:: 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 * <i>)) by Th1
.= ((m -' k) !) * (((m - k) + 1) + (0 * <i>)) 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 * <i>)) * (p2 + (g2 * <i>)) = ((p1 * p2) - (g1 * g2)) + (((p1 * g2) + (p2 * g1)) * <i>) & (p2 + (g2 * <i>)) *' = p2 + ((- g2) * <i>) )
proof
let p1, p2, g1, g2 be Element of REAL ; ::_thesis: ( (p1 + (g1 * <i>)) * (p2 + (g2 * <i>)) = ((p1 * p2) - (g1 * g2)) + (((p1 * g2) + (p2 * g1)) * <i>) & (p2 + (g2 * <i>)) *' = p2 + ((- g2) * <i>) )
thus  (p1 + (g1 * <i>)) * (p2 + (g2 * <i>)) = ((p1 * p2) - (g1 * g2)) + (((p1 * g2) + (p2 * g1)) * <i>) ; ::_thesis: (p2 + (g2 * <i>)) *' = p2 + ((- g2) * <i>)
thus (p2 + (g2 * <i>)) *' = p2 - ((Im (p2 + (g2 * <i>))) * <i>) by COMPLEX1:12
.= p2 - (g2 * <i>) by COMPLEX1:12
.= p2 + ((- g2) * <i>) ; ::_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 * <i>)) & (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 * <i>)) & (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 * <i>)) & (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_*_<i>))
let n be Element of NAT ; ::_thesis: (z ExpSeq) . (n + 1) = (((z ExpSeq) . n) * z) / ((n + 1) + (0 * <i>))
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 * <i>))) by Th1
.= ((z |^ n) / (n !)) * (z / ((n + 1) + (0 * <i>))) by XCMPLX_1:76
.= (((z |^ n) / (n !)) * z) / ((n + 1) + (0 * <i>))
.= (((z ExpSeq) . n) * z) / ((n + 1) + (0 * <i>)) 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 * <i>)).| = n + 1 by ABSVALUE:def_1;
|.((z ExpSeq) . (n + 1)).| = |.((((z ExpSeq) . n) * z) / ((n + 1) + (0 * <i>))).| by A1
.= |.(((z ExpSeq) . n) * z).| / |.((n + 1) + (0 * <i>)).| by COMPLEX1:67
.= (((|.z.| rExpSeq) . n) * |.z.|) / |.((n + 1) + (0 * <i>)).| by A5, COMPLEX1:65
.= (((|.z.| |^ n) / (n !)) * |.z.|) / |.((n + 1) + (0 * <i>)).| by Def5
.= ((|.z.| |^ n) * |.z.|) / ((n !) * |.((n + 1) + (0 * <i>)).|) 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 * <i>)) & (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 * <i>)) = (k !) * (((n -' k) !) * (((n + 1) - k) + (0 * <i>))) ;
A43: (((k -' 1) !) * ((n -' (k -' 1)) !)) * (k + (0 * <i>)) = ((k + (0 * <i>)) * ((k -' 1) !)) * ((n -' (k -' 1)) !)
.= (k !) * (((n + 1) -' k) !) by A33, A39, Th2 ;
 ((n + 1) - k) + (0 * <i>) <> 0c by A26;
then A44: (n !) / ((k !) * ((n -' k) !)) = ((n !) * (((n + 1) - k) + (0 * <i>))) / (((k !) * ((n -' k) !)) * (((n + 1) - k) + (0 * <i>))) by XCMPLX_1:91
.= ((n !) * (((n + 1) - k) + (0 * <i>))) / ((k !) * (((n + 1) -' k) !)) by A35, A42, Th2 ;
 (n !) / (((k -' 1) !) * ((n -' (k -' 1)) !)) = ((n !) * (k + (0 * <i>))) / ((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) * <i>))) / ((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 * <i>))).| 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 * <i>))) 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 * <i>)) = 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 * <i>))) / ((z ExpSeq) . n)).| by Th3
.= |.((((z ExpSeq) . n) * (z / ((n + 1) + (0 * <i>)))) / ((z ExpSeq) . n)).|
.= |.(z / ((n + 1) + (0 * <i>))).| 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 * <i>) ExpSeq));
existence
 ex b1 being Function of REAL,REAL st
for d being Element of REAL holds b1 . d = Im (Sum ((d * <i>) ExpSeq))
proof
deffunc H1( Element of REAL ) ->  Element of REAL = Im (Sum (($1 * <i>) 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 * <i>) ExpSeq))
thus  for d being Element of REAL holds f . d = Im (Sum ((d * <i>) 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 * <i>) ExpSeq)) ) & ( for d being Element of REAL holds b2 . d = Im (Sum ((d * <i>) 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 * <i>) ExpSeq)) ) & ( for d being Element of REAL holds f2 . d = Im (Sum ((d * <i>) ExpSeq)) ) implies f1 = f2 )
assume A2: for d being Element of REAL holds f1 . d = Im (Sum ((d * <i>) ExpSeq)) ; ::_thesis: ( ex d being Element of REAL st not f2 . d = Im (Sum ((d * <i>) ExpSeq)) or f1 = f2 )
assume A3: for d being Element of REAL holds f2 . d = Im (Sum ((d * <i>) 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 * <i>) 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 * <i>) 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 * <i>) ExpSeq));
existence
 ex b1 being Function of REAL,REAL st
for d being Real holds b1 . d = Re (Sum ((d * <i>) ExpSeq))
proof
deffunc H1( Element of REAL ) ->  Element of REAL = Re (Sum (($1 * <i>) 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 * <i>) ExpSeq))
thus  for d being Real holds f . d = Re (Sum ((d * <i>) 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 * <i>) ExpSeq)) ) & ( for d being Real holds b2 . d = Re (Sum ((d * <i>) ExpSeq)) ) holds
b1 = b2
proof
let f1, f2 be Function of REAL,REAL; ::_thesis: ( ( for d being Real holds f1 . d = Re (Sum ((d * <i>) ExpSeq)) ) & ( for d being Real holds f2 . d = Re (Sum ((d * <i>) ExpSeq)) ) implies f1 = f2 )
assume A2: for d being Real holds f1 . d = Re (Sum ((d * <i>) ExpSeq)) ; ::_thesis: ( ex d being Real st not f2 . d = Re (Sum ((d * <i>) ExpSeq)) or f1 = f2 )
assume A3: for d being Real holds f2 . d = Re (Sum ((d * <i>) 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 * <i>) 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 * <i>) 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 * <i>) ExpSeq) = (cos . th) + ((sin . th) * <i>)
proof
let th be Real; ::_thesis: Sum ((th * <i>) ExpSeq) = (cos . th) + ((sin . th) * <i>)
 ( Im (Sum ((th * <i>) ExpSeq)) = sin . th & Re (Sum ((th * <i>) ExpSeq)) = cos . th ) by Def16, Def18;
hence  Sum ((th * <i>) ExpSeq) = (cos . th) + ((sin . th) * <i>) by COMPLEX1:13; ::_thesis: verum
end;

theorem :: SIN_COS:25
for th being Real holds exp (th * <i>) = (cos th) + ((sin th) * <i>)
proof
let th be Real; ::_thesis: exp (th * <i>) = (cos th) + ((sin th) * <i>)
exp (th * <i>) = Sum ((th * <i>) ExpSeq) by Def14
.= (cos th) + ((sin th) * <i>) by Lm3 ;
hence  exp (th * <i>) = (cos th) + ((sin th) * <i>) ; ::_thesis: verum
end;

Lm4:  for th being Real holds (Sum ((th * <i>) ExpSeq)) *' = Sum ((- (th * <i>)) ExpSeq)
proof
let th be Real; ::_thesis: (Sum ((th * <i>) ExpSeq)) *' = Sum ((- (th * <i>)) ExpSeq)
 (Partial_Sums ((th * <i>) ExpSeq)) *' = Partial_Sums ((- (th * <i>)) ExpSeq)
proof
A1: for n being Element of NAT holds (((th * <i>) ExpSeq) . n) *' = ((- (th * <i>)) ExpSeq) . n
proof
defpred S1[ Element of NAT ] means (((th * <i>) ExpSeq) . $1) *' = ((- (th * <i>)) ExpSeq) . $1;
(((th * <i>) ExpSeq) . 0) *' = 1r by Th3, COMPLEX1:30
.= ((- (th * <i>)) 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 * <i>) ExpSeq) . n) *' = ((- (th * <i>)) ExpSeq) . n ; ::_thesis: S1[n + 1]
thus (((th * <i>) ExpSeq) . (n + 1)) *' = (((((th * <i>) ExpSeq) . n) * (th * <i>)) / ((n + 1) + (0 * <i>))) *' by Th3
.= ((((th * <i>) ExpSeq) . n) * ((th * <i>) / ((n + 1) + (0 * <i>)))) *'
.= (((- (th * <i>)) ExpSeq) . n) * (((th * <i>) / (n + 1)) *') by A4, COMPLEX1:35
.= (((- (th * <i>)) ExpSeq) . n) * (((0 + (th * <i>)) *') / ((n + 1) *')) by COMPLEX1:37
.= (((- (th * <i>)) ExpSeq) . n) * ((0 + ((- th) * <i>)) / (((n + 1) + (0 * <i>)) *')) by Lm1
.= (((- (th * <i>)) ExpSeq) . n) * ((0 + ((- th) * <i>)) / ((n + 1) + ((- 0) * <i>))) by Lm1
.= ((((- (th * <i>)) ExpSeq) . n) * (- (th * <i>))) / ((n + 1) + (0 * <i>))
.= ((- (th * <i>)) 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 * <i>) ExpSeq)) *') . $1 = (Partial_Sums ((- (th * <i>)) ExpSeq)) . $1;
((Partial_Sums ((th * <i>) ExpSeq)) *') . 0 = ((Partial_Sums ((th * <i>) ExpSeq)) . 0) *' by COMSEQ_2:def_2
.= (((th * <i>) ExpSeq) . 0) *' by SERIES_1:def_1
.= 1 by Th3, COMPLEX1:30
.= ((- (th * <i>)) ExpSeq) . 0 by Th3
.= (Partial_Sums ((- (th * <i>)) 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 * <i>) ExpSeq)) *') . n = (Partial_Sums ((- (th * <i>)) ExpSeq)) . n ; ::_thesis: S1[n + 1]
thus ((Partial_Sums ((th * <i>) ExpSeq)) *') . (n + 1) = ((Partial_Sums ((th * <i>) ExpSeq)) . (n + 1)) *' by COMSEQ_2:def_2
.= (((Partial_Sums ((th * <i>) ExpSeq)) . n) + (((th * <i>) ExpSeq) . (n + 1))) *' by SERIES_1:def_1
.= (((Partial_Sums ((th * <i>) ExpSeq)) . n) *') + ((((th * <i>) ExpSeq) . (n + 1)) *') by COMPLEX1:32
.= ((Partial_Sums ((- (th * <i>)) ExpSeq)) . n) + ((((th * <i>) ExpSeq) . (n + 1)) *') by A7, COMSEQ_2:def_2
.= ((Partial_Sums ((- (th * <i>)) ExpSeq)) . n) + (((- (th * <i>)) ExpSeq) . (n + 1)) by A1
.= (Partial_Sums ((- (th * <i>)) 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 * <i>) ExpSeq)) *' = Partial_Sums ((- (th * <i>)) ExpSeq) by FUNCT_2:63; ::_thesis: verum
end;
hence  (Sum ((th * <i>) ExpSeq)) *' = Sum ((- (th * <i>)) ExpSeq) by COMSEQ_2:12; ::_thesis: verum
end;

theorem :: SIN_COS:26
for th being Real holds (exp (th * <i>)) *' = exp (- (th * <i>))
proof
let th be Real; ::_thesis: (exp (th * <i>)) *' = exp (- (th * <i>))
(exp (th * <i>)) *' = (Sum ((th * <i>) ExpSeq)) *' by Def14
.= Sum ((- (th * <i>)) ExpSeq) by Lm4
.= exp (- (th * <i>)) by Def14 ;
hence  (exp (th * <i>)) *' = exp (- (th * <i>)) ; ::_thesis: verum
end;

Lm5:  for th being Real
for th1 being real number st th = th1 holds
( |.(Sum ((th * <i>) 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 * <i>) ExpSeq)).| = 1 & abs (sin . th1) <= 1 & abs (cos . th1) <= 1 )
let th1 be real number ; ::_thesis: ( th = th1 implies ( |.(Sum ((th * <i>) ExpSeq)).| = 1 & abs (sin . th1) <= 1 & abs (cos . th1) <= 1 ) )
assume A1: th = th1 ; ::_thesis: ( |.(Sum ((th * <i>) ExpSeq)).| = 1 & abs (sin . th1) <= 1 & abs (cos . th1) <= 1 )
A2: (Sum ((th * <i>) ExpSeq)) * (Sum ((- (th * <i>)) ExpSeq)) = Sum (((th * <i>) + (- (th * <i>))) ExpSeq) by Lm2
.= 1r by Th9 ;
|.(Sum ((th * <i>) ExpSeq)).| * |.(Sum ((th * <i>) ExpSeq)).| = |.((Sum ((th * <i>) ExpSeq)) * (Sum ((th * <i>) ExpSeq))).| by COMPLEX1:65
.= |.((Sum ((th * <i>) ExpSeq)) * ((Sum ((th * <i>) ExpSeq)) *')).| by COMPLEX1:69
.= 1 by A2, Lm4, COMPLEX1:48 ;
then A3: ( |.(Sum ((th * <i>) ExpSeq)).| = 1 / |.(Sum ((th * <i>) ExpSeq)).| & |.(Sum ((th * <i>) ExpSeq)).| <> 0 ) by XCMPLX_1:73;
 |.(Sum ((th * <i>) ExpSeq)).| <> - 1 by COMPLEX1:46;
hence A4: |.(Sum ((th * <i>) ExpSeq)).| = 1 by A3, XCMPLX_1:199; ::_thesis: ( abs (sin . th1) <= 1 & abs (cos . th1) <= 1 )
 ( abs (sin . th) = abs (Im (Sum ((th * <i>) ExpSeq))) & abs (cos . th) = abs (Re (Sum ((th * <i>) 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 * <i>)).| = 1 & ( for th being real number holds
( abs (sin th) <= 1 & abs (cos th) <= 1 ) ) )
proof
let th be Real; ::_thesis: ( |.(exp (th * <i>)).| = 1 & ( for th being real number holds
( abs (sin th) <= 1 & abs (cos th) <= 1 ) ) )
thus |.(exp (th * <i>)).| = |.(Sum ((th * <i>) 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 * <i>) ExpSeq)) * (Sum ((- (th1 * <i>)) ExpSeq)) = Sum (((th1 * <i>) + (- (th1 * <i>))) ExpSeq) by Lm2
.= 1r by Th9 ;
thus ((cos . th) ^2) + ((sin . th) ^2) = ((Re (Sum ((th1 * <i>) ExpSeq))) ^2) + ((sin . th) ^2) by Def18
.= ((Re (Sum ((th1 * <i>) ExpSeq))) ^2) + ((Im (Sum ((th1 * <i>) ExpSeq))) ^2) by Def16
.= |.((Sum ((th1 * <i>) ExpSeq)) * (Sum ((th1 * <i>) ExpSeq))).| by COMPLEX1:68
.= |.((Sum ((th1 * <i>) ExpSeq)) * ((Sum ((th1 * <i>) 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 * <i>) 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 * <i>) 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) * <i>)) ExpSeq)) by Def18
.= Re (Sum ((- (th1 * <i>)) ExpSeq))
.= Re ((Sum ((th1 * <i>) ExpSeq)) *') by Lm4
.= Re (((cos . th) + ((sin . th) * <i>)) *') by Lm3
.= Re ((cos . th) + ((- (sin . th)) * <i>)) by Lm1
.= cos . th by COMPLEX1:12 ; ::_thesis: sin . (- th) = - (sin . th)
thus sin . (- th) = Im (Sum (((- 0) + ((- th1) * <i>)) ExpSeq)) by Def16
.= Im (Sum ((- (th1 * <i>)) ExpSeq))
.= Im ((Sum ((th1 * <i>) ExpSeq)) *') by Lm4
.= Im (((cos . th) + ((sin . th) * <i>)) *') by Lm3
.= Im ((cos . th) + ((- (sin . th)) * <i>)) 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 * <i>) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) & (th * <i>) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * <i> )
proof
let k be Element of NAT ; ::_thesis: for th being Real holds
( (th * <i>) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) & (th * <i>) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * <i> )
let th be Real; ::_thesis: ( (th * <i>) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) & (th * <i>) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * <i> )
A1: (((- 1) |^ 0) * (th |^ (2 * 0))) * <i> = (((- 1) |^ 0) * ((th GeoSeq) . 0)) * <i> by PREPOWER:def_1
.= (((- 1) |^ 0) * 1) * <i> by PREPOWER:3
.= (((- 1) GeoSeq) . 0) * <i> by PREPOWER:def_1
.= 1 * <i> by PREPOWER:3 ;
A2: (th * <i>) |^ ((2 * 0) + 1) = (((th * <i>) GeoSeq) . 0) * (th * <i>) by COMSEQ_3:def_1
.= 1 * (th * <i>) by COMSEQ_3:def_1
.= th * <i> ;
defpred S1[ Element of NAT ] means ( (th * <i>) |^ (2 * $1) = ((- 1) |^ $1) * (th |^ (2 * $1)) & (th * <i>) |^ ((2 * $1) + 1) = (((- 1) |^ $1) * (th |^ ((2 * $1) + 1))) * <i> );
(((- 1) |^ 0) * (th |^ ((2 * 0) + 1))) * <i> = (((- 1) |^ 0) * ((th GeoSeq) . ((2 * 0) + 1))) * <i> by PREPOWER:def_1
.= (((- 1) |^ 0) * (((th GeoSeq) . 0) * th)) * <i> by PREPOWER:3
.= (((- 1) |^ 0) * (1 * th)) * <i> by PREPOWER:3
.= ((((- 1) GeoSeq) . 0) * th) * <i> by PREPOWER:def_1
.= (1 * th) * <i> by PREPOWER:3
.= th * <i> ;
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 * <i>) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) and
A6: (th * <i>) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * <i> ; ::_thesis: S1[k + 1]
A7: (th * <i>) |^ (2 * (k + 1)) = ((th * <i>) |^ 2) |^ (k + 1) by Th32
.= (((th * <i>) |^ 2) |^ k) * ((th * <i>) |^ 2) by COMSEQ_3:def_1
.= (((- 1) |^ k) * (th |^ (2 * k))) * (((th * <i>) GeoSeq) . (1 + 1)) by A5, Th32
.= (((- 1) |^ k) * (th |^ (2 * k))) * ((((th * <i>) GeoSeq) . (0 + 1)) * (th * <i>)) by COMSEQ_3:def_1
.= (((- 1) |^ k) * (th |^ (2 * k))) * (((((th * <i>) GeoSeq) . 0) * (th * <i>)) * (th * <i>)) by COMSEQ_3:def_1
.= (((- 1) |^ k) * (th |^ (2 * k))) * ((1r * (th * <i>)) * (th * <i>)) 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 * <i>) |^ ((2 * (k + 1)) + 1) = (((th * <i>) GeoSeq) . (((2 * k) + 1) + 1)) * (th * <i>) by COMSEQ_3:def_1
.= ((((th * <i>) GeoSeq) . ((2 * k) + 1)) * (th * <i>)) * (th * <i>) by COMSEQ_3:def_1
.= ((((((- 1) |^ k) * (- 1)) * (th |^ ((2 * k) + 1))) * th) * th) * <i> by A6
.= (((((((- 1) GeoSeq) . k) * (- 1)) * (th |^ ((2 * k) + 1))) * th) * th) * <i> by PREPOWER:def_1
.= ((((((- 1) GeoSeq) . (k + 1)) * (th |^ ((2 * k) + 1))) * th) * th) * <i> by PREPOWER:3
.= (((((- 1) |^ (k + 1)) * (th |^ ((2 * k) + 1))) * th) * th) * <i> by PREPOWER:def_1
.= (((((- 1) |^ (k + 1)) * ((th GeoSeq) . ((2 * k) + 1))) * th) * th) * <i> by PREPOWER:def_1
.= ((((- 1) |^ (k + 1)) * (((th GeoSeq) . ((2 * k) + 1)) * th)) * th) * <i>
.= ((((- 1) |^ (k + 1)) * ((th GeoSeq) . (((2 * k) + 1) + 1))) * th) * <i> by PREPOWER:3
.= (((- 1) |^ (k + 1)) * (((th GeoSeq) . (((2 * k) + 1) + 1)) * th)) * <i>
.= (((- 1) |^ (k + 1)) * ((th GeoSeq) . ((2 * (k + 1)) + 1))) * <i> by PREPOWER:3
.= (((- 1) |^ (k + 1)) * (th |^ ((2 * (k + 1)) + 1))) * <i> 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 * <i>) |^ (2 * k) = ((- 1) |^ k) * (th |^ (2 * k)) & (th * <i>) |^ ((2 * k) + 1) = (((- 1) |^ k) * (th |^ ((2 * k) + 1))) * <i> ) ; ::_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 * <i>) ExpSeq))) . ((2 * n) + 1) & (Partial_Sums (th P_cos)) . n = (Partial_Sums (Re ((th * <i>) 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 * <i>) ExpSeq))) . ((2 * n) + 1) & (Partial_Sums (th P_cos)) . n = (Partial_Sums (Re ((th * <i>) ExpSeq))) . (2 * n) )
let th be Real; ::_thesis: ( (Partial_Sums (th P_sin)) . n = (Partial_Sums (Im ((th * <i>) ExpSeq))) . ((2 * n) + 1) & (Partial_Sums (th P_cos)) . n = (Partial_Sums (Re ((th * <i>) 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 * <i>) ExpSeq)) . 0 = Im (((th * <i>) ExpSeq) . 0) & (Im ((th * <i>) ExpSeq)) . 1 = Im (((th * <i>) ExpSeq) . 1) ) by COMSEQ_3:def_6;
A7: th * <i> = 0 + (th * <i>) ;
A8: (Partial_Sums (Im ((th * <i>) ExpSeq))) . ((2 * 0) + 1) = ((Partial_Sums (Im ((th * <i>) ExpSeq))) . 0) + ((Im ((th * <i>) ExpSeq)) . 1) by SERIES_1:def_1
.= ((Im ((th * <i>) ExpSeq)) . 0) + ((Im ((th * <i>) ExpSeq)) . 1) by SERIES_1:def_1
.= 0 + (Im (((th * <i>) ExpSeq) . (0 + 1))) by A6, Th3, COMPLEX1:6
.= 0 + (Im (((((th * <i>) ExpSeq) . 0) * (th * <i>)) / ((0 + 1) + (0 * <i>)))) by Th3
.= Im ((1 * (th * <i>)) / 1) by Th3
.= th by A7, COMPLEX1:12 ;
defpred S1[ Element of NAT ] means ( (Partial_Sums (th P_sin)) . $1 = (Partial_Sums (Im ((th * <i>) ExpSeq))) . ((2 * $1) + 1) & (Partial_Sums (th P_cos)) . $1 = (Partial_Sums (Re ((th * <i>) ExpSeq))) . (2 * $1) );
(Partial_Sums (Re ((th * <i>) ExpSeq))) . (2 * 0) = (Re ((th * <i>) ExpSeq)) . 0 by SERIES_1:def_1
.= Re (((th * <i>) 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 * <i>) ExpSeq))) . ((2 * k) + 1) and
A12: (Partial_Sums (th P_cos)) . k = (Partial_Sums (Re ((th * <i>) ExpSeq))) . (2 * k) ; ::_thesis: S1[k + 1]
(Partial_Sums (Im ((th * <i>) ExpSeq))) . ((2 * (k + 1)) + 1) = ((Partial_Sums (Im ((th * <i>) ExpSeq))) . (((2 * k) + 1) + 1)) + ((Im ((th * <i>) ExpSeq)) . ((2 * (k + 1)) + 1)) by SERIES_1:def_1
.= (((Partial_Sums (th P_sin)) . k) + ((Im ((th * <i>) ExpSeq)) . (2 * (k + 1)))) + ((Im ((th * <i>) ExpSeq)) . ((2 * (k + 1)) + 1)) by A11, SERIES_1:def_1
.= (((Partial_Sums (th P_sin)) . k) + (Im (((th * <i>) ExpSeq) . (2 * (k + 1))))) + ((Im ((th * <i>) ExpSeq)) . ((2 * (k + 1)) + 1)) by COMSEQ_3:def_6
.= (((Partial_Sums (th P_sin)) . k) + (Im (((th * <i>) ExpSeq) . (2 * (k + 1))))) + (Im (((th * <i>) ExpSeq) . ((2 * (k + 1)) + 1))) by COMSEQ_3:def_6
.= (((Partial_Sums (th P_sin)) . k) + (Im (((th * <i>) |^ (2 * (k + 1))) / ((2 * (k + 1)) !)))) + (Im (((th * <i>) ExpSeq) . ((2 * (k + 1)) + 1))) by Def4
.= (((Partial_Sums (th P_sin)) . k) + (Im (((th * <i>) |^ (2 * (k + 1))) / ((2 * (k + 1)) !)))) + (Im (((th * <i>) |^ ((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 * <i>) |^ ((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))) * <i>) / (((2 * (k + 1)) + 1) !))) by Th33 ;
then  (Partial_Sums (Im ((th * <i>) 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)) !)) * <i>)))) + (Im (((((- 1) |^ (k + 1)) * (th |^ ((2 * (k + 1)) + 1))) * <i>) / (((2 * (k + 1)) + 1) !))) ;
then (Partial_Sums (Im ((th * <i>) 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) !)) * <i>))) 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 * <i>) 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 * <i>) ExpSeq))) . (2 * (k + 1)) = ((Partial_Sums (Re ((th * <i>) ExpSeq))) . ((2 * k) + 1)) + ((Re ((th * <i>) ExpSeq)) . (((2 * k) + 1) + 1)) by SERIES_1:def_1
.= (((Partial_Sums (th P_cos)) . k) + ((Re ((th * <i>) ExpSeq)) . ((2 * k) + 1))) + ((Re ((th * <i>) ExpSeq)) . (((2 * k) + 1) + 1)) by A12, SERIES_1:def_1
.= (((Partial_Sums (th P_cos)) . k) + (Re (((th * <i>) ExpSeq) . ((2 * k) + 1)))) + ((Re ((th * <i>) ExpSeq)) . (((2 * k) + 1) + 1)) by COMSEQ_3:def_5
.= (((Partial_Sums (th P_cos)) . k) + (Re (((th * <i>) ExpSeq) . ((2 * k) + 1)))) + (Re (((th * <i>) ExpSeq) . (((2 * k) + 1) + 1))) by COMSEQ_3:def_5
.= (((Partial_Sums (th P_cos)) . k) + (Re (((th * <i>) |^ ((2 * k) + 1)) / (((2 * k) + 1) !)))) + (Re (((th * <i>) ExpSeq) . (((2 * k) + 1) + 1))) by Def4
.= (((Partial_Sums (th P_cos)) . k) + (Re (((th * <i>) |^ ((2 * k) + 1)) / (((2 * k) + 1) !)))) + (Re (((th * <i>) |^ (2 * (k + 1))) / ((((2 * k) + 1) + 1) !))) by Def4
.= (((Partial_Sums (th P_cos)) . k) + (Re (((((- 1) |^ k) * (th |^ ((2 * k) + 1))) * <i>) / (((2 * k) + 1) !)))) + (Re (((th * <i>) |^ (2 * (k + 1))) / ((2 * (k + 1)) !))) by Th33
.= (((Partial_Sums (th P_cos)) . k) + (Re (((((- 1) |^ k) * (th |^ ((2 * k) + 1))) * <i>) / (((2 * k) + 1) !)))) + (Re ((((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1)))) / ((2 * (k + 1)) !))) by Th33 ;
then  (Partial_Sums (Re ((th * <i>) ExpSeq))) . (2 * (k + 1)) = (((Partial_Sums (th P_cos)) . k) + (Re ((0 / (((2 * k) + 1) !)) + (((((- 1) |^ k) * (th |^ ((2 * k) + 1))) / (((2 * k) + 1) !)) * <i>)))) + (Re ((((- 1) |^ (k + 1)) * (th |^ (2 * (k + 1)))) / ((2 * (k + 1)) !))) ;
then (Partial_Sums (Re ((th * <i>) 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)) !)) * <i>))) 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 * <i>) 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 * <i>) ExpSeq))) . ((2 * n) + 1) & (Partial_Sums (th P_cos)) . n = (Partial_Sums (Re ((th * <i>) 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 * <i>) ExpSeq)) & Partial_Sums (th P_cos) is convergent & Sum (th P_cos) = Re (Sum ((th * <i>) ExpSeq)) )
proof
let th be Real; ::_thesis: ( Partial_Sums (th P_sin) is convergent & Sum (th P_sin) = Im (Sum ((th * <i>) ExpSeq)) & Partial_Sums (th P_cos) is convergent & Sum (th P_cos) = Re (Sum ((th * <i>) ExpSeq)) )
A1: Sum ((th * <i>) ExpSeq) = (Sum (Re ((th * <i>) ExpSeq))) + ((Sum (Im ((th * <i>) ExpSeq))) * <i>) by COMSEQ_3:53;
then A2: Sum (Re ((th * <i>) ExpSeq)) = Re (Sum ((th * <i>) ExpSeq)) by COMPLEX1:12;
A3: Sum (Im ((th * <i>) ExpSeq)) = Im (Sum ((th * <i>) ExpSeq)) by A1, COMPLEX1:12;
A4: ( Partial_Sums (Re ((th * <i>) ExpSeq)) is convergent & lim (Partial_Sums (Re ((th * <i>) ExpSeq))) = Re (Sum ((th * <i>) ExpSeq)) ) by A2, SERIES_1:def_2, SERIES_1:def_3;
A5: ( Partial_Sums (Im ((th * <i>) ExpSeq)) is convergent & lim (Partial_Sums (Im ((th * <i>) ExpSeq))) = Im (Sum ((th * <i>) 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_*_<i>)_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 * <i>) 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 * <i>) 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 * <i>) ExpSeq))) . k) - (Re (Sum ((th * <i>) 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 * <i>) ExpSeq)))) < p
now__::_thesis:_for_k_being_Element_of_NAT_st_n_<=_k_holds_
abs_(((Partial_Sums_(th_P_cos))_._k)_-_(Re_(Sum_((th_*_<i>)_ExpSeq))))_<_p
let k be Element of NAT ; ::_thesis: ( n <= k implies abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * <i>) ExpSeq)))) < p )
assume A8: n <= k ; ::_thesis: abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * <i>) ExpSeq)))) < p
 ( abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * <i>) ExpSeq)))) = abs (((Partial_Sums (Re ((th * <i>) ExpSeq))) . (2 * k)) - (Re (Sum ((th * <i>) ExpSeq)))) & 2 * k = k + k ) by Th35;
hence  abs (((Partial_Sums (th P_cos)) . k) - (Re (Sum ((th * <i>) 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 * <i>) 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 * <i>) 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_*_<i>)_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 * <i>) 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 * <i>) 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 * <i>) ExpSeq))) . k) - (Im (Sum ((th * <i>) 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 * <i>) ExpSeq)))) < p
now__::_thesis:_for_k_being_Element_of_NAT_st_n_<=_k_holds_
abs_(((Partial_Sums_(th_P_sin))_._k)_-_(Im_(Sum_((th_*_<i>)_ExpSeq))))_<_p
let k be Element of NAT ; ::_thesis: ( n <= k implies abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * <i>) ExpSeq)))) < p )
assume A12: n <= k ; ::_thesis: abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * <i>) ExpSeq)))) < p
A13: abs (((Partial_Sums (th P_sin)) . k) - (Im (Sum ((th * <i>) ExpSeq)))) = abs (((Partial_Sums (Im ((th * <i>) ExpSeq))) . ((2 * k) + 1)) - (Im (Sum ((th * <i>) 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 * <i>) 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 * <i>) 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 * <i>) ExpSeq)) by A10, SEQ_2:def_7;
hence  ( Partial_Sums (th P_sin) is convergent & Sum (th P_sin) = Im (Sum ((th * <i>) ExpSeq)) & Partial_Sums (th P_cos) is convergent & Sum (th P_cos) = Re (Sum ((th * <i>) 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 * <i>) ExpSeq)) & cos . th = Re (Sum ((th * <i>) 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 * <i>))
.= 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))) * <i>)) 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 * <i>))) by Def4
.= Re (((th |^ n) / (n !)) + (0 * <i>))
.= (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))) * <i>) & 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 * <i>)) 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 * <i>) 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))) * <i>)) * (Sum (q ExpSeq))) by COMPLEX1:13
.= Re (((Re (Sum (p ExpSeq))) + ((Im (Sum (p ExpSeq))) * <i>)) * ((Re (Sum (q ExpSeq))) + ((Im (Sum (q ExpSeq))) * <i>))) by COMPLEX1:13
.= Re (((Sum (p rExpSeq)) + ((Im (Sum (p ExpSeq))) * <i>)) * ((Re (Sum (q ExpSeq))) + ((Im (Sum (q ExpSeq))) * <i>))) by Th45
.= Re (((Sum (p rExpSeq)) + (0 * <i>)) * ((Re (Sum (q ExpSeq))) + ((Im (Sum (q ExpSeq))) * <i>))) by Th42
.= Re ((Sum (p rExpSeq)) * ((Sum (q rExpSeq)) + ((Im (Sum (q ExpSeq))) * <i>))) by Th45
.= Re ((Sum (p rExpSeq)) * ((Sum (q rExpSeq)) + (0 * <i>))) by Th42
.= Re (((Sum (p rExpSeq)) * (Sum (q rExpSeq))) + (0 * <i>))
.= (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 * <i>) * z) = - (p * (Im z)) & Im ((p * <i>) * 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 * <i>) * z) = - (p * (Im z)) & Im ((p * <i>) * z) = p * (Re z) & Re (p * z) = p * (Re z) & Im (p * z) = p * (Im z) )
let z be complex number ; ::_thesis: ( Re ((p * <i>) * z) = - (p * (Im z)) & Im ((p * <i>) * z) = p * (Re z) & Re (p * z) = p * (Re z) & Im (p * z) = p * (Im z) )
A1: (p * <i>) * z = (p * <i>) * ((Re z) + ((Im z) * <i>)) by COMPLEX1:13
.= (- (p * (Im z))) + ((p * (Re z)) * <i>) ;
p * z = p * ((Re z) + ((Im z) * <i>)) by COMPLEX1:13
.= (p * (Re z)) + ((p * (Im z)) * <i>) ;
hence  ( Re ((p * <i>) * z) = - (p * (Im z)) & Im ((p * <i>) * 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 * <i>)) = (Im z) / p & Im (z / (p * <i>)) = - ((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 * <i>)) = (Im z) / p & Im (z / (p * <i>)) = - ((Re z) / p) & |.(z / p).| = |.z.| / p )
let z be complex number ; ::_thesis: ( p > 0 implies ( Re (z / (p * <i>)) = (Im z) / p & Im (z / (p * <i>)) = - ((Re z) / p) & |.(z / p).| = |.z.| / p ) )
assume A1: p > 0 ; ::_thesis: ( Re (z / (p * <i>)) = (Im z) / p & Im (z / (p * <i>)) = - ((Re z) / p) & |.(z / p).| = |.z.| / p )
A2: ( Re (0 + (p * <i>)) = 0 & Im (0 + (p * <i>)) = p ) by COMPLEX1:12;
A3: Im (p + (0 * <i>)) = 0 by COMPLEX1:12;
A4: z / (p * <i>) = ((((Re z) * 0) + ((Im z) * p)) / ((0 ^2) + (p ^2))) + ((((0 * (Im z)) - ((Re z) * p)) / ((0 ^2) + (p ^2))) * <i>) by A2, COMPLEX1:def_10
.= (((Im z) * p) / (p * p)) + ((((- (Re z)) * p) / (p * p)) * <i>)
.= (((Im z) * p) / (p * p)) + (((- (Re z)) / p) * <i>) by A1, XCMPLX_1:91
.= ((Im z) / p) + ((- ((Re z) / p)) * <i>) by A1, XCMPLX_1:91 ;
|.(z / p).| = |.z.| / |.p.| by COMPLEX1:67
.= |.z.| / (sqrt (p ^2)) by A3, COMPLEX1:12 ;
hence  ( Re (z / (p * <i>)) = (Im z) / p & Im (z / (p * <i>)) = - ((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 * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))))
proof
let p, q be Real; ::_thesis: (cos . (p + q)) - (cos . p) = (- (q * (sin . p))) - (q * (Im ((Sum ((q * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))))
(cos . (p + q)) - (cos . p) = (cos . (p + q)) - (Re (Sum ((p * <i>) ExpSeq))) by Def18
.= (Re (Sum (((p + q) * <i>) ExpSeq))) - (Re (Sum ((p * <i>) ExpSeq))) by Def18
.= Re ((Sum (((p * <i>) + (q * <i>)) ExpSeq)) - (Sum ((p * <i>) ExpSeq))) by COMPLEX1:19
.= Re (((Sum ((p * <i>) ExpSeq)) * (q * <i>)) + (((q * <i>) * (Sum ((q * <i>) P_dt))) * (Sum ((p * <i>) ExpSeq)))) by Th59
.= Re ((((cos . p) + ((sin . p) * <i>)) * (q * <i>)) + (((q * <i>) * (Sum ((q * <i>) P_dt))) * (Sum ((p * <i>) ExpSeq)))) by Lm3
.= Re ((((cos . p) + ((sin . p) * <i>)) * (q * <i>)) + (((q * <i>) * (Sum ((q * <i>) P_dt))) * ((cos . p) + ((sin . p) * <i>)))) by Lm3
.= (Re ((- (q * (sin . p))) + ((q * (cos . p)) * <i>))) + (Re (((q * <i>) * (Sum ((q * <i>) P_dt))) * ((cos . p) + ((sin . p) * <i>)))) by COMPLEX1:8
.= (- (q * (sin . p))) + (Re ((q * <i>) * ((Sum ((q * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) by COMPLEX1:12
.= (- (q * (sin . p))) - (q * (Im ((Sum ((q * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) by Lm12 ;
hence  (cos . (p + q)) - (cos . p) = (- (q * (sin . p))) - (q * (Im ((Sum ((q * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) ; ::_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 * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))))
proof
let p, q be Real; ::_thesis: (sin . (p + q)) - (sin . p) = (q * (cos . p)) + (q * (Re ((Sum ((q * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))))
(sin . (p + q)) - (sin . p) = (sin . (p + q)) - (Im (Sum ((p * <i>) ExpSeq))) by Def16
.= (Im (Sum (((p + q) * <i>) ExpSeq))) - (Im (Sum ((p * <i>) ExpSeq))) by Def16
.= Im ((Sum (((p * <i>) + (q * <i>)) ExpSeq)) - (Sum ((p * <i>) ExpSeq))) by COMPLEX1:19
.= Im (((Sum ((p * <i>) ExpSeq)) * (q * <i>)) + (((q * <i>) * (Sum ((q * <i>) P_dt))) * (Sum ((p * <i>) ExpSeq)))) by Th59
.= Im ((((cos . p) + ((sin . p) * <i>)) * (q * <i>)) + (((q * <i>) * (Sum ((q * <i>) P_dt))) * (Sum ((p * <i>) ExpSeq)))) by Lm3
.= Im ((((cos . p) + ((sin . p) * <i>)) * (q * <i>)) + (((q * <i>) * (Sum ((q * <i>) P_dt))) * ((cos . p) + ((sin . p) * <i>)))) by Lm3
.= (Im ((- (q * (sin . p))) + ((q * (cos . p)) * <i>))) + (Im (((q * <i>) * (Sum ((q * <i>) P_dt))) * ((cos . p) + ((sin . p) * <i>)))) by COMPLEX1:8
.= (q * (cos . p)) + (Im ((q * <i>) * ((Sum ((q * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) by COMPLEX1:12
.= (q * (cos . p)) + (q * (Re ((Sum ((q * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) by Lm12 ;
hence  (sin . (p + q)) - (sin . p) = (q * (cos . p)) + (q * (Re ((Sum ((q * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) ; ::_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 * <i>))))) 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 * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))));
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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))
proof
let n be Element of NAT ; ::_thesis: ((hy1 ") (#) (Cr /* hy1)) . n = - (Im ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))
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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))))) by A1
.= - ((((hy1 . n) ") * (hy1 . n)) * (Im ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) ;
 hy1 . n <> 0 by SEQ_1:5;
then ((hy1 ") (#) (Cr /* hy1)) . n = - (1 * (Im ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) by A4, XCMPLX_0:def_7
.= - (Im ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))) ;
hence  ((hy1 ") (#) (Cr /* hy1)) . n = - (Im ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))) ; ::_thesis: verum
end;
deffunc H3( Real) ->  Element of REAL = - (Im ((Sum (((hy1 . $1) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))));
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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>));
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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))).| by A6
.= |.(Sum (((hy1 . m) * <i>) P_dt)).| * |.((cos . p) + ((sin . p) * <i>)).| by COMPLEX1:65
.= |.(Sum (((hy1 . m) * <i>) P_dt)).| * |.(Sum ((p * <i>) ExpSeq)).| by Lm3
.= |.(Sum (((hy1 . m) * <i>) P_dt)).| * 1 by Lm5
.= |.(Sum (((hy1 . m) * <i>) P_dt)).| ;
A15: abs ((hy1 . m) - 0) < r by A11, A13;
 (hy1 . m) * <i> = 0 + ((hy1 . m) * <i>) ;
then  ( Re ((hy1 . m) * <i>) = 0 & Im ((hy1 . m) * <i>) = hy1 . m ) by COMPLEX1:12;
then  |.((hy1 . m) * <i>).| = 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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))
proof
let n be Element of NAT ; ::_thesis: (Im (- cq1)) . n = - (Im ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))
(Im (- cq1)) . n = Im ((- cq1) . n) by COMSEQ_3:def_6
.= Im (- (cq1 . n)) by VALUED_1:8
.= Im (- ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))) by A6
.= - (Im ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))) by COMPLEX1:17 ;
hence  (Im (- cq1)) . n = - (Im ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))) ; ::_thesis: verum
end;
let n be Element of NAT ; ::_thesis: (Im (- cq1)) . n = rseq . n
 rseq . n = - (Im ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))) 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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))) 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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) 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 * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))));
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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))
proof
let n be Element of NAT ; ::_thesis: ((hy1 ") (#) (Cr /* hy1)) . n = Re ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))
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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) by A1
.= (((hy1 . n) ") * (hy1 . n)) * (Re ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))) ;
 hy1 . n <> 0 by SEQ_1:5;
then ((hy1 ") (#) (Cr /* hy1)) . n = 1 * (Re ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)))) by A4, XCMPLX_0:def_7
.= Re ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))) ;
hence  ((hy1 ") (#) (Cr /* hy1)) . n = Re ((Sum (((hy1 . n) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))) ; ::_thesis: verum
end;
deffunc H3( Real) ->  Element of REAL = Re ((Sum (((hy1 . $1) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>)));
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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>));
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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))).| by A6
.= |.(Sum (((hy1 . m) * <i>) P_dt)).| * |.((cos . p) + ((sin . p) * <i>)).| by COMPLEX1:65
.= |.(Sum (((hy1 . m) * <i>) P_dt)).| * |.(Sum ((p * <i>) ExpSeq)).| by Lm3
.= |.(Sum (((hy1 . m) * <i>) P_dt)).| * 1 by Lm5
.= |.(Sum (((hy1 . m) * <i>) P_dt)).| ;
A15: abs ((hy1 . m) - 0) < r by A11, A13;
 (hy1 . m) * <i> = 0 + ((hy1 . m) * <i>) ;
then  ( Re ((hy1 . m) * <i>) = 0 & Im ((hy1 . m) * <i>) = hy1 . m ) by COMPLEX1:12;
then  |.((hy1 . m) * <i>).| = 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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))) 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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))) 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) * <i>) P_dt)) * ((cos . p) + ((sin . p) * <i>))))) 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 * <i>) ;
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) * <i> = (0 + 0) + ((th1 + th2) * <i>) ;
A2: (Sum ((th1 * <i>) ExpSeq)) * (Sum ((th2 * <i>) ExpSeq)) = Sum (((th1 * <i>) + (th2 * <i>)) ExpSeq) by Lm2
.= (cos . (th1 + th2)) + ((sin . (th1 + th2)) * <i>) by A1, Lm3 ;
(Sum ((th1 * <i>) ExpSeq)) * (Sum ((th2 * <i>) ExpSeq)) = ((cos . th1) + ((sin . th1) * <i>)) * (Sum ((th2 * <i>) ExpSeq)) by Lm3
.= ((cos . th1) + ((sin . th1) * <i>)) * ((cos . th2) + ((sin . th2) * <i>)) by Lm3
.= (((cos . th1) * (cos . th2)) - ((sin . th1) * (sin . th2))) + ((((sin . th1) * (cos . th2)) + ((cos . th1) * (sin . th2))) * <i>) ;
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;