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();
 A2: 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: ( (hy1 " ) (#) (Cr /* hy1) is convergent & lim ((hy1 " ) (#) (Cr /* hy1)) = 0 )
A3: 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> ))))
A4: ((hy1 " ) (#) (Cr /* hy1)) . n = ((hy1 " ) . n) * ((Cr /* hy1) . n) by SEQ_1:12
.= ((hy1 . n) " ) * ((Cr /* hy1) . n) by VALUED_1:10 ;
A5: dom Cr = REAL by FUNCT_2:def 1;
A6: rng hy1 c= dom Cr by A5;
A7: ((hy1 " ) (#) (Cr /* hy1)) . n = ((hy1 . n) " ) * (Cr . (hy1 . n)) by A4, A6, FUNCT_2:185
.= ((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> ))))) ;
A8: hy1 is non-empty by FDIFF_1:def 1;
A9: hy1 . n <> 0 by A8, SEQ_1:7;
A10: ((hy1 " ) (#) (Cr /* hy1)) . n = - (1 * (Im ((Sum (((hy1 . n) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> ))))) by A7, A9, XCMPLX_0:def 7
.= - (Im ((Sum (((hy1 . n) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> )))) ;
thus ((hy1 " ) (#) (Cr /* hy1)) . n = - (Im ((Sum (((hy1 . n) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> )))) by A10; :: 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
A11: 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) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> ));
consider cq1 being Complex_Sequence such that
A12: for n being Element of NAT holds cq1 . n = H4(n) from COMSEQ_1:sch 1();
 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: ( 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 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
A15: 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
A16: r > 0 and
A17: for z being complex number st |.z.| < r holds
|.(Sum (z P_dt )).| < q by A14, Th63;
A18: ( hy1 is convergent & lim hy1 = 0 ) by FDIFF_1:def 1;
consider k being Element of NAT such that
A19: for m being Element of NAT st k <= m holds
abs ((hy1 . m) - 0 ) < r by A16, A18, SEQ_2:def 7;
A20: now
let m be Element of NAT ; :: thesis: ( k <= m implies |.((cq1 . m) - 0c ).| < q )
assume A21: k <= m ; :: thesis: |.((cq1 . m) - 0c ).| < q
A22: |.((cq1 . m) - 0c ).| = |.((Sum (((hy1 . m) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> ))).| by A12
.= |.(Sum (((hy1 . m) * <i> ) P_dt )).| * |.((cos . p) + ((sin . p) * <i> )).| by COMPLEX1:151
.= |.(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 )).| ;
A23: abs ((hy1 . m) - 0 ) < r by A19, A21;
A24: (hy1 . m) * <i> = 0 + ((hy1 . m) * <i> ) ;
A25: ( Re ((hy1 . m) * <i> ) = 0 & Im ((hy1 . m) * <i> ) = hy1 . m ) by A24, COMPLEX1:28;
A26: |.((hy1 . m) * <i> ).| = abs (hy1 . m) by A25, COMPLEX1:158;
thus |.((cq1 . m) - 0c ).| < q by A17, A22, A23, A26; :: 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 A20; :: thesis: verum
end;
thus ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.((cq1 . m) - 0c ).| < q by A15; :: thesis: verum
end;
A27: cq1 is convergent by A13, COMSEQ_2:def 4;
A28: lim cq1 = 0c by A13, A27, COMSEQ_2:def 5;
A29: lim (- cq1) = - 0c by A27, A28, COMSEQ_2:22;
A30: for n being Element of NAT holds (Im (- cq1)) . n = rseq . n
proof
A31: 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> ))))
A32: (Im (- cq1)) . n = Im ((- cq1) . n) by COMSEQ_3:def 4
.= Im (- (cq1 . n)) by VALUED_1:8
.= Im (- ((Sum (((hy1 . n) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> )))) by A12
.= - (Im ((Sum (((hy1 . n) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> )))) by COMPLEX1:34 ;
thus (Im (- cq1)) . n = - (Im ((Sum (((hy1 . n) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> )))) by A32; :: thesis: verum
end;
let n be Element of NAT ; :: thesis: (Im (- cq1)) . n = rseq . n
A33: rseq . n = - (Im ((Sum (((hy1 . n) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> )))) by A11;
thus (Im (- cq1)) . n = rseq . n by A31, A33; :: thesis: verum
end;
A34: 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
A35: rseq . n = - (Im ((Sum (((hy1 . n) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> )))) by A11;
thus rseq . n = ((hy1 " ) (#) (Cr /* hy1)) . n by A3, A35; :: thesis: verum
end;
A36: rseq = (hy1 " ) (#) (Cr /* hy1) by A34, FUNCT_2:113;
A37: (hy1 " ) (#) (Cr /* hy1) = Im (- cq1) by A30, A36, FUNCT_2:113;
thus ( (hy1 " ) (#) (Cr /* hy1) is convergent & lim ((hy1 " ) (#) (Cr /* hy1)) = 0 ) by A27, A29, A37, COMPLEX1:12, COMSEQ_3:41; :: thesis: verum
end;
reconsider PR = Cr as REST by A2, FDIFF_1:def 3;
deffunc H3( Element of REAL ) -> Element of REAL = - ($1 * (sin . p));
consider CL being Function of REAL ,REAL such that
A38: for th being Real holds CL . th = H3(th) from FUNCT_2:sch 4();
 A39: ex r being Real st
for q being Real holds CL . q = r * q
proof
A40: for q being Real holds CL . q = (- (sin . p)) * q
proof
let q be Real; :: thesis: CL . q = (- (sin . p)) * q
A41: CL . q = - (q * (sin . p)) by A38
.= (- (sin . p)) * q ;
thus CL . q = (- (sin . p)) * q by A41; :: 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 A40; :: thesis: verum
end;
reconsider PL = CL as LINEAR by A39, FDIFF_1:def 4;
A42: 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
A43: 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)) )
A44: r = p + (r - p) ;
A45: (cos . r) - (cos . p) = (- ((r - p) * (sin . p))) - ((r - p) * (Im ((Sum (((r - p) * <i> ) P_dt )) * ((cos . p) + ((sin . p) * <i> ))))) by A44, Th65
.= (- ((r - p) * (sin . p))) + (Cr . (r - p)) by A1
.= (PL . (r - p)) + (PR . (r - p)) by A38 ;
thus ( r in ].(p - 1),(p + 1).[ implies (cos . r) - (cos . p) = (PL . (r - p)) + (PR . (r - p)) ) by A45; :: 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 A43, Th27, RCOMP_1:def 7; :: thesis: verum
end;
A46: cos is_differentiable_in p by A42, FDIFF_1:def 5;
A47: PL . 1 = - (1 * (sin . p)) by A38
.= - (sin . p) ;
thus ( cos is_differentiable_in p & diff cos ,p = - (sin . p) ) by A42, A46, A47, FDIFF_1:def 6; :: thesis: verum