let p be real number ; :: thesis:
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
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 = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt )))

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) * (exp_R . p)) * (Re (Sum ((hy1 . n) P_dt )))) by A2
.= (((hy1 . n) " ) * (hy1 . n)) * ((exp_R . p) * (Re (Sum ((hy1 . n) P_dt )))) ;
hy1 is non-zero by FDIFF_1:def 1;
then hy1 . n <> 0 by SEQ_1:7;
then ((hy1 " ) (#) (Cr /* hy1)) . n = 1 * ((exp_R . p) * (Re (Sum ((hy1 . n) P_dt )))) by A9, 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:

end;

deffunc H₃( Real) -> Element of REAL = (exp_R . p) * (Re (Sum ((hy1 . $1) P_dt )));
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 )) * (exp_R . 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

exp_R . p > 0 by Th59;
then consider r being Real such that
A15: ( r > 0 & ( for z being complex number st |.z.| < r holds
|.(Sum (z P_dt )).| < q / (exp_R . p) ) ) by Th63, A14;
( 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;

A24: for n being Element of NAT holds (Re cq1) . n = rseq . n

proof

A25: 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 = Re (cq1 . n) by COMSEQ_3:def 3
.= Re ((Sum ((hy1 . n) P_dt )) * (exp_R . p)) by A11
.= (exp_R . p) * (Re (Sum ((hy1 . n) P_dt ))) by Lm17 ;
hence (Re cq1) . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt ))) ; :: thesis:

end;

let n be Element of NAT ; :: thesis:
rseq . n = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt ))) by A10;
hence (Re cq1) . n = rseq . n by A25; :: 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 = (exp_R . p) * (Re (Sum ((hy1 . n) P_dt ))) 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 A24, 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 * (exp_R . p);
consider CL being Function of REAL ,REAL such that
A27: for th being Real holds CL . th = H₃(th) from FUNCT_2:sch 4();
A28: for d being real number holds CL . d = d * (exp_R . p)

proof

let d be real number ; :: thesis:
d is Real by XREAL_0:def 1;
hence CL . d = d * (exp_R . p) by A27; :: thesis:

end;

CL is linear

proof

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

proof

take exp_R . p ; :: thesis:
thus for q being Real holds CL . q = (exp_R . p) * q by A28; :: thesis:

end;

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

end;

then reconsider PL = CL as LINEAR ;
A31: 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

A32: 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:
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 Th67
.= ((r - p) * (exp_R . p)) + (Cr . (r - p)) by A2
.= (PL . (r - p)) + (PR . (r - p)) by A28 ;
hence ( r in ].(p - 1),(p + 1).[ implies (exp_R . r) - (exp_R . 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 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 A32, Th51, RCOMP_1:def 7; :: thesis:

end;

then A33: exp_R is_differentiable_in p by FDIFF_1:def 5;
PL . 1 = 1 * (exp_R . p) by A28;
hence ( exp_R is_differentiable_in p & diff exp_R ,p = exp_R . p ) by A31, A33, FDIFF_1:def 6; :: thesis: