExpSeq) . (((2 * k) + 1) + 1))) by COMSEQ_3:def_5 .= (((Partial_Sums (th P_cos)) . k) + (Re (((th * ) |^ ((2 * k) + 1)) / (((2 * k) + 1) !)))) + (Re (((th * ) ExpSeq) . (((2 * k) + 1) + 1))) by Def4 .= (((Partial_Sums (th P_cos)) . k) + (Re (((th * ) |^ ((2 * k) + 1)) / (((2 * k) + 1) !)))) + (Re (((th * ) |^ (2 * (k + 1))) / ((((2 * k) + 1) + 1) !))) by Def4 .= (((Partial_Sums (th P_cos)) . k) + (Re (((((- 1) |^ k) * (th |^ ((2 * k) + 1))) * ) / (((2 * k) + 1) !)))) + (Re (((th * ) |^ (2 * (k + 1))) / ((2 * (k + 1)) !))) by Th33 .= (((Partial_Sums (th P_cos)) . k) + (Re (((((- 1) |^ k) * (th |^ ((2 * k) + 1))) * ) / (((2 * k) + 1) !)))) + (Re ((((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1)))) / ((2 * (k + 1)) !))) by Th33 ; then (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * (k + 1)) = (((Partial_Sums (th P_cos)) . k) + (Re ((0 / (((2 * k) + 1) !)) + (((((- 1) |^ k) * (th |^ ((2 * k) + 1))) / (((2 * k) + 1) !)) * )))) + (Re ((((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1)))) / ((2 * (k + 1)) !))) ; then (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * (k + 1)) = (((Partial_Sums (th P_cos)) . k) + (0 / (((2 * k) + 1) !))) + (Re (((((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1)))) / ((2 * (k + 1)) !)) + ((0 / ((2 * (k + 1)) !)) * ))) by COMPLEX1:12 .= (((Partial_Sums (th P_cos)) . k) + (0 / (((2 * k) + 1) !))) + ((((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1)))) / ((2 * (k + 1)) !)) by COMPLEX1:12 ; then (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * (k + 1)) = ((Partial_Sums (th P_cos)) . k) + ((th P_cos) . (k + 1)) by Def21 .= (Partial_Sums (th P_cos)) . (k + 1) by SERIES_1:def_1 ; hence S1[k + 1] by A13; ::_thesis: verum end; thus for n being Element of NAT holds S1[n] from NAT_1:sch_1(A9, A10); ::_thesis: verum end; hence ( (Partial_Sums (th P_sin)) . n = (Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * n) + 1) & (Partial_Sums (th P_cos)) . n = (Partial_Sums (Re ((th * ) ExpSeq))) . (2 * n) ) ; ::_thesis: verum end; theorem Th36: :: SIN_COS:36 for th being Real holds ( Partial_Sums (th P_sin) is convergent & Sum (th P_sin) = Im (Sum ((th * ) ExpSeq)) & Partial_Sums (th P_cos) is convergent & Sum (th P_cos) = Re (Sum ((th * ) ExpSeq)) ) proof let th be Real; ::_thesis: ( Partial_Sums (th P_sin) is convergent & Sum (th P_sin) = Im (Sum ((th * ) ExpSeq)) & Partial_Sums (th P_cos) is convergent & Sum (th P_cos) = Re (Sum ((th * ) ExpSeq)) ) A1: Sum ((th * ) ExpSeq) = (Sum (Re ((th * ) ExpSeq))) + ((Sum (Im ((th * ) ExpSeq))) * ) by COMSEQ_3:53; then A2: Sum (Re ((th * ) ExpSeq)) = Re (Sum ((th * ) ExpSeq)) by COMPLEX1:12; A3: Sum (Im ((th * ) ExpSeq)) = Im (Sum ((th * ) ExpSeq)) by A1, COMPLEX1:12; A4: ( Partial_Sums (Re ((th * ) ExpSeq)) is convergent & lim (Partial_Sums (Re ((th * ) ExpSeq))) = Re (Sum ((th * ) ExpSeq)) ) by A2, SERIES_1:def_2, SERIES_1:def_3; A5: ( Partial_Sums (Im ((th * ) ExpSeq)) is convergent & lim (Partial_Sums (Im ((th * ) ExpSeq))) = Im (Sum ((th * ) ExpSeq)) ) by A3, SERIES_1:def_2, SERIES_1:def_3; A6: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_ ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_(((Partial_Sums_(th_P_cos))_._k)_-_(Re_(Sum_((th_*_)_ExpSeq))))_<_p let p be real number ; ::_thesis: ( p > 0 implies ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * ) ExpSeq)))) < p ) assume p > 0 ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * ) ExpSeq)))) < p then consider n being Element of NAT such that A7: for k being Element of NAT st n <= k holds abs (((Partial_Sums (Re ((th * ) ExpSeq))) . k) - (Re (Sum ((th * ) ExpSeq)))) < p by A4, SEQ_2:def_7; take n = n; ::_thesis: for k being Element of NAT st n <= k holds abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * ) ExpSeq)))) < p now__::_thesis:_for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_(((Partial_Sums_(th_P_cos))_._k)_-_(Re_(Sum_((th_*_)_ExpSeq))))_<_p let k be Element of NAT ; ::_thesis: ( n <= k implies abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * ) ExpSeq)))) < p ) assume A8: n <= k ; ::_thesis: abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * ) ExpSeq)))) < p ( abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * ) ExpSeq)))) = abs (((Partial_Sums (Re ((th * ) ExpSeq))) . (2 * k)) - (Re (Sum ((th * ) ExpSeq)))) & 2 * k = k + k ) by Th35; hence abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * ) ExpSeq)))) < p by A7, A8, NAT_1:12; ::_thesis: verum end; hence for k being Element of NAT st n <= k holds abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * ) ExpSeq)))) < p ; ::_thesis: verum end; then Partial_Sums (th P_cos) is convergent by SEQ_2:def_6; then A9: lim (Partial_Sums (th P_cos)) = Re (Sum ((th * ) ExpSeq)) by A6, SEQ_2:def_7; A10: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_ ex_n_being_Element_of_NAT_st_ for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_(((Partial_Sums_(th_P_sin))_._k)_-_(Im_(Sum_((th_*_)_ExpSeq))))_<_p let p be real number ; ::_thesis: ( p > 0 implies ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * ) ExpSeq)))) < p ) assume p > 0 ; ::_thesis: ex n being Element of NAT st for k being Element of NAT st n <= k holds abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * ) ExpSeq)))) < p then consider n being Element of NAT such that A11: for k being Element of NAT st n <= k holds abs (((Partial_Sums (Im ((th * ) ExpSeq))) . k) - (Im (Sum ((th * ) ExpSeq)))) < p by A5, SEQ_2:def_7; take n = n; ::_thesis: for k being Element of NAT st n <= k holds abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * ) ExpSeq)))) < p now__::_thesis:_for_k_being_Element_of_NAT_st_n_<=_k_holds_ abs_(((Partial_Sums_(th_P_sin))_._k)_-_(Im_(Sum_((th_*_)_ExpSeq))))_<_p let k be Element of NAT ; ::_thesis: ( n <= k implies abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * ) ExpSeq)))) < p ) assume A12: n <= k ; ::_thesis: abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * ) ExpSeq)))) < p A13: abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * ) ExpSeq)))) = abs (((Partial_Sums (Im ((th * ) ExpSeq))) . ((2 * k) + 1)) - (Im (Sum ((th * ) ExpSeq)))) by Th35; 2 * k = k + k ; then n <= 2 * k by A12, NAT_1:12; then n < (2 * k) + 1 by NAT_1:13; hence abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * ) ExpSeq)))) < p by A11, A13; ::_thesis: verum end; hence for k being Element of NAT st n <= k holds abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * ) ExpSeq)))) < p ; ::_thesis: verum end; then Partial_Sums (th P_sin) is convergent by SEQ_2:def_6; then lim (Partial_Sums (th P_sin)) = Im (Sum ((th * ) ExpSeq)) by A10, SEQ_2:def_7; hence ( Partial_Sums (th P_sin) is convergent & Sum (th P_sin) = Im (Sum ((th * ) ExpSeq)) & Partial_Sums (th P_cos) is convergent & Sum (th P_cos) = Re (Sum ((th * ) ExpSeq)) ) by A6, A9, A10, SEQ_2:def_6, SERIES_1:def_3; ::_thesis: verum end; theorem Th37: :: SIN_COS:37 for th being real number holds ( cos . th = Sum (th P_cos) & sin . th = Sum (th P_sin) ) proof let th be real number ; ::_thesis: ( cos . th = Sum (th P_cos) & sin . th = Sum (th P_sin) ) reconsider th = th as Real by XREAL_0:def_1; ( sin . th = Im (Sum ((th * ) ExpSeq)) & cos . th = Re (Sum ((th * ) ExpSeq)) ) by Def16, Def18; hence ( cos . th = Sum (th P_cos) & sin . th = Sum (th P_sin) ) by Th36; ::_thesis: verum end; theorem :: SIN_COS:38 for p, th being real number for rseq being Real_Sequence st rseq is convergent & lim rseq = th & ( for n being Element of NAT holds rseq . n >= p ) holds th >= p by PREPOWER:1; deffunc H1( Real) -> Element of REAL = (2 * \$1) + 1; consider bq being Real_Sequence such that Lm6: for n being Element of NAT holds bq . n = H1(n) from SEQ_1:sch_1(); bq is increasing sequence of NAT proof A1: for n, k being Element of NAT st n < k holds bq . n < bq . k proof let n, k be Element of NAT ; ::_thesis: ( n < k implies bq . n < bq . k ) assume n < k ; ::_thesis: bq . n < bq . k then 2 * n < 2 * k by XREAL_1:68; then (2 * n) + 1 < (2 * k) + 1 by XREAL_1:6; then bq . n < (2 * k) + 1 by Lm6; hence bq . n < bq . k by Lm6; ::_thesis: verum end; for n being Element of NAT holds bq . n is Element of NAT proof let n be Element of NAT ; ::_thesis: bq . n is Element of NAT (2 * n) + 1 is Element of NAT ; hence bq . n is Element of NAT by Lm6; ::_thesis: verum end; hence bq is increasing sequence of NAT by A1, SEQM_3:1, SEQM_3:13; ::_thesis: verum end; then reconsider bq = bq as increasing sequence of NAT ; Lm7: for n being Element of NAT for th, th1, th2, th3 being real number holds ( th |^ (2 * n) = (th |^ n) |^ 2 & (- 1) |^ (2 * n) = 1 & (- 1) |^ ((2 * n) + 1) = - 1 ) proof let n be Element of NAT ; ::_thesis: for th, th1, th2, th3 being real number holds ( th |^ (2 * n) = (th |^ n) |^ 2 & (- 1) |^ (2 * n) = 1 & (- 1) |^ ((2 * n) + 1) = - 1 ) let th, th1, th2, th3 be real number ; ::_thesis: ( th |^ (2 * n) = (th |^ n) |^ 2 & (- 1) |^ (2 * n) = 1 & (- 1) |^ ((2 * n) + 1) = - 1 ) thus th |^ (2 * n) = (th |^ n) |^ 2 by NEWTON:9; ::_thesis: ( (- 1) |^ (2 * n) = 1 & (- 1) |^ ((2 * n) + 1) = - 1 ) thus A1: (- 1) |^ (2 * n) = 1 |^ (2 * n) by POWER:1 .= 1 by NEWTON:10 ; ::_thesis: (- 1) |^ ((2 * n) + 1) = - 1 thus (- 1) |^ ((2 * n) + 1) = ((- 1) |^ (2 * n)) * (- 1) by NEWTON:6 .= - 1 by A1 ; ::_thesis: verum end; theorem Th39: :: SIN_COS:39 for n, k being Element of NAT st n <= k holds n ! <= k ! proof let n, k be Element of NAT ; ::_thesis: ( n <= k implies n ! <= k ! ) assume A1: n <= k ; ::_thesis: n ! <= k ! A2: for m being Element of NAT holds 1 <= m + 1 proof let m be Element of NAT ; ::_thesis: 1 <= m + 1 0 + 1 <= m + 1 by XREAL_1:6; hence 1 <= m + 1 ; ::_thesis: verum end; deffunc H2( Element of NAT ) -> Element of REAL = \$1 ! ; consider rseq being Real_Sequence such that A3: for l being Element of NAT holds rseq . l = H2(l) from SEQ_1:sch_1(); for l being Element of NAT holds ( rseq . l <= rseq . (l + 1) & rseq . l > 0 ) proof let l be Element of NAT ; ::_thesis: ( rseq . l <= rseq . (l + 1) & rseq . l > 0 ) defpred S1[ Element of NAT ] means ( rseq . \$1 <= rseq . (\$1 + 1) & rseq . \$1 > 0 ); rseq . (0 + 1) = (0 + 1) ! by A3 .= (0 !) * 1 by NEWTON:15 .= 1 by NEWTON:12 ; then A4: S1[ 0 ] by A3, NEWTON:12; A5: for l being Element of NAT st S1[l] holds S1[l + 1] proof let l be Element of NAT ; ::_thesis: ( S1[l] implies S1[l + 1] ) assume A6: ( rseq . l <= rseq . (l + 1) & rseq . l > 0 ) ; ::_thesis: S1[l + 1] rseq . ((l + 1) + 1) = ((l + 1) + 1) ! by A3 .= ((l + 1) !) * ((l + 1) + 1) by NEWTON:15 .= (rseq . (l + 1)) * ((l + 1) + 1) by A3 ; hence S1[l + 1] by A2, A6, XREAL_1:151; ::_thesis: verum end; for l being Element of NAT holds S1[l] from NAT_1:sch_1(A4, A5); hence ( rseq . l <= rseq . (l + 1) & rseq . l > 0 ) ; ::_thesis: verum end; then A7: rseq is non-decreasing by SEQM_3:def_8; ( n ! = rseq . n & k ! = rseq . k ) by A3; hence n ! <= k ! by A1, A7, SEQM_3:6; ::_thesis: verum end; theorem Th40: :: SIN_COS:40 for n, k being Element of NAT for th being real number st 0 <= th & th <= 1 & n <= k holds th |^ k <= th |^ n proof let n, k be Element of NAT ; ::_thesis: for th being real number st 0 <= th & th <= 1 & n <= k holds th |^ k <= th |^ n let th be real number ; ::_thesis: ( 0 <= th & th <= 1 & n <= k implies th |^ k <= th |^ n ) assume that A1: 0 <= th and A2: th <= 1 and A3: n <= k ; ::_thesis: th |^ k <= th |^ n for m being Element of NAT holds ( (th GeoSeq) . (m + 1) <= (th GeoSeq) . m & (th GeoSeq) . m >= 0 ) proof let m be Element of NAT ; ::_thesis: ( (th GeoSeq) . (m + 1) <= (th GeoSeq) . m & (th GeoSeq) . m >= 0 ) defpred S1[ Element of NAT ] means ( (th GeoSeq) . (\$1 + 1) <= (th GeoSeq) . \$1 & (th GeoSeq) . \$1 >= 0 ); (th GeoSeq) . (0 + 1) = ((th GeoSeq) . 0) * th by PREPOWER:3 .= 1 * th by PREPOWER:3 .= th ; then A4: S1[ 0 ] by A2, PREPOWER:3; A5: for m being Element of NAT st S1[m] holds S1[m + 1] proof let m be Element of NAT ; ::_thesis: ( S1[m] implies S1[m + 1] ) assume that (th GeoSeq) . (m + 1) <= (th GeoSeq) . m and A6: (th GeoSeq) . m >= 0 ; ::_thesis: S1[m + 1] ( (th GeoSeq) . ((m + 1) + 1) = ((th GeoSeq) . (m + 1)) * th & (th GeoSeq) . (m + 1) = ((th GeoSeq) . m) * th ) by PREPOWER:3; hence S1[m + 1] by A1, A2, A6, XREAL_1:153; ::_thesis: verum end; for m being Element of NAT holds S1[m] from NAT_1:sch_1(A4, A5); hence ( (th GeoSeq) . (m + 1) <= (th GeoSeq) . m & (th GeoSeq) . m >= 0 ) ; ::_thesis: verum end; then A7: th GeoSeq is non-increasing by SEQM_3:def_9; ( th |^ k = (th GeoSeq) . k & th |^ n = (th GeoSeq) . n ) by PREPOWER:def_1; hence th |^ k <= th |^ n by A3, A7, SEQM_3:8; ::_thesis: verum end; theorem :: SIN_COS:41 for n being Element of NAT for th being Real holds (th |^ n) / (n !) = (th |^ n) / (n !) ; theorem Th42: :: SIN_COS:42 for p being real number holds Im (Sum (p ExpSeq)) = 0 proof let p be real number ; ::_thesis: Im (Sum (p ExpSeq)) = 0 A1: for n being Element of NAT holds (Partial_Sums (Im (p ExpSeq))) . n = 0 proof let n be Element of NAT ; ::_thesis: (Partial_Sums (Im (p ExpSeq))) . n = 0 defpred S1[ Element of NAT ] means (Partial_Sums (Im (p ExpSeq))) . \$1 = 0 ; (Partial_Sums (Im (p ExpSeq))) . 0 = (Im (p ExpSeq)) . 0 by SERIES_1:def_1 .= Im ((p ExpSeq) . 0) by COMSEQ_3:def_6 .= 0 by Th3, COMPLEX1:6 ; then A2: S1[ 0 ] ; A3: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume (Partial_Sums (Im (p ExpSeq))) . k = 0 ; ::_thesis: S1[k + 1] then (Partial_Sums (Im (p ExpSeq))) . (k + 1) = 0 + ((Im (p ExpSeq)) . (k + 1)) by SERIES_1:def_1 .= Im ((p ExpSeq) . (k + 1)) by COMSEQ_3:def_6 .= Im ((p |^ (k + 1)) / ((k + 1) !)) by Def4 .= Im (((p |^ (k + 1)) / ((k + 1) !)) + (0 * )) .= 0 by COMPLEX1:12 ; hence S1[k + 1] ; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A2, A3); hence (Partial_Sums (Im (p ExpSeq))) . n = 0 ; ::_thesis: verum end; for n, m being Nat holds (Partial_Sums (Im (p ExpSeq))) . n = (Partial_Sums (Im (p ExpSeq))) . m proof let n, m be Nat; ::_thesis: (Partial_Sums (Im (p ExpSeq))) . n = (Partial_Sums (Im (p ExpSeq))) . m n in NAT by ORDINAL1:def_12; then ( m in NAT & (Partial_Sums (Im (p ExpSeq))) . n = 0 ) by A1, ORDINAL1:def_12; hence (Partial_Sums (Im (p ExpSeq))) . n = (Partial_Sums (Im (p ExpSeq))) . m by A1; ::_thesis: verum end; then A4: lim (Partial_Sums (Im (p ExpSeq))) = (Partial_Sums (Im (p ExpSeq))) . 0 by SEQ_4:26, VALUED_0:24 .= 0 by A1 ; Im (Sum (p ExpSeq)) = Im ((Sum (Re (p ExpSeq))) + ((Sum (Im (p ExpSeq))) * )) by COMSEQ_3:53 .= Sum (Im (p ExpSeq)) by COMPLEX1:12 ; hence Im (Sum (p ExpSeq)) = 0 by A4, SERIES_1:def_3; ::_thesis: verum end; theorem Th43: :: SIN_COS:43 ( cos . 1 > 0 & sin . 1 > 0 & cos . 1 < sin . 1 ) proof A1: Partial_Sums (1 P_cos) is convergent by Th36; A2: cos . 1 = Sum (1 P_cos) by Th37; lim ((Partial_Sums (1 P_cos)) * bq) = lim (Partial_Sums (1 P_cos)) by A1, SEQ_4:17; then A3: lim ((Partial_Sums (1 P_cos)) * bq) = cos . 1 by A2, SERIES_1:def_3; for n being Element of NAT holds ((Partial_Sums (1 P_cos)) * bq) . n >= 1 / 2 proof let n be Element of NAT ; ::_thesis: ((Partial_Sums (1 P_cos)) * bq) . n >= 1 / 2 defpred S1[ Element of NAT ] means ((Partial_Sums (1 P_cos)) * bq) . \$1 >= 1 / 2; ((Partial_Sums (1 P_cos)) * bq) . 0 = (Partial_Sums (1 P_cos)) . (bq . 0) by FUNCT_2:15 .= (Partial_Sums (1 P_cos)) . ((2 * 0) + 1) by Lm6 .= ((Partial_Sums (1 P_cos)) . 0) + ((1 P_cos) . (0 + 1)) by SERIES_1:def_1 .= ((1 P_cos) . 0) + ((1 P_cos) . (0 + 1)) by SERIES_1:def_1 .= ((((- 1) |^ 0) * (1 |^ (2 * 0))) / ((2 * 0) !)) + ((1 P_cos) . 1) by Def21 .= ((((- 1) |^ 0) * (1 |^ (2 * 0))) / ((2 * 0) !)) + ((((- 1) |^ 1) * (1 |^ (2 * 1))) / ((2 * 1) !)) by Def21 .= ((1 * (1 |^ (2 * 0))) / ((2 * 0) !)) + ((((- 1) |^ 1) * (1 |^ (2 * 1))) / ((2 * 1) !)) by Lm7 .= (1 / 1) + ((((- 1) |^ 1) * (1 |^ (2 * 1))) / ((2 * 1) !)) by NEWTON:4, NEWTON:12 .= 1 + (((- 1) * (1 |^ (2 * 1))) / ((2 * 1) !)) by NEWTON:5 .= 1 + (((- 1) * 1) / ((2 * 1) !)) by NEWTON:10 .= 1 + ((- 1) / ((1 !) * (1 + 1))) by NEWTON:15 .= 1 + ((- 1) / (((0 !) * (0 + 1)) * 2)) by NEWTON:15 .= 1 / 2 by NEWTON:12 ; then A4: S1[ 0 ] ; A5: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A6: ((Partial_Sums (1 P_cos)) * bq) . k >= 1 / 2 ; ::_thesis: S1[k + 1] ((Partial_Sums (1 P_cos)) * bq) . (k + 1) = (Partial_Sums (1 P_cos)) . (bq . (k + 1)) by FUNCT_2:15 .= (Partial_Sums (1 P_cos)) . ((2 * (k + 1)) + 1) by Lm6 .= ((Partial_Sums (1 P_cos)) . (((2 * k) + 1) + 1)) + ((1 P_cos) . ((2 * (k + 1)) + 1)) by SERIES_1:def_1 .= (((Partial_Sums (1 P_cos)) . ((2 * k) + 1)) + ((1 P_cos) . (((2 * k) + 1) + 1))) + ((1 P_cos) . ((2 * (k + 1)) + 1)) by SERIES_1:def_1 .= (((Partial_Sums (1 P_cos)) . (bq . k)) + ((1 P_cos) . (((2 * k) + 1) + 1))) + ((1 P_cos) . ((2 * (k + 1)) + 1)) by Lm6 .= ((((Partial_Sums (1 P_cos)) * bq) . k) + ((1 P_cos) . (((2 * k) + 1) + 1))) + ((1 P_cos) . ((2 * (k + 1)) + 1)) by FUNCT_2:15 ; then A7: (((Partial_Sums (1 P_cos)) * bq) . (k + 1)) - (((Partial_Sums (1 P_cos)) * bq) . k) = ((1 P_cos) . (((2 * k) + 1) + 1)) + ((1 P_cos) . ((2 * (k + 1)) + 1)) ; A8: (1 P_cos) . (((2 * k) + 1) + 1) = (((- 1) |^ (2 * (k + 1))) * (1 |^ (2 * (2 * (k + 1))))) / ((2 * (2 * (k + 1))) !) by Def21 .= (1 * (1 |^ (2 * (2 * (k + 1))))) / ((2 * (2 * (k + 1))) !) by Lm7 .= 1 / ((2 * (2 * (k + 1))) !) by NEWTON:10 ; A9: (1 P_cos) . ((2 * (k + 1)) + 1) = (((- 1) |^ ((2 * (k + 1)) + 1)) * (1 |^ (2 * ((2 * (k + 1)) + 1)))) / ((2 * ((2 * (k + 1)) + 1)) !) by Def21 .= ((- 1) * (1 |^ (2 * ((2 * (k + 1)) + 1)))) / ((2 * ((2 * (k + 1)) + 1)) !) by Lm7 .= ((- 1) * 1) / ((2 * ((2 * (k + 1)) + 1)) !) by NEWTON:10 .= (- 1) / ((2 * ((2 * (k + 1)) + 1)) !) ; 2 * (2 * (k + 1)) < 2 * ((2 * (k + 1)) + 1) by XREAL_1:29, XREAL_1:68; then (2 * (2 * (k + 1))) ! <= (2 * ((2 * (k + 1)) + 1)) ! by Th39; then 1 / ((2 * (2 * (k + 1))) !) >= 1 / ((2 * ((2 * (k + 1)) + 1)) !) by XREAL_1:85; then (1 / ((2 * (2 * (k + 1))) !)) - (1 / ((2 * ((2 * (k + 1)) + 1)) !)) >= 0 by XREAL_1:48; then ((Partial_Sums (1 P_cos)) * bq) . (k + 1) >= ((Partial_Sums (1 P_cos)) * bq) . k by A7, A8, A9, XREAL_1:49; hence S1[k + 1] by A6, XXREAL_0:2; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A4, A5); hence ((Partial_Sums (1 P_cos)) * bq) . n >= 1 / 2 ; ::_thesis: verum end; then A10: cos . 1 >= 1 / 2 by A1, A3, PREPOWER:1, SEQ_4:16; A11: Partial_Sums (1 P_sin) is convergent by Th36; A12: sin . 1 = Sum (1 P_sin) by Th37; lim ((Partial_Sums (1 P_sin)) * bq) = lim (Partial_Sums (1 P_sin)) by A11, SEQ_4:17; then A13: lim ((Partial_Sums (1 P_sin)) * bq) = sin . 1 by A12, SERIES_1:def_3; for n being Element of NAT holds ((Partial_Sums (1 P_sin)) * bq) . n >= 5 / 6 proof let n be Element of NAT ; ::_thesis: ((Partial_Sums (1 P_sin)) * bq) . n >= 5 / 6 defpred S1[ Element of NAT ] means ((Partial_Sums (1 P_sin)) * bq) . \$1 >= 5 / 6; ((Partial_Sums (1 P_sin)) * bq) . 0 = (Partial_Sums (1 P_sin)) . (bq . 0) by FUNCT_2:15 .= (Partial_Sums (1 P_sin)) . ((2 * 0) + 1) by Lm6 .= ((Partial_Sums (1 P_sin)) . 0) + ((1 P_sin) . (0 + 1)) by SERIES_1:def_1 .= ((1 P_sin) . 0) + ((1 P_sin) . (0 + 1)) by SERIES_1:def_1 .= ((((- 1) |^ 0) * (1 |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !)) + ((1 P_sin) . 1) by Def20 .= ((((- 1) |^ 0) * (1 |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !)) + ((((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) by Def20 .= ((1 * (1 |^ ((2 * 0) + 1))) / (((2 * 0) + 1) !)) + ((((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) by Lm7 .= (1 / ((0 + 1) !)) + ((((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) by NEWTON:5 .= (1 / ((0 !) * 1)) + ((((- 1) |^ 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) by NEWTON:15 .= 1 + (((- 1) * (1 |^ ((2 * 1) + 1))) / (((2 * 1) + 1) !)) by NEWTON:5, NEWTON:12 .= 1 + (((- 1) * 1) / (((2 * 1) + 1) !)) by NEWTON:10 .= 1 + ((- 1) / (((2 * 1) !) * ((2 * 1) + 1))) by NEWTON:15 .= 1 + ((- 1) / (((1 !) * (1 + 1)) * 3)) by NEWTON:15 .= 1 + ((- 1) / (((0 + 1) !) * (2 * 3))) .= 1 + ((- 1) / ((1 * 1) * 6)) by NEWTON:12, NEWTON:15 .= 5 / 6 ; then A14: S1[ 0 ] ; A15: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A16: ((Partial_Sums (1 P_sin)) * bq) . k >= 5 / 6 ; ::_thesis: S1[k + 1] ((Partial_Sums (1 P_sin)) * bq) . (k + 1) = (Partial_Sums (1 P_sin)) . (bq . (k + 1)) by FUNCT_2:15 .= (Partial_Sums (1 P_sin)) . ((2 * (k + 1)) + 1) by Lm6 .= ((Partial_Sums (1 P_sin)) . (((2 * k) + 1) + 1)) + ((1 P_sin) . ((2 * (k + 1)) + 1)) by SERIES_1:def_1 .= (((Partial_Sums (1 P_sin)) . ((2 * k) + 1)) + ((1 P_sin) . (((2 * k) + 1) + 1))) + ((1 P_sin) . ((2 * (k + 1)) + 1)) by SERIES_1:def_1 .= (((Partial_Sums (1 P_sin)) . (bq . k)) + ((1 P_sin) . (((2 * k) + 1) + 1))) + ((1 P_sin) . ((2 * (k + 1)) + 1)) by Lm6 .= ((((Partial_Sums (1 P_sin)) * bq) . k) + ((1 P_sin) . (((2 * k) + 1) + 1))) + ((1 P_sin) . ((2 * (k + 1)) + 1)) by FUNCT_2:15 ; then A17: (((Partial_Sums (1 P_sin)) * bq) . (k + 1)) - (((Partial_Sums (1 P_sin)) * bq) . k) = ((1 P_sin) . (((2 * k) + 1) + 1)) + ((1 P_sin) . ((2 * (k + 1)) + 1)) ; A18: (1 P_sin) . (((2 * k) + 1) + 1) = (((- 1) |^ (2 * (k + 1))) * (1 |^ ((2 * (2 * (k + 1))) + 1))) / (((2 * (2 * (k + 1))) + 1) !) by Def20 .= (1 * (1 |^ ((2 * (2 * (k + 1))) + 1))) / (((2 * (2 * (k + 1))) + 1) !) by Lm7 .= 1 / (((2 * (2 * (k + 1))) + 1) !) by NEWTON:10 ; A19: (1 P_sin) . ((2 * (k + 1)) + 1) = (((- 1) |^ ((2 * (k + 1)) + 1)) * (1 |^ ((2 * ((2 * (k + 1)) + 1)) + 1))) / (((2 * ((2 * (k + 1)) + 1)) + 1) !) by Def20 .= ((- 1) * (1 |^ ((2 * ((2 * (k + 1)) + 1)) + 1))) / (((2 * ((2 * (k + 1)) + 1)) + 1) !) by Lm7 .= ((- 1) * 1) / (((2 * ((2 * (k + 1)) + 1)) + 1) !) by NEWTON:10 .= (- 1) / (((2 * ((2 * (k + 1)) + 1)) + 1) !) ; 2 * (2 * (k + 1)) < 2 * ((2 * (k + 1)) + 1) by XREAL_1:29, XREAL_1:68; then (2 * (2 * (k + 1))) + 1 < (2 * ((2 * (k + 1)) + 1)) + 1 by XREAL_1:6; then ((2 * (2 * (k + 1))) + 1) ! <= ((2 * ((2 * (k + 1)) + 1)) + 1) ! by Th39; then 1 / (((2 * (2 * (k + 1))) + 1) !) >= 1 / (((2 * ((2 * (k + 1)) + 1)) + 1) !) by XREAL_1:85; then (1 / (((2 * (2 * (k + 1))) + 1) !)) - (1 / (((2 * ((2 * (k + 1)) + 1)) + 1) !)) >= 0 by XREAL_1:48; then ((Partial_Sums (1 P_sin)) * bq) . (k + 1) >= ((Partial_Sums (1 P_sin)) * bq) . k by A17, A18, A19, XREAL_1:49; hence S1[k + 1] by A16, XXREAL_0:2; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A14, A15); hence ((Partial_Sums (1 P_sin)) * bq) . n >= 5 / 6 ; ::_thesis: verum end; then A20: sin . 1 >= 5 / 6 by A11, A13, PREPOWER:1, SEQ_4:16; A21: ((cos . 1) ^2) + ((sin . 1) ^2) = 1 by Th28; A22: (sin . 1) ^2 >= (5 / 6) ^2 by A20, SQUARE_1:15; then 1 - (1 - ((cos . 1) ^2)) <= 1 - (25 / 36) by A21, XREAL_1:10; then (cos . 1) ^2 < 25 / 36 by XXREAL_0:2; then (sin . 1) ^2 > (cos . 1) ^2 by A22, XXREAL_0:2; then A23: sqrt ((cos . 1) ^2) < sqrt ((sin . 1) ^2) by SQUARE_1:27, XREAL_1:63; sqrt ((cos . 1) ^2) = cos . 1 by A10, SQUARE_1:22; hence ( cos . 1 > 0 & sin . 1 > 0 & cos . 1 < sin . 1 ) by A10, A20, A23, SQUARE_1:22; ::_thesis: verum end; theorem Th44: :: SIN_COS:44 for th being Real holds th rExpSeq = Re (th ExpSeq) proof let th be Real; ::_thesis: th rExpSeq = Re (th ExpSeq) for n being Element of NAT holds (th rExpSeq) . n = (Re (th ExpSeq)) . n proof let n be Element of NAT ; ::_thesis: (th rExpSeq) . n = (Re (th ExpSeq)) . n (Re (th ExpSeq)) . n = Re ((th ExpSeq) . n) by COMSEQ_3:def_5 .= Re ((th |^ n) / ((n !) + (0 * ))) by Def4 .= Re (((th |^ n) / (n !)) + (0 * )) .= (th |^ n) / (n !) by COMPLEX1:12 .= (th rExpSeq) . n by Def5 ; hence (th rExpSeq) . n = (Re (th ExpSeq)) . n ; ::_thesis: verum end; hence th rExpSeq = Re (th ExpSeq) by FUNCT_2:63; ::_thesis: verum end; theorem Th45: :: SIN_COS:45 for th being Real holds ( th rExpSeq is summable & Sum (th rExpSeq) = Re (Sum (th ExpSeq)) ) proof let th be Real; ::_thesis: ( th rExpSeq is summable & Sum (th rExpSeq) = Re (Sum (th ExpSeq)) ) ( Sum (th ExpSeq) = (Sum (Re (th ExpSeq))) + ((Sum (Im (th ExpSeq))) * ) & Sum (th rExpSeq) = Sum (Re (th ExpSeq)) ) by Th44, COMSEQ_3:53; hence ( th rExpSeq is summable & Sum (th rExpSeq) = Re (Sum (th ExpSeq)) ) by Th44, COMPLEX1:12; ::_thesis: verum end; Lm8: for n being Element of NAT for z being complex number holds ( (z ExpSeq) . 1 = z & (z ExpSeq) . 0 = 1r & |.(z |^ n).| = |.z.| |^ n ) proof let n be Element of NAT ; ::_thesis: for z being complex number holds ( (z ExpSeq) . 1 = z & (z ExpSeq) . 0 = 1r & |.(z |^ n).| = |.z.| |^ n ) let z be complex number ; ::_thesis: ( (z ExpSeq) . 1 = z & (z ExpSeq) . 0 = 1r & |.(z |^ n).| = |.z.| |^ n ) A1: z in COMPLEX by XCMPLX_0:def_2; z |^ 1 = z by NEWTON:5; then A2: (z ExpSeq) . 1 = z / (1 !) by Def4 .= z by NEWTON:13 ; A3: (z ExpSeq) . 0 = (z |^ 0) / (0 !) by Def4 .= 1r by A1, Th1, COMSEQ_3:11 ; |.(z |^ n).| = |.z.| |^ n proof defpred S1[ Element of NAT ] means |.(z |^ \$1).| = |.z.| |^ \$1; |.(z |^ 0).| = 1 by COMPLEX1:48, COMSEQ_3:def_1; then A4: S1[ 0 ] by NEWTON:4; A5: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A6: |.(z |^ k).| = |.z.| |^ k ; ::_thesis: S1[k + 1] |.(z |^ (k + 1)).| = |.(z * ((z GeoSeq) . k)).| by COMSEQ_3:def_1 .= (|.z.| |^ k) * |.z.| by A6, COMPLEX1:65 .= |.z.| |^ (k + 1) by NEWTON:6 ; hence S1[k + 1] ; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A4, A5); hence |.(z |^ n).| = |.z.| |^ n ; ::_thesis: verum end; hence ( (z ExpSeq) . 1 = z & (z ExpSeq) . 0 = 1r & |.(z |^ n).| = |.z.| |^ n ) by A2, A3; ::_thesis: verum end; Lm9: for th being Real holds Sum (th ExpSeq) = Sum (th rExpSeq) proof let th be Real; ::_thesis: Sum (th ExpSeq) = Sum (th rExpSeq) A1: for n being Nat holds (Im (Partial_Sums (th ExpSeq))) . n = 0 proof defpred S1[ Nat] means (Im (Partial_Sums (th ExpSeq))) . \$1 = 0 ; (Im (Partial_Sums (th ExpSeq))) . 0 = Im ((Partial_Sums (th ExpSeq)) . 0) by COMSEQ_3:def_6 .= Im ((th ExpSeq) . 0) by SERIES_1:def_1 .= 0 by Lm8, COMPLEX1:6 ; then A2: S1[ 0 ] ; A3: for n being Nat st S1[n] holds S1[n + 1] proof let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A4: (Im (Partial_Sums (th ExpSeq))) . n = 0 ; ::_thesis: S1[n + 1] A5: n in NAT by ORDINAL1:def_12; (Im (Partial_Sums (th ExpSeq))) . (n + 1) = Im ((Partial_Sums (th ExpSeq)) . (n + 1)) by COMSEQ_3:def_6 .= Im (((Partial_Sums (th ExpSeq)) . n) + ((th ExpSeq) . (n + 1))) by A5, SERIES_1:def_1 .= (Im ((Partial_Sums (th ExpSeq)) . n)) + (Im ((th ExpSeq) . (n + 1))) by COMPLEX1:8 .= 0 + (Im ((th ExpSeq) . (n + 1))) by A4, A5, COMSEQ_3:def_6 .= Im (((th |^ (n + 1)) / ((n + 1) !)) + (0 * )) by Def4 .= 0 by COMPLEX1:12 ; hence S1[n + 1] ; ::_thesis: verum end; for n being Nat holds S1[n] from NAT_1:sch_2(A2, A3); hence for n being Nat holds (Im (Partial_Sums (th ExpSeq))) . n = 0 ; ::_thesis: verum end; then Im (Partial_Sums (th ExpSeq)) is constant by VALUED_0:def_18; then lim (Im (Partial_Sums (th ExpSeq))) = (Im (Partial_Sums (th ExpSeq))) . 0 by SEQ_4:26 .= 0 by A1 ; then Im (Sum (th ExpSeq)) = 0 by COMSEQ_3:41; then Sum (th ExpSeq) = (Re (Sum (th ExpSeq))) + (0 * ) by COMPLEX1:13 .= Sum (th rExpSeq) by Th45 ; hence Sum (th ExpSeq) = Sum (th rExpSeq) ; ::_thesis: verum end; theorem Th46: :: SIN_COS:46 for p, q being real number holds Sum ((p + q) rExpSeq) = (Sum (p rExpSeq)) * (Sum (q rExpSeq)) proof let p, q be real number ; ::_thesis: Sum ((p + q) rExpSeq) = (Sum (p rExpSeq)) * (Sum (q rExpSeq)) reconsider p = p, q = q as Real by XREAL_0:def_1; Sum ((p + q) rExpSeq) = Re (Sum ((p + q) ExpSeq)) by Th45 .= Re ((Sum (p ExpSeq)) * (Sum (q ExpSeq))) by Lm2 .= Re (((Re (Sum (p ExpSeq))) + ((Im (Sum (p ExpSeq))) * )) * (Sum (q ExpSeq))) by COMPLEX1:13 .= Re (((Re (Sum (p ExpSeq))) + ((Im (Sum (p ExpSeq))) * )) * ((Re (Sum (q ExpSeq))) + ((Im (Sum (q ExpSeq))) * ))) by COMPLEX1:13 .= Re (((Sum (p rExpSeq)) + ((Im (Sum (p ExpSeq))) * )) * ((Re (Sum (q ExpSeq))) + ((Im (Sum (q ExpSeq))) * ))) by Th45 .= Re (((Sum (p rExpSeq)) + (0 * )) * ((Re (Sum (q ExpSeq))) + ((Im (Sum (q ExpSeq))) * ))) by Th42 .= Re ((Sum (p rExpSeq)) * ((Sum (q rExpSeq)) + ((Im (Sum (q ExpSeq))) * ))) by Th45 .= Re ((Sum (p rExpSeq)) * ((Sum (q rExpSeq)) + (0 * ))) by Th42 .= Re (((Sum (p rExpSeq)) * (Sum (q rExpSeq))) + (0 * )) .= (Sum (p rExpSeq)) * (Sum (q rExpSeq)) by COMPLEX1:12 ; hence Sum ((p + q) rExpSeq) = (Sum (p rExpSeq)) * (Sum (q rExpSeq)) ; ::_thesis: verum end; definition func exp_R -> Function of REAL,REAL means :Def22: :: SIN_COS:def 22 for d being real number holds it . d = Sum (d rExpSeq); existence ex b1 being Function of REAL,REAL st for d being real number holds b1 . d = Sum (d rExpSeq) proof deffunc H2( Element of REAL ) -> Element of REAL = Sum (\$1 rExpSeq); consider f being Function of REAL,REAL such that A1: for d being Element of REAL holds f . d = H2(d) from FUNCT_2:sch_4(); take f ; ::_thesis: for d being real number holds f . d = Sum (d rExpSeq) let d be real number ; ::_thesis: f . d = Sum (d rExpSeq) d in REAL by XREAL_0:def_1; hence f . d = Sum (d rExpSeq) by A1; ::_thesis: verum end; uniqueness for b1, b2 being Function of REAL,REAL st ( for d being real number holds b1 . d = Sum (d rExpSeq) ) & ( for d being real number holds b2 . d = Sum (d rExpSeq) ) holds b1 = b2 proof let f1, f2 be Function of REAL,REAL; ::_thesis: ( ( for d being real number holds f1 . d = Sum (d rExpSeq) ) & ( for d being real number holds f2 . d = Sum (d rExpSeq) ) implies f1 = f2 ) assume A2: for d being real number holds f1 . d = Sum (d rExpSeq) ; ::_thesis: ( ex d being real number st not f2 . d = Sum (d rExpSeq) or f1 = f2 ) assume A3: for d being real number holds f2 . d = Sum (d rExpSeq) ; ::_thesis: f1 = f2 for d being Element of REAL holds f1 . d = f2 . d proof let d be Element of REAL ; ::_thesis: f1 . d = f2 . d thus f1 . d = Sum (d rExpSeq) by A2 .= f2 . d by A3 ; ::_thesis: verum end; hence f1 = f2 by FUNCT_2:63; ::_thesis: verum end; end; :: deftheorem Def22 defines exp_R SIN_COS:def_22_:_ for b1 being Function of REAL,REAL holds ( b1 = exp_R iff for d being real number holds b1 . d = Sum (d rExpSeq) ); definition let th be real number ; func exp_R th -> set equals :: SIN_COS:def 23 exp_R . th; coherence exp_R . th is set ; end; :: deftheorem defines exp_R SIN_COS:def_23_:_ for th being real number holds exp_R th = exp_R . th; registration let th be real number ; cluster exp_R th -> real ; coherence exp_R th is real ; end; definition let th be Real; :: original: exp_R redefine func exp_R th -> Real; coherence exp_R th is Real ; end; theorem Th47: :: SIN_COS:47 dom exp_R = REAL by FUNCT_2:def_1; theorem Th48: :: SIN_COS:48 for th being Real holds exp_R . th = Re (Sum (th ExpSeq)) proof let th be Real; ::_thesis: exp_R . th = Re (Sum (th ExpSeq)) exp_R . th = Sum (th rExpSeq) by Def22; hence exp_R . th = Re (Sum (th ExpSeq)) by Th45; ::_thesis: verum end; theorem :: SIN_COS:49 for th being Real holds exp th = exp_R th proof let th be Real; ::_thesis: exp th = exp_R th thus exp th = Sum (th ExpSeq) by Def14 .= Sum (th rExpSeq) by Lm9 .= exp_R th by Def22 ; ::_thesis: verum end; Lm10: for p, q being real number holds exp_R . (p + q) = (exp_R . p) * (exp_R . q) proof let p, q be real number ; ::_thesis: exp_R . (p + q) = (exp_R . p) * (exp_R . q) exp_R . (p + q) = Sum ((p + q) rExpSeq) by Def22 .= (Sum (p rExpSeq)) * (Sum (q rExpSeq)) by Th46 .= (exp_R . p) * (Sum (q rExpSeq)) by Def22 .= (exp_R . p) * (exp_R . q) by Def22 ; hence exp_R . (p + q) = (exp_R . p) * (exp_R . q) ; ::_thesis: verum end; theorem :: SIN_COS:50 for p, q being real number holds exp_R (p + q) = (exp_R p) * (exp_R q) by Lm10; Lm11: exp_R . 0 = 1 proof exp_R . 0 = Sum (0 rExpSeq) by Def22 .= 1 by Th9, Th45, COMPLEX1:6 ; hence exp_R . 0 = 1 ; ::_thesis: verum end; theorem :: SIN_COS:51 exp_R 0 = 1 by Lm11; theorem Th52: :: SIN_COS:52 for th being real number st th > 0 holds exp_R . th >= 1 proof let th be real number ; ::_thesis: ( th > 0 implies exp_R . th >= 1 ) assume A1: th > 0 ; ::_thesis: exp_R . th >= 1 A2: for n being Element of NAT holds (Partial_Sums (th rExpSeq)) . n >= 1 proof defpred S1[ Element of NAT ] means (Partial_Sums (th rExpSeq)) . \$1 >= 1; (Partial_Sums (th rExpSeq)) . 0 = (th rExpSeq) . 0 by SERIES_1:def_1 .= (th |^ 0) / (0 !) by Def5 .= 1 by NEWTON:4, NEWTON:12 ; then A3: S1[ 0 ] ; A4: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A5: (Partial_Sums (th rExpSeq)) . n >= 1 ; ::_thesis: S1[n + 1] A6: (Partial_Sums (th rExpSeq)) . (n + 1) = ((Partial_Sums (th rExpSeq)) . n) + ((th rExpSeq) . (n + 1)) by SERIES_1:def_1 .= ((Partial_Sums (th rExpSeq)) . n) + ((th |^ (n + 1)) / ((n + 1) !)) by Def5 ; ( th |^ (n + 1) > 0 & (n + 1) ! > 0 ) by A1, PREPOWER:6; then ((Partial_Sums (th rExpSeq)) . n) + ((th |^ (n + 1)) / ((n + 1) !)) > (Partial_Sums (th rExpSeq)) . n by XREAL_1:29; hence S1[n + 1] by A5, A6, XXREAL_0:2; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A3, A4); hence for n being Element of NAT holds (Partial_Sums (th rExpSeq)) . n >= 1 ; ::_thesis: verum end; th is Real by XREAL_0:def_1; then th rExpSeq is summable by Th45; then A7: Partial_Sums (th rExpSeq) is convergent by SERIES_1:def_2; lim (Partial_Sums (th rExpSeq)) = Sum (th rExpSeq) by SERIES_1:def_3; then Sum (th rExpSeq) >= 1 by A2, A7, PREPOWER:1; hence exp_R . th >= 1 by Def22; ::_thesis: verum end; theorem Th53: :: SIN_COS:53 for th being real number st th < 0 holds ( 0 < exp_R . th & exp_R . th <= 1 ) proof let th be real number ; ::_thesis: ( th < 0 implies ( 0 < exp_R . th & exp_R . th <= 1 ) ) assume th < 0 ; ::_thesis: ( 0 < exp_R . th & exp_R . th <= 1 ) then A1: exp_R . (- th) >= 1 by Th52; A2: (exp_R . (- th)) * (exp_R . th) = exp_R . ((- th) + th) by Lm10 .= 1 by Lm11 ; then A3: exp_R . th = 1 / (exp_R . (- th)) by XCMPLX_1:73; thus 0 < exp_R . th by A1, A2; ::_thesis: exp_R . th <= 1 thus exp_R . th <= 1 by A1, A3, XREAL_1:211; ::_thesis: verum end; theorem Th54: :: SIN_COS:54 for th being real number holds exp_R . th > 0 proof let th be real number ; ::_thesis: exp_R . th > 0 now__::_thesis:_exp_R_._th_>_0 percases ( th = 0 or th <> 0 ) ; suppose th = 0 ; ::_thesis: exp_R . th > 0 hence exp_R . th > 0 by Lm11; ::_thesis: verum end; supposeA1: th <> 0 ; ::_thesis: exp_R . th > 0 now__::_thesis:_exp_R_._th_>_0 percases ( th < 0 or th > 0 ) by A1; suppose th < 0 ; ::_thesis: exp_R . th > 0 hence exp_R . th > 0 by Th53; ::_thesis: verum end; suppose th > 0 ; ::_thesis: exp_R . th > 0 hence exp_R . th > 0 by Th52; ::_thesis: verum end; end; end; hence exp_R . th > 0 ; ::_thesis: verum end; end; end; hence exp_R . th > 0 ; ::_thesis: verum end; theorem :: SIN_COS:55 for th being real number holds exp_R th > 0 by Th54; begin definition let z be complex number ; deffunc H2( Element of NAT ) -> Element of COMPLEX = (z #N (\$1 + 1)) / ((\$1 + 2) !); funcz P_dt -> Complex_Sequence means :Def24: :: SIN_COS:def 24 for n being Element of NAT holds it . n = (z |^ (n + 1)) / ((n + 2) !); existence ex b1 being Complex_Sequence st for n being Element of NAT holds b1 . n = (z |^ (n + 1)) / ((n + 2) !) proof thus ex s being Complex_Sequence st for n being Element of NAT holds s . n = H2(n) from COMSEQ_1:sch_1(); ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st ( for n being Element of NAT holds b1 . n = (z |^ (n + 1)) / ((n + 2) !) ) & ( for n being Element of NAT holds b2 . n = (z |^ (n + 1)) / ((n + 2) !) ) holds b1 = b2 proof thus for s1, s2 being Complex_Sequence st ( for x being Element of NAT holds s1 . x = H2(x) ) & ( for x being Element of NAT holds s2 . x = H2(x) ) holds s1 = s2 from BINOP_2:sch_1(); ::_thesis: verum end; deffunc H3( Element of NAT ) -> Element of COMPLEX = (z #N \$1) / ((\$1 + 2) !); funcz P_t -> Complex_Sequence means :: SIN_COS:def 25 for n being Element of NAT holds it . n = (z #N n) / ((n + 2) !); existence ex b1 being Complex_Sequence st for n being Element of NAT holds b1 . n = (z #N n) / ((n + 2) !) proof thus ex s being Complex_Sequence st for n being Element of NAT holds s . n = H3(n) from COMSEQ_1:sch_1(); ::_thesis: verum end; uniqueness for b1, b2 being Complex_Sequence st ( for n being Element of NAT holds b1 . n = (z #N n) / ((n + 2) !) ) & ( for n being Element of NAT holds b2 . n = (z #N n) / ((n + 2) !) ) holds b1 = b2 proof thus for s1, s2 being Complex_Sequence st ( for x being Element of NAT holds s1 . x = H3(x) ) & ( for x being Element of NAT holds s2 . x = H3(x) ) holds s1 = s2 from BINOP_2:sch_1(); ::_thesis: verum end; end; :: deftheorem Def24 defines P_dt SIN_COS:def_24_:_ for z being complex number for b2 being Complex_Sequence holds ( b2 = z P_dt iff for n being Element of NAT holds b2 . n = (z |^ (n + 1)) / ((n + 2) !) ); :: deftheorem defines P_t SIN_COS:def_25_:_ for z being complex number for b2 being Complex_Sequence holds ( b2 = z P_t iff for n being Element of NAT holds b2 . n = (z #N n) / ((n + 2) !) ); Lm12: for p being Real for z being complex number holds ( Re ((p * ) * z) = - (p * (Im z)) & Im ((p * ) * z) = p * (Re z) & Re (p * z) = p * (Re z) & Im (p * z) = p * (Im z) ) proof let p be Real; ::_thesis: for z being complex number holds ( Re ((p * ) * z) = - (p * (Im z)) & Im ((p * ) * z) = p * (Re z) & Re (p * z) = p * (Re z) & Im (p * z) = p * (Im z) ) let z be complex number ; ::_thesis: ( Re ((p * ) * z) = - (p * (Im z)) & Im ((p * ) * z) = p * (Re z) & Re (p * z) = p * (Re z) & Im (p * z) = p * (Im z) ) A1: (p * ) * z = (p * ) * ((Re z) + ((Im z) * )) by COMPLEX1:13 .= (- (p * (Im z))) + ((p * (Re z)) * ) ; p * z = p * ((Re z) + ((Im z) * )) by COMPLEX1:13 .= (p * (Re z)) + ((p * (Im z)) * ) ; hence ( Re ((p * ) * z) = - (p * (Im z)) & Im ((p * ) * z) = p * (Re z) & Re (p * z) = p * (Re z) & Im (p * z) = p * (Im z) ) by A1, COMPLEX1:12; ::_thesis: verum end; Lm13: for p being real number for z being complex number st p > 0 holds ( Re (z / (p * )) = (Im z) / p & Im (z / (p * )) = - ((Re z) / p) & |.(z / p).| = |.z.| / p ) proof let p be real number ; ::_thesis: for z being complex number st p > 0 holds ( Re (z / (p * )) = (Im z) / p & Im (z / (p * )) = - ((Re z) / p) & |.(z / p).| = |.z.| / p ) let z be complex number ; ::_thesis: ( p > 0 implies ( Re (z / (p * )) = (Im z) / p & Im (z / (p * )) = - ((Re z) / p) & |.(z / p).| = |.z.| / p ) ) assume A1: p > 0 ; ::_thesis: ( Re (z / (p * )) = (Im z) / p & Im (z / (p * )) = - ((Re z) / p) & |.(z / p).| = |.z.| / p ) A2: ( Re (0 + (p * )) = 0 & Im (0 + (p * )) = p ) by COMPLEX1:12; A3: Im (p + (0 * )) = 0 by COMPLEX1:12; A4: z / (p * ) = ((((Re z) * 0) + ((Im z) * p)) / ((0 ^2) + (p ^2))) + ((((0 * (Im z)) - ((Re z) * p)) / ((0 ^2) + (p ^2))) * ) by A2, COMPLEX1:def_10 .= (((Im z) * p) / (p * p)) + ((((- (Re z)) * p) / (p * p)) * ) .= (((Im z) * p) / (p * p)) + (((- (Re z)) / p) * ) by A1, XCMPLX_1:91 .= ((Im z) / p) + ((- ((Re z) / p)) * ) by A1, XCMPLX_1:91 ; |.(z / p).| = |.z.| / |.p.| by COMPLEX1:67 .= |.z.| / (sqrt (p ^2)) by A3, COMPLEX1:12 ; hence ( Re (z / (p * )) = (Im z) / p & Im (z / (p * )) = - ((Re z) / p) & |.(z / p).| = |.z.| / p ) by A1, A4, COMPLEX1:12, SQUARE_1:22; ::_thesis: verum end; theorem Th56: :: SIN_COS:56 for z being complex number holds z P_dt is absolutely_summable proof let z be complex number ; ::_thesis: z P_dt is absolutely_summable ex r being Real st for n being Element of NAT holds (Partial_Sums |.(z P_dt).|) . n < r proof A1: for n being Element of NAT holds (Partial_Sums |.(z P_dt).|) . n < (Partial_Sums |.(z ExpSeq).|) . (n + 1) proof let n be Element of NAT ; ::_thesis: (Partial_Sums |.(z P_dt).|) . n < (Partial_Sums |.(z ExpSeq).|) . (n + 1) (Partial_Sums |.(z P_dt).|) . 0 = |.(z P_dt).| . 0 by SERIES_1:def_1 .= |.((z P_dt) . 0).| by VALUED_1:18 .= |.((z |^ (0 + 1)) / ((0 + 2) !)).| by Def24 .= |.(z / 2).| by NEWTON:5, NEWTON:14 ; then A2: (Partial_Sums |.(z P_dt).|) . 0 = |.z.| / 2 by Lm13; (Partial_Sums |.(z ExpSeq).|) . (0 + 1) = ((Partial_Sums |.(z ExpSeq).|) . 0) + (|.(z ExpSeq).| . (0 + 1)) by SERIES_1:def_1 .= (|.(z ExpSeq).| . 0) + (|.(z ExpSeq).| . 1) by SERIES_1:def_1 .= (|.(z ExpSeq).| . 0) + |.((z ExpSeq) . 1).| by VALUED_1:18 .= |.((z ExpSeq) . 0).| + |.((z ExpSeq) . 1).| by VALUED_1:18 .= |.((z ExpSeq) . 0).| + |.z.| by Lm8 .= 1 + |.z.| by Lm8, COMPLEX1:48 ; then A3: ((Partial_Sums |.(z ExpSeq).|) . (0 + 1)) - ((Partial_Sums |.(z P_dt).|) . 0) = 1 + (|.z.| - (|.z.| / 2)) by A2; defpred S1[ Element of NAT ] means (Partial_Sums |.(z P_dt).|) . \$1 < (Partial_Sums |.(z ExpSeq).|) . (\$1 + 1); 0 <= |.z.| by COMPLEX1:46; then A4: S1[ 0 ] by A3, XREAL_1:47; A5: for n being Element of NAT st S1[n] holds S1[n + 1] proof let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] ) assume A6: (Partial_Sums |.(z P_dt).|) . n < (Partial_Sums |.(z ExpSeq).|) . (n + 1) ; ::_thesis: S1[n + 1] (Partial_Sums |.(z P_dt).|) . (n + 1) = ((Partial_Sums |.(z P_dt).|) . n) + (|.(z P_dt).| . (n + 1)) by SERIES_1:def_1 .= ((Partial_Sums |.(z P_dt).|) . n) + |.((z P_dt) . (n + 1)).| by VALUED_1:18 .= ((Partial_Sums |.(z P_dt).|) . n) + |.((z |^ ((n + 1) + 1)) / (((n + 1) + 2) !)).| by Def24 .= ((Partial_Sums |.(z P_dt).|) . n) + |.((z |^ (n + 2)) / ((n + 3) !)).| ; then A7: (Partial_Sums |.(z P_dt).|) . (n + 1) = ((Partial_Sums |.(z P_dt).|) . n) + (|.(z #N (n + 2)).| / ((n + 3) !)) by Lm13; (Partial_Sums |.(z ExpSeq).|) . ((n + 1) + 1) = ((Partial_Sums |.(z ExpSeq).|) . (n + 1)) + (|.(z ExpSeq).| . (n + (1 + 1))) by SERIES_1:def_1 .= ((Partial_Sums |.(z ExpSeq).|) . (n + 1)) + |.((z ExpSeq) . (n + 2)).| by VALUED_1:18 .= ((Partial_Sums |.(z ExpSeq).|) . (n + 1)) + |.((z |^ (n + 2)) / ((n + 2) !)).| by Def4 .= ((Partial_Sums |.(z ExpSeq).|) . (n + 1)) + |.((z |^ (n + 2)) / ((n + 2) !)).| ; then A8: (Partial_Sums |.(z ExpSeq).|) . ((n + 1) + 1) = ((Partial_Sums |.(z ExpSeq).|) . (n + 1)) + (|.(z #N (n + 2)).| / ((n + 2) !)) by Lm13; n + 2 < n + 3 by XREAL_1:6; then A9: (n + 2) ! <= (n + 3) ! by Th39; |.(z |^ (n + 2)).| >= 0 by COMPLEX1:46; then |.(z |^ (n + 2)).| / ((n + 2) !) >= |.(z |^ (n + 2)).| / ((n + 3) !) by A9, XREAL_1:118; then A10: ((Partial_Sums |.(z P_dt).|) . n) + (|.(z |^ (n + 2)).| / ((n + 3) !)) <= ((Partial_Sums |.(z P_dt).|) . n) + (|.(z |^ (n + 2)).| / ((n + 2) !)) by XREAL_1:6; ((Partial_Sums |.(z P_dt).|) . n) + (|.(z |^ (n + 2)).| / ((n + 2) !)) < ((Partial_Sums |.(z ExpSeq).|) . (n + 1)) + (|.(z |^ (n + 2)).| / ((n + 2) !)) by A6, XREAL_1:6; hence S1[n + 1] by A7, A8, A10, XXREAL_0:2; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A4, A5); hence (Partial_Sums |.(z P_dt).|) . n < (Partial_Sums |.(z ExpSeq).|) . (n + 1) ; ::_thesis: verum end; consider r being real number such that A11: for n being Element of NAT holds (Partial_Sums |.(z ExpSeq).|) . n < r by SEQ_2:def_3; A12: for n being Element of NAT holds (Partial_Sums |.(z P_dt).|) . n < r proof let n be Element of NAT ; ::_thesis: (Partial_Sums |.(z P_dt).|) . n < r (Partial_Sums |.(z P_dt).|) . n < (Partial_Sums |.(z ExpSeq).|) . (n + 1) by A1; hence (Partial_Sums |.(z P_dt).|) . n < r by A11, XXREAL_0:2; ::_thesis: verum end; take r ; ::_thesis: ( r is Real & ( for n being Element of NAT holds (Partial_Sums |.(z P_dt).|) . n < r ) ) thus ( r is Real & ( for n being Element of NAT holds (Partial_Sums |.(z P_dt).|) . n < r ) ) by A12, XREAL_0:def_1; ::_thesis: verum end; then Partial_Sums |.(z P_dt).| is bounded_above by SEQ_2:def_3; hence z P_dt is absolutely_summable by COMSEQ_3:61; ::_thesis: verum end; theorem Th57: :: SIN_COS:57 for z being complex number holds z * (Sum (z P_dt)) = ((Sum (z ExpSeq)) - 1r) - z proof let z be complex number ; ::_thesis: z * (Sum (z P_dt)) = ((Sum (z ExpSeq)) - 1r) - z A1: for z being complex number holds z (#) (z P_dt) = (z ExpSeq) ^\ 2 proof let z be complex number ; ::_thesis: z (#) (z P_dt) = (z ExpSeq) ^\ 2 for n being Element of NAT holds (z (#) (z P_dt)) . n = ((z ExpSeq) ^\ 2) . n proof let n be Element of NAT ; ::_thesis: (z (#) (z P_dt)) . n = ((z ExpSeq) ^\ 2) . n (z (#) (z P_dt)) . n = z * ((z P_dt) . n) by VALUED_1:6 .= z * ((z |^ (n + 1)) / ((n + 2) !)) by Def24 ; then (z (#) (z P_dt)) . n = (z * (z |^ (n + 1))) / ((n + 2) !) .= (z |^ ((n + 1) + 1)) / ((n + 2) !) by NEWTON:6 .= (z ExpSeq) . (n + 2) by Def4 .= ((z ExpSeq) ^\ 2) . n by NAT_1:def_3 ; hence (z (#) (z P_dt)) . n = ((z ExpSeq) ^\ 2) . n ; ::_thesis: verum end; hence z (#) (z P_dt) = (z ExpSeq) ^\ 2 by FUNCT_2:63; ::_thesis: verum end; Sum (z ExpSeq) = ((Partial_Sums (z ExpSeq)) . 1) + (Sum ((z ExpSeq) ^\ (1 + 1))) by COMSEQ_3:60 .= ((Partial_Sums (z ExpSeq)) . 1) + (Sum ((z ExpSeq) ^\ 2)) ; then A2: Sum (z (#) (z P_dt)) = (Sum (z ExpSeq)) - ((Partial_Sums (z ExpSeq)) . (0 + 1)) by A1 .= (Sum (z ExpSeq)) - (((Partial_Sums (z ExpSeq)) . 0) + ((z ExpSeq) . 1)) by SERIES_1:def_1 .= (Sum (z ExpSeq)) - (((z ExpSeq) . 0) + ((z ExpSeq) . 1)) by SERIES_1:def_1 .= (Sum (z ExpSeq)) - (1r + ((z ExpSeq) . 1)) by Lm8 .= (Sum (z ExpSeq)) - (1r + z) by Lm8 .= ((Sum (z ExpSeq)) - 1r) - z ; reconsider BBB = z P_dt as absolutely_summable Complex_Sequence by Th56; BBB is summable ; hence z * (Sum (z P_dt)) = ((Sum (z ExpSeq)) - 1r) - z by A2, COMSEQ_3:56; ::_thesis: verum end; theorem Th58: :: SIN_COS:58 for p being real number st p > 0 holds ex q being Real st ( q > 0 & ( for z being complex number st |.z.| < q holds |.(Sum (z P_dt)).| < p ) ) proof A1: for z being Element of COMPLEX holds |.(Sum (z P_dt)).| <= Sum |.(z P_dt).| proof let z be Element of COMPLEX ; ::_thesis: |.(Sum (z P_dt)).| <= Sum |.(z P_dt).| A2: for k being Element of NAT holds |.(Partial_Sums (z P_dt)).| . k <= (Partial_Sums |.(z P_dt).|) . k proof let k be Element of NAT ; ::_thesis: |.(Partial_Sums (z P_dt)).| . k <= (Partial_Sums |.(z P_dt).|) . k |.((Partial_Sums (z P_dt)) . k).| = |.(Partial_Sums (z P_dt)).| . k by VALUED_1:18; hence |.(Partial_Sums (z P_dt)).| . k <= (Partial_Sums |.(z P_dt).|) . k by COMSEQ_3:30; ::_thesis: verum end; A3: z P_dt is absolutely_summable by Th56; A4: z P_dt is summable by Th56, COMSEQ_3:63; A5: lim |.(Partial_Sums (z P_dt)).| = |.(lim (Partial_Sums (z P_dt))).| by A4, SEQ_2:27; lim |.(Partial_Sums (z P_dt)).| <= lim (Partial_Sums |.(z P_dt).|) by A2, A3, SEQ_2:18; hence |.(Sum (z P_dt)).| <= Sum |.(z P_dt).| by A5, SERIES_1:def_3; ::_thesis: verum end; A6: for z being Element of COMPLEX for n being Element of NAT holds |.(z P_dt).| . n <= |.z.| * ((|.z.| GeoSeq) . n) proof let z be Element of COMPLEX ; ::_thesis: for n being Element of NAT holds |.(z P_dt).| . n <= |.z.| * ((|.z.| GeoSeq) . n) let n be Element of NAT ; ::_thesis: |.(z P_dt).| . n <= |.z.| * ((|.z.| GeoSeq) . n) |.(z P_dt).| . n = |.((z P_dt) . n).| by VALUED_1:18 .= |.((z |^ (n + 1)) / ((n + 2) !)).| by Def24 .= |.((z |^ (n + 1)) / ((n + 2) !)).| ; then A7: |.(z P_dt).| . n = |.(z #N (n + 1)).| / ((n + 2) !) by Lm13 .= (|.z.| |^ (n + 1)) / ((n + 2) !) by Lm8 ; A8: |.z.| * ((|.z.| GeoSeq) . n) = |.z.| * (|.z.| |^ n) by PREPOWER:def_1 .= |.z.| |^ (n + 1) by NEWTON:6 ; ( (n + 2) ! >= 1 & |.z.| |^ (n + 1) >= 0 ) by Th39, COMPLEX1:46, NEWTON:12, POWER:3; then (|.z.| |^ (n + 1)) / 1 >= (|.z.| |^ (n + 1)) / ((n + 2) !) by XREAL_1:118; hence |.(z P_dt).| . n <= |.z.| * ((|.z.| GeoSeq) . n) by A7, A8; ::_thesis: verum end; let p0 be real number ; ::_thesis: ( p0 > 0 implies ex q being Real st ( q > 0 & ( for z being complex number st |.z.| < q holds |.(Sum (z P_dt)).| < p0 ) ) ) assume A9: p0 > 0 ; ::_thesis: ex q being Real st ( q > 0 & ( for z being complex number st |.z.| < q holds |.(Sum (z P_dt)).| < p0 ) ) reconsider p = p0 as Real by XREAL_0:def_1; consider q being Real such that A10: q = p / (p + 1) ; p + 1 > p by XREAL_1:29; then A11: q < 1 by A9, A10, XREAL_1:189; A12: for z being Element of COMPLEX st |.z.| < q holds |.(Sum (z P_dt)).| < p proof let z be Element of COMPLEX ; ::_thesis: ( |.z.| < q implies |.(Sum (z P_dt)).| < p ) assume A13: |.z.| < q ; ::_thesis: |.(Sum (z P_dt)).| < p then A14: |.z.| < 1 by A11, XXREAL_0:2; A15: abs |.z.| < 1 by A11, A13, XXREAL_0:2; then A16: |.z.| GeoSeq is summable by SERIES_1:24; A17: Sum (|.z.| GeoSeq) = 1 / (1 - |.z.|) by A15, SERIES_1:24; A18: |.z.| (#) (|.z.| GeoSeq) is summable by A16, SERIES_1:10; A19: for n being Element of NAT holds |.(z P_dt).| . n <= (|.z.| (#) (|.z.| GeoSeq)) . n proof let n be Element of NAT ; ::_thesis: |.(z P_dt).| . n <= (|.z.| (#) (|.z.| GeoSeq)) . n |.(z P_dt).| . n <= |.z.| * ((|.z.| GeoSeq) . n) by A6; hence |.(z P_dt).| . n <= (|.z.| (#) (|.z.| GeoSeq)) . n by SEQ_1:9; ::_thesis: verum end; for n being Element of NAT holds 0 <= |.(z P_dt).| . n proof let n be Element of NAT ; ::_thesis: 0 <= |.(z P_dt).| . n |.(z P_dt).| . n = |.((z P_dt) . n).| by VALUED_1:18; hence 0 <= |.(z P_dt).| . n by COMPLEX1:46; ::_thesis: verum end; then A20: Sum |.(z P_dt).| <= Sum (|.z.| (#) (|.z.| GeoSeq)) by A18, A19, SERIES_1:20; A21: Sum (|.z.| (#) (|.z.| GeoSeq)) = |.z.| / (1 - |.z.|) by A16, A17, SERIES_1:10; A22: |.z.| * (p + 1) < (p / (p + 1)) * (p + 1) by A9, A10, A13, XREAL_1:68; (p / (p + 1)) * (p + 1) = p by A9, XCMPLX_1:87; then A23: ((p * |.z.|) + |.z.|) - (p * |.z.|) < p - (p * |.z.|) by A22, XREAL_1:9; A24: 1 - |.z.| > 0 by A14, XREAL_1:50; then |.z.| / (1 - |.z.|) < (p * (1 - |.z.|)) / (1 - |.z.|) by A23, XREAL_1:74; then |.z.| / (1 - |.z.|) < p by A24, XCMPLX_1:89; then A25: Sum |.(z P_dt).| < p by A20, A21, XXREAL_0:2; |.(Sum (z P_dt)).| <= Sum |.(z P_dt).| by A1; hence |.(Sum (z P_dt)).| < p by A25, XXREAL_0:2; ::_thesis: verum end; take q ; ::_thesis: ( q > 0 & ( for z being complex number st |.z.| < q holds |.(Sum (z P_dt)).| < p0 ) ) thus q > 0 by A9, A10; ::_thesis: for z being complex number st |.z.| < q holds |.(Sum (z P_dt)).| < p0 let z be complex number ; ::_thesis: ( |.z.| < q implies |.(Sum (z P_dt)).| < p0 ) z in COMPLEX by XCMPLX_0:def_2; hence ( |.z.| < q implies |.(Sum (z P_dt)).| < p0 ) by A12; ::_thesis: verum end; theorem Th59: :: SIN_COS:59 for z, z1 being complex number holds (Sum ((z1 + z) ExpSeq)) - (Sum (z1 ExpSeq)) = ((Sum (z1 ExpSeq)) * z) + ((z * (Sum (z P_dt))) * (Sum (z1 ExpSeq))) proof let z, z1 be complex number ; ::_thesis: (Sum ((z1 + z) ExpSeq)) - (Sum (z1 ExpSeq)) = ((Sum (z1 ExpSeq)) * z) + ((z * (Sum (z P_dt))) * (Sum (z1 ExpSeq))) (Sum ((z1 + z) ExpSeq)) - (Sum (z1 ExpSeq)) = ((Sum (z1 ExpSeq)) * (Sum (z ExpSeq))) - ((Sum (z1 ExpSeq)) * 1r) by Lm2 .= (Sum (z1 ExpSeq)) * ((((Sum (z ExpSeq)) - 1r) - z) + z) .= (Sum (z1 ExpSeq)) * ((z * (Sum (z P_dt))) + z) by Th57 .= ((Sum (z1 ExpSeq)) * z) + ((z * (Sum (z P_dt))) * (Sum (z1 ExpSeq))) ; hence (Sum ((z1 + z) ExpSeq)) - (Sum (z1 ExpSeq)) = ((Sum (z1 ExpSeq)) * z) + ((z * (Sum (z P_dt))) * (Sum (z1 ExpSeq))) ; ::_thesis: verum end; theorem Th60: :: SIN_COS:60 for p, q being Real holds (cos . (p + q)) - (cos . p) = (- (q * (sin . p))) - (q * (Im ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) proof let p, q be Real; ::_thesis: (cos . (p + q)) - (cos . p) = (- (q * (sin . p))) - (q * (Im ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) (cos . (p + q)) - (cos . p) = (cos . (p + q)) - (Re (Sum ((p * ) ExpSeq))) by Def18 .= (Re (Sum (((p + q) * ) ExpSeq))) - (Re (Sum ((p * ) ExpSeq))) by Def18 .= Re ((Sum (((p * ) + (q * )) ExpSeq)) - (Sum ((p * ) ExpSeq))) by COMPLEX1:19 .= Re (((Sum ((p * ) ExpSeq)) * (q * )) + (((q * ) * (Sum ((q * ) P_dt))) * (Sum ((p * ) ExpSeq)))) by Th59 .= Re ((((cos . p) + ((sin . p) * )) * (q * )) + (((q * ) * (Sum ((q * ) P_dt))) * (Sum ((p * ) ExpSeq)))) by Lm3 .= Re ((((cos . p) + ((sin . p) * )) * (q * )) + (((q * ) * (Sum ((q * ) P_dt))) * ((cos . p) + ((sin . p) * )))) by Lm3 .= (Re ((- (q * (sin . p))) + ((q * (cos . p)) * ))) + (Re (((q * ) * (Sum ((q * ) P_dt))) * ((cos . p) + ((sin . p) * )))) by COMPLEX1:8 .= (- (q * (sin . p))) + (Re ((q * ) * ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) by COMPLEX1:12 .= (- (q * (sin . p))) - (q * (Im ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) by Lm12 ; hence (cos . (p + q)) - (cos . p) = (- (q * (sin . p))) - (q * (Im ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) ; ::_thesis: verum end; theorem Th61: :: SIN_COS:61 for p, q being Real holds (sin . (p + q)) - (sin . p) = (q * (cos . p)) + (q * (Re ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) proof let p, q be Real; ::_thesis: (sin . (p + q)) - (sin . p) = (q * (cos . p)) + (q * (Re ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) (sin . (p + q)) - (sin . p) = (sin . (p + q)) - (Im (Sum ((p * ) ExpSeq))) by Def16 .= (Im (Sum (((p + q) * ) ExpSeq))) - (Im (Sum ((p * ) ExpSeq))) by Def16 .= Im ((Sum (((p * ) + (q * )) ExpSeq)) - (Sum ((p * ) ExpSeq))) by COMPLEX1:19 .= Im (((Sum ((p * ) ExpSeq)) * (q * )) + (((q * ) * (Sum ((q * ) P_dt))) * (Sum ((p * ) ExpSeq)))) by Th59 .= Im ((((cos . p) + ((sin . p) * )) * (q * )) + (((q * ) * (Sum ((q * ) P_dt))) * (Sum ((p * ) ExpSeq)))) by Lm3 .= Im ((((cos . p) + ((sin . p) * )) * (q * )) + (((q * ) * (Sum ((q * ) P_dt))) * ((cos . p) + ((sin . p) * )))) by Lm3 .= (Im ((- (q * (sin . p))) + ((q * (cos . p)) * ))) + (Im (((q * ) * (Sum ((q * ) P_dt))) * ((cos . p) + ((sin . p) * )))) by COMPLEX1:8 .= (q * (cos . p)) + (Im ((q * ) * ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) by COMPLEX1:12 .= (q * (cos . p)) + (q * (Re ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) by Lm12 ; hence (sin . (p + q)) - (sin . p) = (q * (cos . p)) + (q * (Re ((Sum ((q * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) ; ::_thesis: verum end; theorem Th62: :: SIN_COS:62 for p, q being Real holds (exp_R . (p + q)) - (exp_R . p) = (q * (exp_R . p)) + ((q * (exp_R . p)) * (Re (Sum (q P_dt)))) proof let p, q be Real; ::_thesis: (exp_R . (p + q)) - (exp_R . p) = (q * (exp_R . p)) + ((q * (exp_R . p)) * (Re (Sum (q P_dt)))) (exp_R . (p + q)) - (exp_R . p) = (exp_R . (p + q)) - (Re (Sum (p ExpSeq))) by Th48 .= (Re (Sum ((p + q) ExpSeq))) - (Re (Sum (p ExpSeq))) by Th48 .= Re ((Sum ((p + q) ExpSeq)) - (Sum (p ExpSeq))) by COMPLEX1:19 .= Re (((Sum (p ExpSeq)) * q) + ((q * (Sum (q P_dt))) * (Sum (p ExpSeq)))) by Th59 .= (Re ((Sum (p ExpSeq)) * q)) + (Re ((q * (Sum (q P_dt))) * (Sum (p ExpSeq)))) by COMPLEX1:8 .= (q * (Re (Sum (p ExpSeq)))) + (Re ((q * (Sum (q P_dt))) * (Sum (p ExpSeq)))) by Lm12 .= (q * (exp_R . p)) + (Re (q * ((Sum (q P_dt)) * (Sum (p ExpSeq))))) by Th48 .= (q * (exp_R . p)) + (q * (Re ((Sum (q P_dt)) * (Sum (p ExpSeq))))) by Lm12 .= (q * (exp_R . p)) + (q * (Re ((Sum (q P_dt)) * (Sum (p rExpSeq))))) by Lm9 .= (q * (exp_R . p)) + (q * (Re ((Sum (q P_dt)) * ((exp_R . p) + (0 * ))))) by Def22 .= (q * (exp_R . p)) + (q * ((exp_R . p) * (Re (Sum (q P_dt))))) by Lm12 .= (q * (exp_R . p)) + ((q * (exp_R . p)) * (Re (Sum (q P_dt)))) ; hence (exp_R . (p + q)) - (exp_R . p) = (q * (exp_R . p)) + ((q * (exp_R . p)) * (Re (Sum (q P_dt)))) ; ::_thesis: verum end; theorem Th63: :: SIN_COS:63 for p being real number holds ( cos is_differentiable_in p & diff (cos,p) = - (sin . p) ) proof let p be real number ; ::_thesis: ( cos is_differentiable_in p & diff (cos,p) = - (sin . p) ) reconsider p = p as Real by XREAL_0:def_1; deffunc H2( Element of REAL ) -> Element of REAL = - (\$1 * (Im ((Sum ((\$1 * ) P_dt)) * ((cos . p) + ((sin . p) * ))))); consider Cr being Function of REAL,REAL such that A1: for th being Real holds Cr . th = H2(th) from FUNCT_2:sch_4(); for hy1 being non-zero 0 -convergent Real_Sequence holds ( (hy1 ") (#) (Cr /* hy1) is convergent & lim ((hy1 ") (#) (Cr /* hy1)) = 0 ) proof let hy1 be non-zero 0 -convergent Real_Sequence; ::_thesis: ( (hy1 ") (#) (Cr /* hy1) is convergent & lim ((hy1 ") (#) (Cr /* hy1)) = 0 ) A2: for n being Element of NAT holds ((hy1 ") (#) (Cr /* hy1)) . n = - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) proof let n be Element of NAT ; ::_thesis: ((hy1 ") (#) (Cr /* hy1)) . n = - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) A3: ((hy1 ") (#) (Cr /* hy1)) . n = ((hy1 ") . n) * ((Cr /* hy1) . n) by SEQ_1:8 .= ((hy1 . n) ") * ((Cr /* hy1) . n) by VALUED_1:10 ; dom Cr = REAL by FUNCT_2:def_1; then rng hy1 c= dom Cr ; then A4: ((hy1 ") (#) (Cr /* hy1)) . n = ((hy1 . n) ") * (Cr . (hy1 . n)) by A3, FUNCT_2:108 .= ((hy1 . n) ") * (- ((hy1 . n) * (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))))) by A1 .= - ((((hy1 . n) ") * (hy1 . n)) * (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) ; hy1 . n <> 0 by SEQ_1:5; then ((hy1 ") (#) (Cr /* hy1)) . n = - (1 * (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) by A4, XCMPLX_0:def_7 .= - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) ; hence ((hy1 ") (#) (Cr /* hy1)) . n = - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) ; ::_thesis: verum end; deffunc H3( Real) -> Element of REAL = - (Im ((Sum (((hy1 . \$1) * ) P_dt)) * ((cos . p) + ((sin . p) * )))); consider rseq being Real_Sequence such that A5: for n being Element of NAT holds rseq . n = H3(n) from SEQ_1:sch_1(); deffunc H4( Element of NAT ) -> set = (Sum (((hy1 . \$1) * ) P_dt)) * ((cos . p) + ((sin . p) * )); consider cq1 being Complex_Sequence such that A6: for n being Element of NAT holds cq1 . n = H4(n) from COMSEQ_1:sch_1(); A7: for q being Real st q > 0 holds ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q proof let q be Real; ::_thesis: ( q > 0 implies ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q ) assume A8: q > 0 ; ::_thesis: ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q proof consider r being Real such that A9: r > 0 and A10: for z being complex number st |.z.| < r holds |.(Sum (z P_dt)).| < q by A8, Th58; ( hy1 is convergent & lim hy1 = 0 ) ; then consider k being Element of NAT such that A11: for m being Element of NAT st k <= m holds abs ((hy1 . m) - 0) < r by A9, SEQ_2:def_7; A12: now__::_thesis:_for_m_being_Element_of_NAT_st_k_<=_m_holds_ |.((cq1_._m)_-_0c).|_<_q let m be Element of NAT ; ::_thesis: ( k <= m implies |.((cq1 . m) - 0c).| < q ) assume A13: k <= m ; ::_thesis: |.((cq1 . m) - 0c).| < q A14: |.((cq1 . m) - 0c).| = |.((Sum (((hy1 . m) * ) P_dt)) * ((cos . p) + ((sin . p) * ))).| by A6 .= |.(Sum (((hy1 . m) * ) P_dt)).| * |.((cos . p) + ((sin . p) * )).| by COMPLEX1:65 .= |.(Sum (((hy1 . m) * ) P_dt)).| * |.(Sum ((p * ) ExpSeq)).| by Lm3 .= |.(Sum (((hy1 . m) * ) P_dt)).| * 1 by Lm5 .= |.(Sum (((hy1 . m) * ) P_dt)).| ; A15: abs ((hy1 . m) - 0) < r by A11, A13; (hy1 . m) * = 0 + ((hy1 . m) * ) ; then ( Re ((hy1 . m) * ) = 0 & Im ((hy1 . m) * ) = hy1 . m ) by COMPLEX1:12; then |.((hy1 . m) * ).| = abs (hy1 . m) by COMPLEX1:72; hence |.((cq1 . m) - 0c).| < q by A10, A14, A15; ::_thesis: verum end; take k ; ::_thesis: for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q thus for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q by A12; ::_thesis: verum end; hence ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q ; ::_thesis: verum end; then A16: cq1 is convergent by COMSEQ_2:def_5; then lim cq1 = 0c by A7, COMSEQ_2:def_6; then A17: lim (- cq1) = - 0c by A16, COMSEQ_2:22; A18: for n being Element of NAT holds (Im (- cq1)) . n = rseq . n proof A19: for n being Element of NAT holds (Im (- cq1)) . n = - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) proof let n be Element of NAT ; ::_thesis: (Im (- cq1)) . n = - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) (Im (- cq1)) . n = Im ((- cq1) . n) by COMSEQ_3:def_6 .= Im (- (cq1 . n)) by VALUED_1:8 .= Im (- ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) by A6 .= - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) by COMPLEX1:17 ; hence (Im (- cq1)) . n = - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) ; ::_thesis: verum end; let n be Element of NAT ; ::_thesis: (Im (- cq1)) . n = rseq . n rseq . n = - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) by A5; hence (Im (- cq1)) . n = rseq . n by A19; ::_thesis: verum end; for n being Element of NAT holds rseq . n = ((hy1 ") (#) (Cr /* hy1)) . n proof let n be Element of NAT ; ::_thesis: rseq . n = ((hy1 ") (#) (Cr /* hy1)) . n rseq . n = - (Im ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) by A5; hence rseq . n = ((hy1 ") (#) (Cr /* hy1)) . n by A2; ::_thesis: verum end; then rseq = (hy1 ") (#) (Cr /* hy1) by FUNCT_2:63; then (hy1 ") (#) (Cr /* hy1) = Im (- cq1) by A18, FUNCT_2:63; hence ( (hy1 ") (#) (Cr /* hy1) is convergent & lim ((hy1 ") (#) (Cr /* hy1)) = 0 ) by A16, A17, COMPLEX1:4, COMSEQ_3:41; ::_thesis: verum end; then reconsider PR = Cr as RestFunc by FDIFF_1:def_2; deffunc H3( Element of REAL ) -> Element of REAL = - (\$1 * (sin . p)); consider CL being Function of REAL,REAL such that A20: for th being Real holds CL . th = H3(th) from FUNCT_2:sch_4(); ex r being Real st for q being Real holds CL . q = r * q proof A21: for q being Real holds CL . q = (- (sin . p)) * q proof let q be Real; ::_thesis: CL . q = (- (sin . p)) * q CL . q = - (q * (sin . p)) by A20 .= (- (sin . p)) * q ; hence CL . q = (- (sin . p)) * q ; ::_thesis: verum end; take - (sin . p) ; ::_thesis: for q being Real holds CL . q = (- (sin . p)) * q thus for q being Real holds CL . q = (- (sin . p)) * q by A21; ::_thesis: verum end; then reconsider PL = CL as LinearFunc by FDIFF_1:def_3; A22: ex N being Neighbourhood of p st ( N c= dom cos & ( for r being Real st r in N holds (cos . r) - (cos . p) = (PL . (r - p)) + (PR . (r - p)) ) ) proof A23: for r being Real st r in ].(p - 1),(p + 1).[ holds (cos . r) - (cos . p) = (PL . (r - p)) + (PR . (r - p)) proof let r be Real; ::_thesis: ( r in ].(p - 1),(p + 1).[ implies (cos . r) - (cos . p) = (PL . (r - p)) + (PR . (r - p)) ) r = p + (r - p) ; then (cos . r) - (cos . p) = (- ((r - p) * (sin . p))) - ((r - p) * (Im ((Sum (((r - p) * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) by Th60 .= (- ((r - p) * (sin . p))) + (Cr . (r - p)) by A1 .= (PL . (r - p)) + (PR . (r - p)) by A20 ; hence ( r in ].(p - 1),(p + 1).[ implies (cos . r) - (cos . p) = (PL . (r - p)) + (PR . (r - p)) ) ; ::_thesis: verum end; take ].(p - 1),(p + 1).[ ; ::_thesis: ( ].(p - 1),(p + 1).[ is Neighbourhood of p & ].(p - 1),(p + 1).[ c= dom cos & ( for r being Real st r in ].(p - 1),(p + 1).[ holds (cos . r) - (cos . p) = (PL . (r - p)) + (PR . (r - p)) ) ) thus ( ].(p - 1),(p + 1).[ is Neighbourhood of p & ].(p - 1),(p + 1).[ c= dom cos & ( for r being Real st r in ].(p - 1),(p + 1).[ holds (cos . r) - (cos . p) = (PL . (r - p)) + (PR . (r - p)) ) ) by A23, Th24, RCOMP_1:def_6; ::_thesis: verum end; then A24: cos is_differentiable_in p by FDIFF_1:def_4; PL . 1 = - (1 * (sin . p)) by A20 .= - (sin . p) ; hence ( cos is_differentiable_in p & diff (cos,p) = - (sin . p) ) by A22, A24, FDIFF_1:def_5; ::_thesis: verum end; theorem Th64: :: SIN_COS:64 for p being real number holds ( sin is_differentiable_in p & diff (sin,p) = cos . p ) proof let p be real number ; ::_thesis: ( sin is_differentiable_in p & diff (sin,p) = cos . p ) reconsider p = p as Real by XREAL_0:def_1; deffunc H2( Element of REAL ) -> Element of REAL = \$1 * (Re ((Sum ((\$1 * ) P_dt)) * ((cos . p) + ((sin . p) * )))); consider Cr being Function of REAL,REAL such that A1: for th being Real holds Cr . th = H2(th) from FUNCT_2:sch_4(); for hy1 being non-zero 0 -convergent Real_Sequence holds ( (hy1 ") (#) (Cr /* hy1) is convergent & lim ((hy1 ") (#) (Cr /* hy1)) = 0 ) proof let hy1 be non-zero 0 -convergent Real_Sequence; ::_thesis: ( (hy1 ") (#) (Cr /* hy1) is convergent & lim ((hy1 ") (#) (Cr /* hy1)) = 0 ) A2: for n being Element of NAT holds ((hy1 ") (#) (Cr /* hy1)) . n = Re ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * ))) proof let n be Element of NAT ; ::_thesis: ((hy1 ") (#) (Cr /* hy1)) . n = Re ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * ))) A3: ((hy1 ") (#) (Cr /* hy1)) . n = ((hy1 ") . n) * ((Cr /* hy1) . n) by SEQ_1:8 .= ((hy1 . n) ") * ((Cr /* hy1) . n) by VALUED_1:10 ; dom Cr = REAL by FUNCT_2:def_1; then rng hy1 c= dom Cr ; then A4: ((hy1 ") (#) (Cr /* hy1)) . n = ((hy1 . n) ") * (Cr . (hy1 . n)) by A3, FUNCT_2:108 .= ((hy1 . n) ") * ((hy1 . n) * (Re ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) by A1 .= (((hy1 . n) ") * (hy1 . n)) * (Re ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) ; hy1 . n <> 0 by SEQ_1:5; then ((hy1 ") (#) (Cr /* hy1)) . n = 1 * (Re ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * )))) by A4, XCMPLX_0:def_7 .= Re ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * ))) ; hence ((hy1 ") (#) (Cr /* hy1)) . n = Re ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * ))) ; ::_thesis: verum end; deffunc H3( Real) -> Element of REAL = Re ((Sum (((hy1 . \$1) * ) P_dt)) * ((cos . p) + ((sin . p) * ))); consider rseq being Real_Sequence such that A5: for n being Element of NAT holds rseq . n = H3(n) from SEQ_1:sch_1(); deffunc H4( Element of NAT ) -> set = (Sum (((hy1 . \$1) * ) P_dt)) * ((cos . p) + ((sin . p) * )); consider cq1 being Complex_Sequence such that A6: for n being Element of NAT holds cq1 . n = H4(n) from COMSEQ_1:sch_1(); A7: for q being Real st q > 0 holds ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q proof let q be Real; ::_thesis: ( q > 0 implies ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q ) assume A8: q > 0 ; ::_thesis: ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q proof consider r being Real such that A9: r > 0 and A10: for z being complex number st |.z.| < r holds |.(Sum (z P_dt)).| < q by A8, Th58; ( hy1 is convergent & lim hy1 = 0 ) ; then consider k being Element of NAT such that A11: for m being Element of NAT st k <= m holds abs ((hy1 . m) - 0) < r by A9, SEQ_2:def_7; A12: now__::_thesis:_for_m_being_Element_of_NAT_st_k_<=_m_holds_ |.((cq1_._m)_-_0c).|_<_q let m be Element of NAT ; ::_thesis: ( k <= m implies |.((cq1 . m) - 0c).| < q ) assume A13: k <= m ; ::_thesis: |.((cq1 . m) - 0c).| < q A14: |.((cq1 . m) - 0c).| = |.((Sum (((hy1 . m) * ) P_dt)) * ((cos . p) + ((sin . p) * ))).| by A6 .= |.(Sum (((hy1 . m) * ) P_dt)).| * |.((cos . p) + ((sin . p) * )).| by COMPLEX1:65 .= |.(Sum (((hy1 . m) * ) P_dt)).| * |.(Sum ((p * ) ExpSeq)).| by Lm3 .= |.(Sum (((hy1 . m) * ) P_dt)).| * 1 by Lm5 .= |.(Sum (((hy1 . m) * ) P_dt)).| ; A15: abs ((hy1 . m) - 0) < r by A11, A13; (hy1 . m) * = 0 + ((hy1 . m) * ) ; then ( Re ((hy1 . m) * ) = 0 & Im ((hy1 . m) * ) = hy1 . m ) by COMPLEX1:12; then |.((hy1 . m) * ).| = abs (hy1 . m) by COMPLEX1:72; hence |.((cq1 . m) - 0c).| < q by A10, A14, A15; ::_thesis: verum end; take k ; ::_thesis: for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q thus for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q by A12; ::_thesis: verum end; hence ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q ; ::_thesis: verum end; then A16: cq1 is convergent by COMSEQ_2:def_5; then A17: lim cq1 = 0c by A7, COMSEQ_2:def_6; A18: for n being Element of NAT holds (Re cq1) . n = rseq . n proof let n be Element of NAT ; ::_thesis: (Re cq1) . n = rseq . n (Re cq1) . n = Re (cq1 . n) by COMSEQ_3:def_5 .= Re ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * ))) by A6 ; hence (Re cq1) . n = rseq . n by A5; ::_thesis: verum end; for n being Element of NAT holds rseq . n = ((hy1 ") (#) (Cr /* hy1)) . n proof let n be Element of NAT ; ::_thesis: rseq . n = ((hy1 ") (#) (Cr /* hy1)) . n rseq . n = Re ((Sum (((hy1 . n) * ) P_dt)) * ((cos . p) + ((sin . p) * ))) by A5; hence rseq . n = ((hy1 ") (#) (Cr /* hy1)) . n by A2; ::_thesis: verum end; then rseq = (hy1 ") (#) (Cr /* hy1) by FUNCT_2:63; then (hy1 ") (#) (Cr /* hy1) = Re cq1 by A18, FUNCT_2:63; hence ( (hy1 ") (#) (Cr /* hy1) is convergent & lim ((hy1 ") (#) (Cr /* hy1)) = 0 ) by A16, A17, COMPLEX1:4, COMSEQ_3:41; ::_thesis: verum end; then reconsider PR = Cr as RestFunc by FDIFF_1:def_2; deffunc H3( Element of REAL ) -> Element of REAL = \$1 * (cos . p); consider CL being Function of REAL,REAL such that A19: for th being Real holds CL . th = H3(th) from FUNCT_2:sch_4(); A20: for d being real number holds CL . d = d * (cos . p) proof let d be real number ; ::_thesis: CL . d = d * (cos . p) d is Real by XREAL_0:def_1; hence CL . d = d * (cos . p) by A19; ::_thesis: verum end; ex r being Real st for q being Real holds CL . q = r * q proof take cos . p ; ::_thesis: for q being Real holds CL . q = (cos . p) * q thus for q being Real holds CL . q = (cos . p) * q by A20; ::_thesis: verum end; then reconsider PL = CL as LinearFunc by FDIFF_1:def_3; A21: ex N being Neighbourhood of p st ( N c= dom sin & ( for r being Real st r in N holds (sin . r) - (sin . p) = (PL . (r - p)) + (PR . (r - p)) ) ) proof A22: for r being Real st r in ].(p - 1),(p + 1).[ holds (sin . r) - (sin . p) = (PL . (r - p)) + (PR . (r - p)) proof let r be Real; ::_thesis: ( r in ].(p - 1),(p + 1).[ implies (sin . r) - (sin . p) = (PL . (r - p)) + (PR . (r - p)) ) r = p + (r - p) ; then (sin . r) - (sin . p) = ((r - p) * (cos . p)) + ((r - p) * (Re ((Sum (((r - p) * ) P_dt)) * ((cos . p) + ((sin . p) * ))))) by Th61 .= ((r - p) * (cos . p)) + (Cr . (r - p)) by A1 .= (PL . (r - p)) + (PR . (r - p)) by A20 ; hence ( r in ].(p - 1),(p + 1).[ implies (sin . r) - (sin . p) = (PL . (r - p)) + (PR . (r - p)) ) ; ::_thesis: verum end; take ].(p - 1),(p + 1).[ ; ::_thesis: ( ].(p - 1),(p + 1).[ is Neighbourhood of p & ].(p - 1),(p + 1).[ c= dom sin & ( for r being Real st r in ].(p - 1),(p + 1).[ holds (sin . r) - (sin . p) = (PL . (r - p)) + (PR . (r - p)) ) ) thus ( ].(p - 1),(p + 1).[ is Neighbourhood of p & ].(p - 1),(p + 1).[ c= dom sin & ( for r being Real st r in ].(p - 1),(p + 1).[ holds (sin . r) - (sin . p) = (PL . (r - p)) + (PR . (r - p)) ) ) by A22, Th24, RCOMP_1:def_6; ::_thesis: verum end; then A23: sin is_differentiable_in p by FDIFF_1:def_4; PL . 1 = 1 * (cos . p) by A20; hence ( sin is_differentiable_in p & diff (sin,p) = cos . p ) by A21, A23, FDIFF_1:def_5; ::_thesis: verum end; theorem Th65: :: SIN_COS:65 for p being real number holds ( exp_R is_differentiable_in p & diff (exp_R,p) = exp_R . p ) proof let p be real number ; ::_thesis: ( exp_R is_differentiable_in p & diff (exp_R,p) = exp_R . p ) deffunc H2( Element of REAL ) -> Element of REAL = (\$1 * (exp_R . p)) * (Re (Sum (\$1 P_dt))); consider Cr being Function of REAL,REAL such that A1: for th being Real holds Cr . th = H2(th) from FUNCT_2:sch_4(); for hy1 being non-zero 0 -convergent Real_Sequence holds ( (hy1 ") (#) (Cr /* hy1) is convergent & lim ((hy1 ") (#) (Cr /* hy1)) = 0 ) proof let hy1 be non-zero 0 -convergent Real_Sequence; ::_thesis: ( (hy1 ") (#) (Cr /* hy1) is convergent & lim ((hy1 ") (#) (Cr /* hy1)) = 0 ) A2: for n being Element of NAT holds ((hy1 ") (#) (Cr /* hy1)) . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) proof let n be Element of NAT ; ::_thesis: ((hy1 ") (#) (Cr /* hy1)) . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) A3: ((hy1 ") (#) (Cr /* hy1)) . n = ((hy1 ") . n) * ((Cr /* hy1) . n) by SEQ_1:8 .= ((hy1 . n) ") * ((Cr /* hy1) . n) by VALUED_1:10 ; dom Cr = REAL by FUNCT_2:def_1; then rng hy1 c= dom Cr ; then A4: ((hy1 ") (#) (Cr /* hy1)) . n = ((hy1 . n) ") * (Cr . (hy1 . n)) by A3, FUNCT_2:108 .= ((hy1 . n) ") * (((hy1 . n) * (exp_R . p)) * (Re (Sum ((hy1 . n) P_dt)))) by A1 .= (((hy1 . n) ") * (hy1 . n)) * ((exp_R . p) * (Re (Sum ((hy1 . n) P_dt)))) ; hy1 . n <> 0 by SEQ_1:5; then ((hy1 ") (#) (Cr /* hy1)) . n = 1 * ((exp_R . p) * (Re (Sum ((hy1 . n) P_dt)))) by A4, XCMPLX_0:def_7 .= (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) ; hence ((hy1 ") (#) (Cr /* hy1)) . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) ; ::_thesis: verum end; deffunc H3( Real) -> Element of REAL = (exp_R . p) * (Re (Sum ((hy1 . \$1) P_dt))); consider rseq being Real_Sequence such that A5: for n being Element of NAT holds rseq . n = H3(n) from SEQ_1:sch_1(); deffunc H4( Element of NAT ) -> set = (Sum ((hy1 . \$1) P_dt)) * (exp_R . p); consider cq1 being Complex_Sequence such that A6: for n being Element of NAT holds cq1 . n = H4(n) from COMSEQ_1:sch_1(); A7: for q being Real st q > 0 holds ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q proof let q be Real; ::_thesis: ( q > 0 implies ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q ) assume A8: q > 0 ; ::_thesis: ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q proof exp_R . p > 0 by Th54; then consider r being Real such that A9: r > 0 and A10: for z being complex number st |.z.| < r holds |.(Sum (z P_dt)).| < q / (exp_R . p) by A8, Th58; ( hy1 is convergent & lim hy1 = 0 ) ; then consider k being Element of NAT such that A11: for m being Element of NAT st k <= m holds abs ((hy1 . m) - 0) < r by A9, SEQ_2:def_7; A12: now__::_thesis:_for_m_being_Element_of_NAT_st_k_<=_m_holds_ |.((cq1_._m)_-_0c).|_<_q let m be Element of NAT ; ::_thesis: ( k <= m implies |.((cq1 . m) - 0c).| < q ) assume A13: k <= m ; ::_thesis: |.((cq1 . m) - 0c).| < q A14: |.((cq1 . m) - 0c).| = |.((Sum ((hy1 . m) P_dt)) * (exp_R . p)).| by A6 .= |.(Sum ((hy1 . m) P_dt)).| * |.(exp_R . p).| by COMPLEX1:65 .= |.(Sum ((hy1 . m) P_dt)).| * (sqrt (((Re (exp_R . p)) ^2) + ((Im (exp_R . p)) ^2))) ; exp_R . p = (exp_R . p) + (0 * ) ; then A15: ( Re (exp_R . p) = exp_R . p & Im (exp_R . p) = 0 ) by COMPLEX1:12; A16: exp_R . p > 0 by Th54; then A17: |.((cq1 . m) - 0c).| = |.(Sum ((hy1 . m) P_dt)).| * (exp_R . p) by A14, A15, SQUARE_1:22; abs ((hy1 . m) - 0) < r by A11, A13; then |.((cq1 . m) - 0c).| < (q / (exp_R . p)) * (exp_R . p) by A10, A16, A17, XREAL_1:68; hence |.((cq1 . m) - 0c).| < q by A16, XCMPLX_1:87; ::_thesis: verum end; take k ; ::_thesis: for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q thus for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q by A12; ::_thesis: verum end; hence ex k being Element of NAT st for m being Element of NAT st k <= m holds |.((cq1 . m) - 0c).| < q ; ::_thesis: verum end; then A18: cq1 is convergent by COMSEQ_2:def_5; then A19: lim cq1 = 0c by A7, COMSEQ_2:def_6; A20: for n being Element of NAT holds (Re cq1) . n = rseq . n proof A21: for n being Element of NAT holds (Re cq1) . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) proof let n be Element of NAT ; ::_thesis: (Re cq1) . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) (Re cq1) . n = Re (cq1 . n) by COMSEQ_3:def_5 .= Re ((Sum ((hy1 . n) P_dt)) * (exp_R . p)) by A6 .= (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) by Lm12 ; hence (Re cq1) . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) ; ::_thesis: verum end; let n be Element of NAT ; ::_thesis: (Re cq1) . n = rseq . n rseq . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) by A5; hence (Re cq1) . n = rseq . n by A21; ::_thesis: verum end; for n being Element of NAT holds rseq . n = ((hy1 ") (#) (Cr /* hy1)) . n proof let n be Element of NAT ; ::_thesis: rseq . n = ((hy1 ") (#) (Cr /* hy1)) . n rseq . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt))) by A5; hence rseq . n = ((hy1 ") (#) (Cr /* hy1)) . n by A2; ::_thesis: verum end; then rseq = (hy1 ") (#) (Cr /* hy1) by FUNCT_2:63; then (hy1 ") (#) (Cr /* hy1) = Re cq1 by A20, FUNCT_2:63; hence ( (hy1 ") (#) (Cr /* hy1) is convergent & lim ((hy1 ") (#) (Cr /* hy1)) = 0 ) by A18, A19, COMPLEX1:4, COMSEQ_3:41; ::_thesis: verum end; then reconsider PR = Cr as RestFunc by FDIFF_1:def_2; deffunc H3( Element of REAL ) -> Element of REAL = \$1 * (exp_R . p); consider CL being Function of REAL,REAL such that A22: for th being Real holds CL . th = H3(th) from FUNCT_2:sch_4(); A23: for d being real number holds CL . d = d * (exp_R . p) proof let d be real number ; ::_thesis: CL . d = d * (exp_R . p) d is Real by XREAL_0:def_1; hence CL . d = d * (exp_R . p) by A22; ::_thesis: verum end; ex r being Real st for q being Real holds CL . q = r * q proof take exp_R . p ; ::_thesis: for q being Real holds CL . q = (exp_R . p) * q thus for q being Real holds CL . q = (exp_R . p) * q by A23; ::_thesis: verum end; then reconsider PL = CL as LinearFunc by FDIFF_1:def_3; A24: ex N being Neighbourhood of p st ( N c= dom exp_R & ( for r being Real st r in N holds (exp_R . r) - (exp_R . p) = (PL . (r - p)) + (PR . (r - p)) ) ) proof A25: for r being Real st r in ].(p - 1),(p + 1).[ holds (exp_R . r) - (exp_R . p) = (PL . (r - p)) + (PR . (r - p)) proof let r be Real; ::_thesis: ( r in ].(p - 1),(p + 1).[ implies (exp_R . r) - (exp_R . p) = (PL . (r - p)) + (PR . (r - p)) ) reconsider p = p as Real by XREAL_0:def_1; r = p + (r - p) ; then (exp_R . r) - (exp_R . p) = ((r - p) * (exp_R . p)) + (((r - p) * (exp_R . p)) * (Re (Sum ((r - p) P_dt)))) by Th62 .= ((r - p) * (exp_R . p)) + (Cr . (r - p)) by A1 .= (PL . (r - p)) + (PR . (r - p)) by A23 ; hence ( r in ].(p - 1),(p + 1).[ implies (exp_R . r) - (exp_R . p) = (PL . (r - p)) + (PR . (r - p)) ) ; ::_thesis: verum end; take ].(p - 1),(p + 1).[ ; ::_thesis: ( ].(p - 1),(p + 1).[ is Neighbourhood of p & ].(p - 1),(p + 1).[ c= dom exp_R & ( for r being Real st r in ].(p - 1),(p + 1).[ holds (exp_R . r) - (exp_R . p) = (PL . (r - p)) + (PR . (r - p)) ) ) thus ( ].(p - 1),(p + 1).[ is Neighbourhood of p & ].(p - 1),(p + 1).[ c= dom exp_R & ( for r being Real st r in ].(p - 1),(p + 1).[ holds (exp_R . r) - (exp_R . p) = (PL . (r - p)) + (PR . (r - p)) ) ) by A25, Th47, RCOMP_1:def_6; ::_thesis: verum end; then A26: exp_R is_differentiable_in p by FDIFF_1:def_4; PL . 1 = 1 * (exp_R . p) by A23; hence ( exp_R is_differentiable_in p & diff (exp_R,p) = exp_R . p ) by A24, A26, FDIFF_1:def_5; ::_thesis: verum end; theorem Th66: :: SIN_COS:66 for th being real number holds ( exp_R is_differentiable_on REAL & diff (exp_R,th) = exp_R . th ) proof let th be real number ; ::_thesis: ( exp_R is_differentiable_on REAL & diff (exp_R,th) = exp_R . th ) A1: ( [#] REAL is open Subset of REAL & REAL c= dom exp_R ) by FUNCT_2:def_1; for r being Real st r in REAL holds exp_R is_differentiable_in r by Th65; hence ( exp_R is_differentiable_on REAL & diff (exp_R,th) = exp_R . th ) by A1, Th65, FDIFF_1:9; ::_thesis: verum end; theorem Th67: :: SIN_COS:67 for th being real number holds ( cos is_differentiable_on REAL & diff (cos,th) = - (sin . th) ) proof let th be real number ; ::_thesis: ( cos is_differentiable_on REAL & diff (cos,th) = - (sin . th) ) A1: ( [#] REAL is open Subset of REAL & REAL c= dom cos ) by FUNCT_2:def_1; for r being Real st r in REAL holds cos is_differentiable_in r by Th63; hence ( cos is_differentiable_on REAL & diff (cos,th) = - (sin . th) ) by A1, Th63, FDIFF_1:9; ::_thesis: verum end; theorem Th68: :: SIN_COS:68 for th being real number holds ( sin is_differentiable_on REAL & diff (sin,th) = cos . th ) proof let th be real number ; ::_thesis: ( sin is_differentiable_on REAL & diff (sin,th) = cos . th ) A1: ( [#] REAL is open Subset of REAL & REAL c= dom sin ) by FUNCT_2:def_1; for r being Real st r in REAL holds sin is_differentiable_in r by Th64; hence ( sin is_differentiable_on REAL & diff (sin,th) = cos . th ) by A1, Th64, FDIFF_1:9; ::_thesis: verum end; theorem Th69: :: SIN_COS:69 for th being real number st th in [.0,1.] holds ( 0 < cos . th & cos . th >= 1 / 2 ) proof let th be real number ; ::_thesis: ( th in [.0,1.] implies ( 0 < cos . th & cos . th >= 1 / 2 ) ) assume th in [.0,1.] ; ::_thesis: ( 0 < cos . th & cos . th >= 1 / 2 ) then A1: ( 0 <= th & th <= 1 ) by XXREAL_1:1; th is Real by XREAL_0:def_1; then A2: Partial_Sums (th P_cos) is convergent by Th36; A3: cos . th = Sum (th P_cos) by Th37; lim ((Partial_Sums (th P_cos)) * bq) = lim (Partial_Sums (th P_cos)) by A2, SEQ_4:17; then A4: lim ((Partial_Sums (th P_cos)) * bq) = cos . th by A3, SERIES_1:def_3; for n being Element of NAT holds ((Partial_Sums (th P_cos)) * bq) . n >= 1 / 2 proof let n be Element of NAT ; ::_thesis: ((Partial_Sums (th P_cos)) * bq) . n >= 1 / 2 A5: ((Partial_Sums (th P_cos)) * bq) . 0 = (Partial_Sums (th P_cos)) . (bq . 0) by FUNCT_2:15 .= (Partial_Sums (th P_cos)) . ((2 * 0) + 1) by Lm6 .= ((Partial_Sums (th P_cos)) . 0) + ((th P_cos) . (0 + 1)) by SERIES_1:def_1 .= ((th P_cos) . 0) + ((th P_cos) . (0 + 1)) by SERIES_1:def_1 .= ((((- 1) |^ 0) * (th |^ (2 * 0))) / ((2 * 0) !)) + ((th P_cos) . 1) by Def21 .= ((((- 1) |^ 0) * (th |^ (2 * 0))) / ((2 * 0) !)) + ((((- 1) |^ 1) * (th |^ (2 * 1))) / ((2 * 1) !)) by Def21 .= ((1 * (th |^ (2 * 0))) / ((2 * 0) !)) + ((((- 1) |^ 1) * (th |^ (2 * 1))) / ((2 * 1) !)) by Lm7 .= (1 / 1) + ((((- 1) |^ 1) * (th |^ (2 * 1))) / ((2 * 1) !)) by NEWTON:4, NEWTON:12 .= 1 + (((- 1) * (th |^ (2 * 1))) / ((2 * 1) !)) by NEWTON:5 .= 1 + (((- 1) * (th * th)) / ((2 * 1) !)) by NEWTON:81 .= 1 - ((th ^2) / 2) by NEWTON:14 ; defpred S1[ Element of NAT ] means ((Partial_Sums (th P_cos)) * bq) . \$1 >= 1 / 2; th ^2 <= 1 ^2 by A1, SQUARE_1:15; then ( 1 - (1 / 2) = 1 / 2 & (th ^2) / 2 <= 1 / 2 ) by XREAL_1:72; then A6: S1[ 0 ] by A5, XREAL_1:10; A7: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A8: ((Partial_Sums (th P_cos)) * bq) . k >= 1 / 2 ; ::_thesis: S1[k + 1] ((Partial_Sums (th P_cos)) * bq) . (k + 1) = (Partial_Sums (th P_cos)) . (bq . (k + 1)) by FUNCT_2:15 .= (Partial_Sums (th P_cos)) . ((2 * (k + 1)) + 1) by Lm6 .= ((Partial_Sums (th P_cos)) . (((2 * k) + 1) + 1)) + ((th P_cos) . ((2 * (k + 1)) + 1)) by SERIES_1:def_1 .= (((Partial_Sums (th P_cos)) . ((2 * k) + 1)) + ((th P_cos) . (((2 * k) + 1) + 1))) + ((th P_cos) . ((2 * (k + 1)) + 1)) by SERIES_1:def_1 .= (((Partial_Sums (th P_cos)) . (bq . k)) + ((th P_cos) . (((2 * k) + 1) + 1))) + ((th P_cos) . ((2 * (k + 1)) + 1)) by Lm6 .= ((((Partial_Sums (th P_cos)) * bq) . k) + ((th P_cos) . (((2 * k) + 1) + 1))) + ((th P_cos) . ((2 * (k + 1)) + 1)) by FUNCT_2:15 ; then A9: (((Partial_Sums (th P_cos)) * bq) . (k + 1)) - (((Partial_Sums (th P_cos)) * bq) . k) = ((th P_cos) . (((2 * k) + 1) + 1)) + ((th P_cos) . ((2 * (k + 1)) + 1)) ; A10: (th P_cos) . (((2 * k) + 1) + 1) = (((- 1) |^ (2 * (k + 1))) * (th |^ (2 * (2 * (k + 1))))) / ((2 * (2 * (k + 1))) !) by Def21 .= (1 * (th |^ (2 * (2 * (k + 1))))) / ((2 * (2 * (k + 1))) !) by Lm7 .= (th |^ (2 * (2 * (k + 1)))) / ((2 * (2 * (k + 1))) !) ; A11: (th P_cos) . ((2 * (k + 1)) + 1) = (((- 1) |^ ((2 * (k + 1)) + 1)) * (th |^ (2 * ((2 * (k + 1)) + 1)))) / ((2 * ((2 * (k + 1)) + 1)) !) by Def21 .= ((- 1) * (th |^ (2 * ((2 * (k + 1)) + 1)))) / ((2 * ((2 * (k + 1)) + 1)) !) by Lm7 ; A12: 2 * (2 * (k + 1)) < 2 * ((2 * (k + 1)) + 1) by XREAL_1:29, XREAL_1:68; then A13: (2 * (2 * (k + 1))) ! <= (2 * ((2 * (k + 1)) + 1)) ! by Th39; A14: th |^ (2 * ((2 * (k + 1)) + 1)) <= th |^ (2 * (2 * (k + 1))) by A1, A12, Th40; A15: ( 0 <= th |^ (2 * ((2 * (k + 1)) + 1)) & (2 * ((2 * (k + 1)) + 1)) ! > 0 ) by POWER:3; 1 / ((2 * ((2 * (k + 1)) + 1)) !) <= 1 / ((2 * (2 * (k + 1))) !) by A13, XREAL_1:85; then (th |^ (2 * ((2 * (k + 1)) + 1))) * (1 / ((2 * ((2 * (k + 1)) + 1)) !)) <= (th |^ (2 * (2 * (k + 1)))) * (1 / ((2 * (2 * (k + 1))) !)) by A14, A15, XREAL_1:66; then (((th |^ (2 * (2 * (k + 1)))) * 1) / ((2 * (2 * (k + 1))) !)) - (((th |^ (2 * ((2 * (k + 1)) + 1))) * 1) / ((2 * ((2 * (k + 1)) + 1)) !)) >= 0 by XREAL_1:48; then ((Partial_Sums (th P_cos)) * bq) . (k + 1) >= ((Partial_Sums (th P_cos)) * bq) . k by A9, A10, A11, XREAL_1:49; hence S1[k + 1] by A8, XXREAL_0:2; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A6, A7); hence ((Partial_Sums (th P_cos)) * bq) . n >= 1 / 2 ; ::_thesis: verum end; hence ( 0 < cos . th & cos . th >= 1 / 2 ) by A2, A4, PREPOWER:1, SEQ_4:16; ::_thesis: verum end; definition func tan -> PartFunc of REAL,REAL equals :: SIN_COS:def 26 sin / cos; coherence sin / cos is PartFunc of REAL,REAL ; func cot -> PartFunc of REAL,REAL equals :: SIN_COS:def 27 cos / sin; coherence cos / sin is PartFunc of REAL,REAL ; end; :: deftheorem defines tan SIN_COS:def_26_:_ tan = sin / cos; :: deftheorem defines cot SIN_COS:def_27_:_ cot = cos / sin; theorem Th70: :: SIN_COS:70 ( [.0,1.] c= dom tan & ].0,1.[ c= dom tan ) proof A1: [.0,1.] \ (cos " {0}) c= (dom cos) \ (cos " {0}) by Th24, XBOOLE_1:33; [.0,1.] /\ (cos " {0}) = {} proof assume [.0,1.] /\ (cos " {0}) <> {} ; ::_thesis: contradiction then consider rr being set such that A2: rr in [.0,1.] /\ (cos " {0}) by XBOOLE_0:def_1; A3: rr in [.0,1.] by A2, XBOOLE_0:def_4; A4: rr in cos " {0} by A2, XBOOLE_0:def_4; A5: cos . rr <> 0 by A3, Th69; cos . rr in {0} by A4, FUNCT_1:def_7; hence contradiction by A5, TARSKI:def_1; ::_thesis: verum end; then [.0,1.] misses cos " {0} by XBOOLE_0:def_7; then [.0,1.] c= (dom cos) \ (cos " {0}) by A1, XBOOLE_1:83; then [.0,1.] c= (dom sin) /\ ((dom cos) \ (cos " {0})) by Th24, XBOOLE_1:19; then A6: [.0,1.] c= dom tan by RFUNCT_1:def_1; ].0,1.[ c= [.0,1.] by XXREAL_1:25; hence ( [.0,1.] c= dom tan & ].0,1.[ c= dom tan ) by A6, XBOOLE_1:1; ::_thesis: verum end; Lm14: ( dom (tan | [.0,1.]) = [.0,1.] & ( for th being real number st th in [.0,1.] holds (tan | [.0,1.]) . th = tan . th ) ) proof dom (tan | [.0,1.]) = (dom tan) /\ [.0,1.] by RELAT_1:61; then dom (tan | [.0,1.]) = [.0,1.] by Th70, XBOOLE_1:28; hence ( dom (tan | [.0,1.]) = [.0,1.] & ( for th being real number st th in [.0,1.] holds (tan | [.0,1.]) . th = tan . th ) ) by FUNCT_1:47; ::_thesis: verum end; Lm15: ( tan is_differentiable_on ].0,1.[ & ( for th being real number st th in ].0,1.[ holds diff (tan,th) > 0 ) ) proof A1: ( sin is_differentiable_on ].0,1.[ & cos is_differentiable_on ].0,1.[ ) by Th67, Th68, FDIFF_1:26; A2: for th being Real st th in ].0,1.[ holds cos . th <> 0 proof let th be Real; ::_thesis: ( th in ].0,1.[ implies cos . th <> 0 ) assume A3: th in ].0,1.[ ; ::_thesis: cos . th <> 0 ].0,1.[ c= [.0,1.] by XXREAL_1:25; hence cos . th <> 0 by A3, Th69; ::_thesis: verum end; for th being real number st th in ].0,1.[ holds diff (tan,th) > 0 proof let th be real number ; ::_thesis: ( th in ].0,1.[ implies diff (tan,th) > 0 ) assume A4: th in ].0,1.[ ; ::_thesis: diff (tan,th) > 0 A5: ( th is Real & cos is_differentiable_in th ) by Th63, XREAL_0:def_1; A6: sin is_differentiable_in th by Th64; A7: cos . th <> 0 by A2, A4; then A8: diff (tan,th) = (((diff (sin,th)) * (cos . th)) - ((diff (cos,th)) * (sin . th))) / ((cos . th) ^2) by A5, A6, FDIFF_2:14 .= (((cos . th) * (cos . th)) - ((diff (cos,th)) * (sin . th))) / ((cos . th) ^2) by Th64 .= (((cos . th) * (cos . th)) - ((- (sin . th)) * (sin . th))) / ((cos . th) ^2) by Th63 .= (((cos . th) ^2) + ((sin . th) * (sin . th))) / ((cos . th) ^2) .= 1 / ((cos . th) ^2) by Th28 ; (cos . th) ^2 > 0 by A7, SQUARE_1:12; hence diff (tan,th) > 0 by A8; ::_thesis: verum end; hence ( tan is_differentiable_on ].0,1.[ & ( for th being real number st th in ].0,1.[ holds diff (tan,th) > 0 ) ) by A1, A2, FDIFF_2:21; ::_thesis: verum end; theorem Th71: :: SIN_COS:71 tan | [.0,1.] is continuous proof for th being real number st th in dom (tan | [.0,1.]) holds tan | [.0,1.] is_continuous_in th proof let th be real number ; ::_thesis: ( th in dom (tan | [.0,1.]) implies tan | [.0,1.] is_continuous_in th ) assume A1: th in dom (tan | [.0,1.]) ; ::_thesis: tan | [.0,1.] is_continuous_in th then A2: th in [.0,1.] by RELAT_1:57; dom sin = REAL by FUNCT_2:def_1; then A3: th in dom sin by XREAL_0:def_1; A4: th in [#] REAL by XREAL_0:def_1; then sin is_differentiable_in th by Th68, FDIFF_1:9; then A5: sin is_continuous_in th by FDIFF_1:24; cos is_differentiable_in th by A4, Th67, FDIFF_1:9; then A6: cos is_continuous_in th by FDIFF_1:24; cos . th <> 0 by A2, Th69; then A7: tan is_continuous_in th by A3, A5, A6, FCONT_1:11; now__::_thesis:_for_rseq_being_Real_Sequence_st_rng_rseq_c=_dom_(tan_|_[.0,1.])_&_rseq_is_convergent_&_lim_rseq_=_th_holds_ (_(tan_|_[.0,1.])_/*_rseq_is_convergent_&_(tan_|_[.0,1.])_._th_=_lim_((tan_|_[.0,1.])_/*_rseq)_) let rseq be Real_Sequence; ::_thesis: ( rng rseq c= dom (tan | [.0,1.]) & rseq is convergent & lim rseq = th implies ( (tan | [.0,1.]) /* rseq is convergent & (tan | [.0,1.]) . th = lim ((tan | [.0,1.]) /* rseq) ) ) assume that A8: rng rseq c= dom (tan | [.0,1.]) and A9: ( rseq is convergent & lim rseq = th ) ; ::_thesis: ( (tan | [.0,1.]) /* rseq is convergent & (tan | [.0,1.]) . th = lim ((tan | [.0,1.]) /* rseq) ) A10: rng rseq c= dom tan by A8, Lm14, Th70, XBOOLE_1:1; then A11: tan . th = lim (tan /* rseq) by A7, A9, FCONT_1:def_1; now__::_thesis:_for_k1_being_Element_of_NAT_holds_((tan_|_[.0,1.])_/*_rseq)_._k1_=_(tan_/*_rseq)_._k1 let k1 be Element of NAT ; ::_thesis: ((tan | [.0,1.]) /* rseq) . k1 = (tan /* rseq) . k1 dom rseq = NAT by SEQ_1:1; then rseq . k1 in rng rseq by FUNCT_1:def_3; then A12: (tan | [.0,1.]) . (rseq . k1) = tan . (rseq . k1) by A8, Lm14; (tan | [.0,1.]) . (rseq . k1) = ((tan | [.0,1.]) /* rseq) . k1 by A8, FUNCT_2:108; hence ((tan | [.0,1.]) /* rseq) . k1 = (tan /* rseq) . k1 by A8, A12, Lm14, Th70, FUNCT_2:108, XBOOLE_1:1; ::_thesis: verum end; then (tan | [.0,1.]) /* rseq = tan /* rseq by FUNCT_2:63; hence ( (tan | [.0,1.]) /* rseq is convergent & (tan | [.0,1.]) . th = lim ((tan | [.0,1.]) /* rseq) ) by A1, A7, A9, A10, A11, Lm14, FCONT_1:def_1; ::_thesis: verum end; hence tan | [.0,1.] is_continuous_in th by FCONT_1:def_1; ::_thesis: verum end; hence tan | [.0,1.] is continuous by FCONT_1:def_2; ::_thesis: verum end; theorem Th72: :: SIN_COS:72 for th1, th2 being real number st th1 in ].0,1.[ & th2 in ].0,1.[ & tan . th1 = tan . th2 holds th1 = th2 proof let th1, th2 be real number ; ::_thesis: ( th1 in ].0,1.[ & th2 in ].0,1.[ & tan . th1 = tan . th2 implies th1 = th2 ) assume that A1: th1 in ].0,1.[ and A2: th2 in ].0,1.[ and A3: tan . th1 = tan . th2 ; ::_thesis: th1 = th2 A4: 0 < th1 by A1, XXREAL_1:4; A5: th1 < 1 by A1, XXREAL_1:4; A6: 0 < th2 by A2, XXREAL_1:4; A7: th2 < 1 by A2, XXREAL_1:4; assume A8: th1 <> th2 ; ::_thesis: contradiction now__::_thesis:_ex_th_being_Real_st_ (_th_in_].0,1.[_&_diff_(tan,th)_=_0_) percases ( th1 < th2 or th2 < th1 ) by A8, XXREAL_0:1; supposeA9: th1 < th2 ; ::_thesis: ex th being Real st ( th in ].0,1.[ & diff (tan,th) = 0 ) A10: for th being Real st th in ].th1,th2.[ holds th in ].0,1.[ proof let th be Real; ::_thesis: ( th in ].th1,th2.[ implies th in ].0,1.[ ) assume A11: th in ].th1,th2.[ ; ::_thesis: th in ].0,1.[ then A12: th1 < th by XXREAL_1:4; th < th2 by A11, XXREAL_1:4; then th < 1 by A7, XXREAL_0:2; hence th in ].0,1.[ by A4, A12, XXREAL_1:4; ::_thesis: verum end; A13: for th being Real st th in [.th1,th2.] holds th in [.0,1.] proof let th be Real; ::_thesis: ( th in [.th1,th2.] implies th in [.0,1.] ) assume A14: th in [.th1,th2.] ; ::_thesis: th in [.0,1.] then A15: th1 <= th by XXREAL_1:1; th <= th2 by A14, XXREAL_1:1; then th <= 1 by A7, XXREAL_0:2; hence th in [.0,1.] by A4, A15, XXREAL_1:1; ::_thesis: verum end; ].th1,th2.[ c= ].0,1.[ by A10, SUBSET_1:2; then A16: tan is_differentiable_on ].th1,th2.[ by Lm15, FDIFF_1:26; ( [.th1,th2.] c= [.0,1.] & tan | [.th1,th2.] is continuous ) by A13, Th71, FCONT_1:16, SUBSET_1:2; then consider r being Real such that A17: ( r in ].th1,th2.[ & diff (tan,r) = 0 ) by A1, A2, A3, A9, A16, Th70, ROLLE:1, XBOOLE_1:1; take th = r; ::_thesis: ( th in ].0,1.[ & diff (tan,th) = 0 ) thus ( th in ].0,1.[ & diff (tan,th) = 0 ) by A10, A17; ::_thesis: verum end; supposeA18: th2 < th1 ; ::_thesis: ex th being Real st ( th in ].0,1.[ & diff (tan,th) = 0 ) A19: for th being Real st th in ].th2,th1.[ holds th in ].0,1.[ proof let th be Real; ::_thesis: ( th in ].th2,th1.[ implies th in ].0,1.[ ) assume A20: th in ].th2,th1.[ ; ::_thesis: th in ].0,1.[ then A21: th2 < th by XXREAL_1:4; th < th1 by A20, XXREAL_1:4; then th < 1 by A5, XXREAL_0:2; hence th in ].0,1.[ by A6, A21, XXREAL_1:4; ::_thesis: verum end; A22: for th being Real st th in [.th2,th1.] holds th in [.0,1.] proof let th be Real; ::_thesis: ( th in [.th2,th1.] implies th in [.0,1.] ) assume A23: th in [.th2,th1.] ; ::_thesis: th in [.0,1.] then A24: th2 <= th by XXREAL_1:1; th <= th1 by A23, XXREAL_1:1; then th <= 1 by A5, XXREAL_0:2; hence th in [.0,1.] by A6, A24, XXREAL_1:1; ::_thesis: verum end; ].th2,th1.[ c= ].0,1.[ by A19, SUBSET_1:2; then A25: tan is_differentiable_on ].th2,th1.[ by Lm15, FDIFF_1:26; ( [.th2,th1.] c= [.0,1.] & tan | [.th2,th1.] is continuous ) by A22, Th71, FCONT_1:16, SUBSET_1:2; then consider r being Real such that A26: ( r in ].th2,th1.[ & diff (tan,r) = 0 ) by A1, A2, A3, A18, A25, Th70, ROLLE:1, XBOOLE_1:1; take th = r; ::_thesis: ( th in ].0,1.[ & diff (tan,th) = 0 ) thus ( th in ].0,1.[ & diff (tan,th) = 0 ) by A19, A26; ::_thesis: verum end; end; end; hence contradiction by Lm15; ::_thesis: verum end; Lm16: ( tan . 0 = 0 & tan . 1 > 1 ) proof 0 in [.0,1.] by XXREAL_1:1; then A1: tan . 0 = (sin . 0) * ((cos . 0) ") by Th70, RFUNCT_1:def_1 .= 0 by Th30 ; 1 in [.0,1.] by XXREAL_1:1; then tan . 1 = (sin . 1) / (cos . 1) by Th70, RFUNCT_1:def_1; hence ( tan . 0 = 0 & tan . 1 > 1 ) by A1, Th43, XREAL_1:187; ::_thesis: verum end; begin definition func PI -> real number means :Def28: :: SIN_COS:def 28 ( tan . (it / 4) = 1 & it in ].0,4.[ ); existence ex b1 being real number st ( tan . (b1 / 4) = 1 & b1 in ].0,4.[ ) proof A1: [.(tan . 0),(tan . 1).] = { r where r is Real : ( tan . 0 <= r & r <= tan . 1 ) } by RCOMP_1:def_1; [.(tan . 1),(tan . 0).] = {} by Lm16, XXREAL_1:29; then 1 in [.(tan . 0),(tan . 1).] \/ [.(tan . 1),(tan . 0).] by A1, Lm16; then consider th being Real such that A2: th in [.0,1.] and A3: tan . th = 1 by Th70, Th71, FCONT_2:15; A4: 0 <= th by A2, XXREAL_1:1; A5: th <= 1 by A2, XXREAL_1:1; A6: 0 < th by A3, A4, Lm16; A7: th < 1 by A3, A5, Lm16, XXREAL_0:1; take th1 = th * 4; ::_thesis: ( tan . (th1 / 4) = 1 & th1 in ].0,4.[ ) thus tan . (th1 / 4) = 1 by A3; ::_thesis: th1 in ].0,4.[ th * 4 < 1 * 4 by A7, XREAL_1:68; hence th1 in ].0,4.[ by A6, XXREAL_1:4; ::_thesis: verum end; uniqueness for b1, b2 being real number st tan . (b1 / 4) = 1 & b1 in ].0,4.[ & tan . (b2 / 4) = 1 & b2 in ].0,4.[ holds b1 = b2 proof let th1, th2 be real number ; ::_thesis: ( tan . (th1 / 4) = 1 & th1 in ].0,4.[ & tan . (th2 / 4) = 1 & th2 in ].0,4.[ implies th1 = th2 ) assume that A8: tan . (th1 / 4) = 1 and A9: th1 in ].0,4.[ and A10: tan . (th2 / 4) = 1 and A11: th2 in ].0,4.[ ; ::_thesis: th1 = th2 A12: 0 < th1 by A9, XXREAL_1:4; th1 < 4 by A9, XXREAL_1:4; then th1 / 4 < 4 / 4 by XREAL_1:74; then A13: th1 / 4 in ].0,1.[ by A12, XXREAL_1:4; A14: 0 < th2 by A11, XXREAL_1:4; th2 < 4 by A11, XXREAL_1:4; then th2 / 4 < 4 / 4 by XREAL_1:74; then th2 / 4 in ].0,1.[ by A14, XXREAL_1:4; then th1 / 4 = th2 / 4 by A8, A10, A13, Th72; hence th1 = th2 ; ::_thesis: verum end; end; :: deftheorem Def28 defines PI SIN_COS:def_28_:_ for b1 being real number holds ( b1 = PI iff ( tan . (b1 / 4) = 1 & b1 in ].0,4.[ ) ); definition :: original: PI redefine func PI -> Real; coherence PI is Real by XREAL_0:def_1; end; theorem Th73: :: SIN_COS:73 sin . (PI / 4) = cos . (PI / 4) proof A1: PI in ].0,4.[ by Def28; then A2: 0 < PI by XXREAL_1:4; PI < 4 by A1, XXREAL_1:4; then PI / 4 < 4 / 4 by XREAL_1:74; then A3: PI / 4 in ].0,1.[ by A2, XXREAL_1:4; tan . (PI / 4) = 1 by Def28; then (sin . (PI / 4)) * ((cos . (PI / 4)) ") = 1 by A3, Th70, RFUNCT_1:def_1; hence sin . (PI / 4) = cos . (PI / 4) by XCMPLX_1:209; ::_thesis: verum end; begin theorem Th74: :: SIN_COS:74 for th1, th2 being real number holds ( sin . (th1 + th2) = ((sin . th1) * (cos . th2)) + ((cos . th1) * (sin . th2)) & cos . (th1 + th2) = ((cos . th1) * (cos . th2)) - ((sin . th1) * (sin . th2)) ) proof let th1, th2 be real number ; ::_thesis: ( sin . (th1 + th2) = ((sin . th1) * (cos . th2)) + ((cos . th1) * (sin . th2)) & cos . (th1 + th2) = ((cos . th1) * (cos . th2)) - ((sin . th1) * (sin . th2)) ) reconsider th1 = th1, th2 = th2 as Real by XREAL_0:def_1; A1: (th1 + th2) * = (0 + 0) + ((th1 + th2) * ) ; A2: (Sum ((th1 * ) ExpSeq)) * (Sum ((th2 * ) ExpSeq)) = Sum (((th1 * ) + (th2 * )) ExpSeq) by Lm2 .= (cos . (th1 + th2)) + ((sin . (th1 + th2)) * ) by A1, Lm3 ; (Sum ((th1 * ) ExpSeq)) * (Sum ((th2 * ) ExpSeq)) = ((cos . th1) + ((sin . th1) * )) * (Sum ((th2 * ) ExpSeq)) by Lm3 .= ((cos . th1) + ((sin . th1) * )) * ((cos . th2) + ((sin . th2) * )) by Lm3 .= (((cos . th1) * (cos . th2)) - ((sin . th1) * (sin . th2))) + ((((sin . th1) * (cos . th2)) + ((cos . th1) * (sin . th2))) * ) ; hence ( sin . (th1 + th2) = ((sin . th1) * (cos . th2)) + ((cos . th1) * (sin . th2)) & cos . (th1 + th2) = ((cos . th1) * (cos . th2)) - ((sin . th1) * (sin . th2)) ) by A2, COMPLEX1:77; ::_thesis: verum end; theorem :: SIN_COS:75 for th1, th2 being real number holds ( sin (th1 + th2) = ((sin th1) * (cos th2)) + ((cos th1) * (sin th2)) & cos (th1 + th2) = ((cos th1) * (cos th2)) - ((sin th1) * (sin th2)) ) by Th74; theorem Th76: :: SIN_COS:76 ( cos . (PI / 2) = 0 & sin . (PI / 2) = 1 & cos . PI = - 1 & sin . PI = 0 & cos . (PI + (PI / 2)) = 0 & sin . (PI + (PI / 2)) = - 1 & cos . (2 * PI) = 1 & sin . (2 * PI) = 0 ) proof thus A1: cos . (PI / 2) = cos . ((PI / 4) + (PI / 4)) .= ((cos . (PI / 4)) * (cos . (PI / 4))) - ((cos . (PI / 4)) * (cos . (PI / 4))) by Th73, Th74 .= 0 ; ::_thesis: ( sin . (PI / 2) = 1 & cos . PI = - 1 & sin . PI = 0 & cos . (PI + (PI / 2)) = 0 & sin . (PI + (PI / 2)) = - 1 & cos . (2 * PI) = 1 & sin . (2 * PI) = 0 ) thus A2: sin . (PI / 2) = sin . ((PI / 4) + (PI / 4)) .= ((cos . (PI / 4)) * (cos . (PI / 4))) + ((sin . (PI / 4)) * (sin . (PI / 4))) by Th73, Th74 .= 1 by Th28 ; ::_thesis: ( cos . PI = - 1 & sin . PI = 0 & cos . (PI + (PI / 2)) = 0 & sin . (PI + (PI / 2)) = - 1 & cos . (2 * PI) = 1 & sin . (2 * PI) = 0 ) thus A3: cos . PI = cos . ((PI / 2) + (PI / 2)) .= (0 * 0) - ((sin . (PI / 2)) * (sin . (PI / 2))) by A1, Th74 .= - 1 by A2 ; ::_thesis: ( sin . PI = 0 & cos . (PI + (PI / 2)) = 0 & sin . (PI + (PI / 2)) = - 1 & cos . (2 * PI) = 1 & sin . (2 * PI) = 0 ) thus A4: sin . PI = sin . ((PI / 2) + (PI / 2)) .= ((sin . (PI / 2)) * (cos . (PI / 2))) + ((cos . (PI / 2)) * (sin . (PI / 2))) by Th74 .= 0 by A1 ; ::_thesis: ( cos . (PI + (PI / 2)) = 0 & sin . (PI + (PI / 2)) = - 1 & cos . (2 * PI) = 1 & sin . (2 * PI) = 0 ) thus cos . (PI + (PI / 2)) = ((cos . PI) * (cos . (PI / 2))) - ((sin . PI) * (sin . (PI / 2))) by Th74 .= 0 by A1, A4 ; ::_thesis: ( sin . (PI + (PI / 2)) = - 1 & cos . (2 * PI) = 1 & sin . (2 * PI) = 0 ) thus sin . (PI + (PI / 2)) = ((sin . PI) * (cos . (PI / 2))) + ((cos . PI) * (sin . (PI / 2))) by Th74 .= - 1 by A1, A2, A3 ; ::_thesis: ( cos . (2 * PI) = 1 & sin . (2 * PI) = 0 ) thus cos . (2 * PI) = cos . (PI + PI) .= ((- 1) * (- 1)) - ((sin . PI) * (sin . PI)) by A3, Th74 .= 1 by A4 ; ::_thesis: sin . (2 * PI) = 0 thus sin . (2 * PI) = sin . (PI + PI) .= ((sin . PI) * (cos . PI)) + ((cos . PI) * (sin . PI)) by Th74 .= 0 by A4 ; ::_thesis: verum end; theorem :: SIN_COS:77 ( cos (PI / 2) = 0 & sin (PI / 2) = 1 & cos PI = - 1 & sin PI = 0 & cos (PI + (PI / 2)) = 0 & sin (PI + (PI / 2)) = - 1 & cos (2 * PI) = 1 & sin (2 * PI) = 0 ) by Th76; theorem Th78: :: SIN_COS:78 for th being real number holds ( sin . (th + (2 * PI)) = sin . th & cos . (th + (2 * PI)) = cos . th & sin . ((PI / 2) - th) = cos . th & cos . ((PI / 2) - th) = sin . th & sin . ((PI / 2) + th) = cos . th & cos . ((PI / 2) + th) = - (sin . th) & sin . (PI + th) = - (sin . th) & cos . (PI + th) = - (cos . th) ) proof let th be real number ; ::_thesis: ( sin . (th + (2 * PI)) = sin . th & cos . (th + (2 * PI)) = cos . th & sin . ((PI / 2) - th) = cos . th & cos . ((PI / 2) - th) = sin . th & sin . ((PI / 2) + th) = cos . th & cos . ((PI / 2) + th) = - (sin . th) & sin . (PI + th) = - (sin . th) & cos . (PI + th) = - (cos . th) ) thus sin . (th + (2 * PI)) = ((sin . th) * 1) + ((cos . th) * 0) by Th74, Th76 .= sin . th ; ::_thesis: ( cos . (th + (2 * PI)) = cos . th & sin . ((PI / 2) - th) = cos . th & cos . ((PI / 2) - th) = sin . th & sin . ((PI / 2) + th) = cos . th & cos . ((PI / 2) + th) = - (sin . th) & sin . (PI + th) = - (sin . th) & cos . (PI + th) = - (cos . th) ) thus cos . (th + (2 * PI)) = ((cos . th) * 1) - ((sin . th) * 0) by Th74, Th76 .= cos . th ; ::_thesis: ( sin . ((PI / 2) - th) = cos . th & cos . ((PI / 2) - th) = sin . th & sin . ((PI / 2) + th) = cos . th & cos . ((PI / 2) + th) = - (sin . th) & sin . (PI + th) = - (sin . th) & cos . (PI + th) = - (cos . th) ) thus sin . ((PI / 2) - th) = ((sin . (PI / 2)) * (cos . (- th))) + ((cos . (PI / 2)) * (sin . (- th))) by Th74 .= cos . th by Th30, Th76 ; ::_thesis: ( cos . ((PI / 2) - th) = sin . th & sin . ((PI / 2) + th) = cos . th & cos . ((PI / 2) + th) = - (sin . th) & sin . (PI + th) = - (sin . th) & cos . (PI + th) = - (cos . th) ) thus cos . ((PI / 2) - th) = ((cos . (PI / 2)) * (cos . (- th))) - ((sin . (PI / 2)) * (sin . (- th))) by Th74 .= 0 - (1 * (- (sin . th))) by Th30, Th76 .= sin . th ; ::_thesis: ( sin . ((PI / 2) + th) = cos . th & cos . ((PI / 2) + th) = - (sin . th) & sin . (PI + th) = - (sin . th) & cos . (PI + th) = - (cos . th) ) thus sin . ((PI / 2) + th) = (1 * (cos . th)) + (0 * (sin . th)) by Th74, Th76 .= cos . th ; ::_thesis: ( cos . ((PI / 2) + th) = - (sin . th) & sin . (PI + th) = - (sin . th) & cos . (PI + th) = - (cos . th) ) thus cos . ((PI / 2) + th) = ((cos . (PI / 2)) * (cos . th)) - ((sin . (PI / 2)) * (sin . th)) by Th74 .= - (sin . th) by Th76 ; ::_thesis: ( sin . (PI + th) = - (sin . th) & cos . (PI + th) = - (cos . th) ) thus sin . (PI + th) = ((sin . PI) * (cos . th)) + ((cos . PI) * (sin . th)) by Th74 .= - (sin . th) by Th76 ; ::_thesis: cos . (PI + th) = - (cos . th) thus cos . (PI + th) = ((- 1) * (cos . th)) - (0 * (sin . th)) by Th74, Th76 .= - (cos . th) ; ::_thesis: verum end; theorem :: SIN_COS:79 for th being real number holds ( sin (th + (2 * PI)) = sin th & cos (th + (2 * PI)) = cos th & sin ((PI / 2) - th) = cos th & cos ((PI / 2) - th) = sin th & sin ((PI / 2) + th) = cos th & cos ((PI / 2) + th) = - (sin th) & sin (PI + th) = - (sin th) & cos (PI + th) = - (cos th) ) by Th78; Lm17: for th being real number st th in [.0,1.] holds sin . th >= 0 proof let th be real number ; ::_thesis: ( th in [.0,1.] implies sin . th >= 0 ) assume A1: th in [.0,1.] ; ::_thesis: sin . th >= 0 then A2: th <= 1 by XXREAL_1:1; sin . th >= sin . 0 proof now__::_thesis:_sin_._th_>=_sin_._0 percases ( th = 0 or 0 < th ) by A1, XXREAL_1:1; suppose th = 0 ; ::_thesis: sin . th >= sin . 0 hence sin . th >= sin . 0 ; ::_thesis: verum end; supposeA3: 0 < th ; ::_thesis: sin . th >= sin . 0 sin | REAL is continuous by Th68, FDIFF_1:25; then sin | [.0,th.] is continuous by FCONT_1:16; then consider r being Real such that A4: r in ].0,th.[ and A5: diff (sin,r) = ((sin . th) - (sin . 0)) / (th - 0) by A3, Th24, Th68, FDIFF_1:26, ROLLE:3; A6: r < th by A4, XXREAL_1:4; A7: 0 < r by A4, XXREAL_1:4; r < 1 by A2, A6, XXREAL_0:2; then r in [.0,1.] by A7, XXREAL_1:1; then cos . r > 0 by Th69; then (sin . th) - (sin . 0) > 0 by A3, A5, Th68; hence sin . th >= sin . 0 by XREAL_1:47; ::_thesis: verum end; end; end; hence sin . th >= sin . 0 ; ::_thesis: verum end; hence sin . th >= 0 by Th30; ::_thesis: verum end; theorem Th80: :: SIN_COS:80 for th being real number st th in ].0,(PI / 2).[ holds cos . th > 0 proof let th be real number ; ::_thesis: ( th in ].0,(PI / 2).[ implies cos . th > 0 ) assume that A1: th in ].0,(PI / 2).[ and A2: cos . th <= 0 ; ::_thesis: contradiction cos | REAL is continuous by Th67, FDIFF_1:25; then A3: cos | [.0,th.] is continuous by FCONT_1:16; A4: 0 < th by A1, XXREAL_1:4; A5: th is Real by XREAL_0:def_1; ( [.(cos . 0),(cos . th).] \/ [.(cos . th),(cos . 0).] = [.(cos . th),(cos . 0).] & 0 in [.(cos . th),(cos . 0).] ) by A2, Th30, XXREAL_1:1, XXREAL_1:222; then ex th2 being Real st ( th2 in [.0,th.] & cos . th2 = 0 ) by A3, A4, A5, Th24, FCONT_2:15; then consider th2 being Real such that A6: th2 in [.0,th.] and 0 < th and A7: cos . th2 = 0 by A4; A8: 0 <= th2 by A6, XXREAL_1:1; A9: th2 <= th by A6, XXREAL_1:1; A10: th < PI / 2 by A1, XXREAL_1:4; A11: 0 < th2 by A7, A8, Th30; th2 < PI / 2 by A9, A10, XXREAL_0:2; then A12: th2 / 2 < (PI / 2) / 2 by XREAL_1:74; PI in ].0,4.[ by Def28; then PI < 4 by XXREAL_1:4; then PI / 4 < 4 / 4 by XREAL_1:74; then A13: th2 / 2 < 1 by A12, XXREAL_0:2; 0 = cos . ((th2 / 2) + (th2 / 2)) by A7 .= ((cos . (th2 / 2)) ^2) - ((sin . (th2 / 2)) * (sin . (th2 / 2))) by Th74 .= ((cos . (th2 / 2)) - (sin . (th2 / 2))) * ((cos . (th2 / 2)) + (sin . (th2 / 2))) ; then A14: ( (cos . (th2 / 2)) - (sin . (th2 / 2)) = 0 or (cos . (th2 / 2)) + (sin . (th2 / 2)) = 0 ) ; A15: th2 / 2 in ].0,1.[ by A11, A13, XXREAL_1:4; ].0,1.[ c= [.0,1.] by XXREAL_1:25; then A16: ( cos . (th2 / 2) > 0 & sin . (th2 / 2) >= - 0 ) by A15, Lm17, Th69; 4 * (th2 / 2) < 4 * 1 by A13, XREAL_1:68; then A17: 2 * th2 in ].0,4.[ by A11, XXREAL_1:4; (sin . (th2 / 2)) * ((cos . (th2 / 2)) ") = 1 by A14, A16, XCMPLX_0:def_7; then tan . ((2 * th2) / 4) = 1 by A15, Th70, RFUNCT_1:def_1; then 2 * th2 = PI by A17, Def28; hence contradiction by A1, A9, XXREAL_1:4; ::_thesis: verum end; theorem :: SIN_COS:81 for th being real number st th in ].0,(PI / 2).[ holds cos th > 0 by Th80; begin theorem :: SIN_COS:82 for a, b being real number holds sin (a - b) = ((sin a) * (cos b)) - ((cos a) * (sin b)) proof let a, b be real number ; ::_thesis: sin (a - b) = ((sin a) * (cos b)) - ((cos a) * (sin b)) thus sin (a - b) = ((sin . a) * (cos . (- b))) + ((cos . a) * (sin . (- b))) by Th74 .= ((sin . a) * (cos . b)) + ((cos . a) * (sin . (- b))) by Th30 .= ((sin . a) * (cos . b)) + ((cos . a) * (- (sin . b))) by Th30 .= ((sin a) * (cos b)) - ((cos a) * (sin b)) ; ::_thesis: verum end; theorem :: SIN_COS:83 for a, b being real number holds cos (a - b) = ((cos a) * (cos b)) + ((sin a) * (sin b)) proof let a, b be real number ; ::_thesis: cos (a - b) = ((cos a) * (cos b)) + ((sin a) * (sin b)) thus cos (a - b) = ((cos . a) * (cos . (- b))) - ((sin . a) * (sin . (- b))) by Th74 .= ((cos . a) * (cos . b)) - ((sin . a) * (sin . (- b))) by Th30 .= ((cos . a) * (cos . b)) - ((sin . a) * (- (sin . b))) by Th30 .= ((cos a) * (cos b)) + ((sin a) * (sin b)) ; ::_thesis: verum end; registration cluster sin -> continuous ; coherence sin is continuous proof ( dom sin = REAL & sin | REAL is continuous ) by Th68, FDIFF_1:25, FUNCT_2:def_1; hence sin is continuous by RELAT_1:69; ::_thesis: verum end; cluster cos -> continuous ; coherence cos is continuous proof ( dom cos = REAL & cos | REAL is continuous ) by Th67, FDIFF_1:25, FUNCT_2:def_1; hence cos is continuous by RELAT_1:69; ::_thesis: verum end; cluster exp_R -> continuous ; coherence exp_R is continuous proof ( dom exp_R = REAL & exp_R | REAL is continuous ) by Th66, FDIFF_1:25, FUNCT_2:def_1; hence exp_R is continuous by RELAT_1:69; ::_thesis: verum end; end;