let p be real number ; :: thesis:
reconsider p = p as Real by XREAL_0:def 1;
deffunc H₂( Element of REAL ) -> Element of REAL = $1 * (Re ((Sum (($1 * ) P_dt )) * ((cos . p) + ((sin . p) * ))));
consider Cr being Function of REAL ,REAL such that
A2: for th being Real holds Cr . th = H₂(th) from FUNCT_2:sch 4();
Cr is REST-like

proof

for hy1 being convergent_to_0 Real_Sequence holds
( (hy1 " ) (#) (Cr /* hy1) is convergent & lim ((hy1 " ) (#) (Cr /* hy1)) = 0 )

proof

let hy1 be convergent_to_0 Real_Sequence; :: thesis:
A7: for n being Element of NAT holds ((hy1 " ) (#) (Cr /* hy1)) . n = Re ((Sum (((hy1 . n) * ) P_dt )) * ((cos . p) + ((sin . p) * )))

proof

let n be Element of NAT ; :: thesis:
A8: ((hy1 " ) (#) (Cr /* hy1)) . n = ((hy1 " ) . n) * ((Cr /* hy1) . n) by SEQ_1:12
.= ((hy1 . n) " ) * ((Cr /* hy1) . n) by VALUED_1:10 ;
dom Cr = REAL by FUNCT_2:def 1;
then rng hy1 c= dom Cr ;
then A9: ((hy1 " ) (#) (Cr /* hy1)) . n = ((hy1 . n) " ) * (Cr . (hy1 . n)) by A8, FUNCT_2:185
.= ((hy1 . n) " ) * ((hy1 . n) * (Re ((Sum (((hy1 . n) * ) P_dt )) * ((cos . p) + ((sin . p) * ))))) by A2
.= (((hy1 . n) " ) * (hy1 . n)) * (Re ((Sum (((hy1 . n) * ) P_dt )) * ((cos . p) + ((sin . p) * )))) ;
hy1 is non-zero by FDIFF_1:def 1;
then hy1 . n <> 0 by SEQ_1:7;
then ((hy1 " ) (#) (Cr /* hy1)) . n = 1 * (Re ((Sum (((hy1 . n) * ) P_dt )) * ((cos . p) + ((sin . p) * )))) by A9, XCMPLX_0:def 7
.= Re ((Sum (((hy1 . n) * ) P_dt )) * ((cos . p) + ((sin . p) * ))) ;
hence ((hy1 " ) (#) (Cr /* hy1)) . n = Re ((Sum (((hy1 . n) * ) P_dt )) * ((cos . p) + ((sin . p) * ))) ; :: thesis:

end;

deffunc H₃( Real) -> Element of REAL = Re ((Sum (((hy1 . $1) * ) P_dt )) * ((cos . p) + ((sin . p) * )));
consider rseq being Real_Sequence such that
A10: for n being Element of NAT holds rseq . n = H₃(n) from SEQ_1:sch 1();
deffunc H₄( Element of NAT ) -> Element of COMPLEX = (Sum (((hy1 . $1) * ) P_dt )) * ((cos . p) + ((sin . p) * ));
consider cq1 being Complex_Sequence such that
A11: for n being Element of NAT holds cq1 . n = H₄(n) from COMSEQ_1:sch 1();
A12: ( cq1 is convergent & lim cq1 = 0c )

proof

A13: 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:
assume A14: q > 0 ; :: thesis:
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
A15: ( r > 0 & ( for z being complex number st |.z.| < r holds
|.(Sum (z P_dt )).| < q ) ) by A14, Th63;
( hy1 is convergent & lim hy1 = 0 ) by FDIFF_1:def 1;
then consider k being Element of NAT such that
A16: for m being Element of NAT st k <= m holds
abs ((hy1 . m) - 0 ) < r by A15, SEQ_2:def 7;

A17: now

end;

take k ; :: thesis:
thus for m being Element of NAT st k <= m holds
|.((cq1 . m) - 0c ).| < q by A17; :: thesis:

end;

hence ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.((cq1 . m) - 0c ).| < q ; :: thesis:

end;

then cq1 is convergent by COMSEQ_2:def 4;
hence ( cq1 is convergent & lim cq1 = 0c ) by A13, COMSEQ_2:def 5; :: thesis:

end;

A21: for n being Element of NAT holds (Re cq1) . n = rseq . n

proof

let n be Element of NAT ; :: thesis:
(Re cq1) . n = Re (cq1 . n) by COMSEQ_3:def 3
.= Re ((Sum (((hy1 . n) * ) P_dt )) * ((cos . p) + ((sin . p) * ))) by A11 ;
hence (Re cq1) . n = rseq . n by A10; :: thesis:

end;

rseq = (hy1 " ) (#) (Cr /* hy1)

proof

for n being Element of NAT holds rseq . n = ((hy1 " ) (#) (Cr /* hy1)) . n

proof

let n be Element of NAT ; :: thesis:
rseq . n = Re ((Sum (((hy1 . n) * ) P_dt )) * ((cos . p) + ((sin . p) * ))) by A10;
hence rseq . n = ((hy1 " ) (#) (Cr /* hy1)) . n by A7; :: thesis:

end;

hence rseq = (hy1 " ) (#) (Cr /* hy1) by FUNCT_2:113; :: thesis:

end;

then (hy1 " ) (#) (Cr /* hy1) = Re cq1 by A21, FUNCT_2:113;
hence ( (hy1 " ) (#) (Cr /* hy1) is convergent & lim ((hy1 " ) (#) (Cr /* hy1)) = 0 ) by A12, COMPLEX1:12, COMSEQ_3:41; :: thesis:

end;

hence Cr is REST-like by FDIFF_1:def 3; :: thesis:

end;

then reconsider PR = Cr as REST ;
deffunc H₃( Element of REAL ) -> Element of REAL = $1 * (cos . p);
consider CL being Function of REAL ,REAL such that
A23: for th being Real holds CL . th = H₃(th) from FUNCT_2:sch 4();
A24: for d being real number holds CL . d = d * (cos . p)

proof

let d be real number ; :: thesis:
d is Real by XREAL_0:def 1;
hence CL . d = d * (cos . p) by A23; :: thesis:

end;

CL is linear

proof

ex r being Real st
for q being Real holds CL . q = r * q

proof

take cos . p ; :: thesis:
thus for q being Real holds CL . q = (cos . p) * q by A24; :: thesis:

end;

hence CL is linear by FDIFF_1:def 4; :: thesis:

end;

then reconsider PL = CL as LINEAR ;
A27: 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

A28: 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 = p + (r - p) ;
then (sin . r) - (sin . p) = ((r - p) * (cos . p)) + ((r - p) * (Re ((Sum (((r - p) * ) P_dt )) * ((cos . p) + ((sin . p) * ))))) by Th66
.= ((r - p) * (cos . p)) + (Cr . (r - p)) by A2
.= (PL . (r - p)) + (PR . (r - p)) by A24 ;
hence ( r in ].(p - 1),(p + 1).[ implies (sin . r) - (sin . p) = (PL . (r - p)) + (PR . (r - p)) ) ; :: thesis:

end;

take ].(p - 1),(p + 1).[ ; :: thesis:
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 A28, Th27, RCOMP_1:def 7; :: thesis:

end;

then A29: sin is_differentiable_in p by FDIFF_1:def 5;
PL . 1 = 1 * (cos . p) by A24;
hence ( sin is_differentiable_in p & diff sin ,p = cos . p ) by A27, A29, FDIFF_1:def 6; :: thesis: