reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;
reserve d for Real;
reserve th,th1,th2 for Real;

theorem Th35:
  for th 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;
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,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,def 3;
A6: now
    let p be Real;
    assume p>0;
    then consider n such that
A7: for k st n <= k holds |.Partial_Sums(Re ((th*<i>) ExpSeq)).k-
    Re (Sum(((th*<i>) ExpSeq))).| < p by A4,SEQ_2:def 7;
    take n;
 now
      let k such that
A8:  n <= k;
|.Partial_Sums(th P_cos).k-Re (Sum(((th*<i>) ExpSeq))).| =|.Partial_Sums(
Re ((th*<i>) ExpSeq)).(2*k)- Re (Sum(((th*<i>) ExpSeq))).| & 2*k=k+k by Th34;
      hence |.Partial_Sums(th P_cos).k-Re (Sum(((th*<i>) ExpSeq))).|
      < p by A7,A8,NAT_1:12;
    end;
    hence for k st n <= k holds
    |.Partial_Sums(th P_cos).k-Re (Sum(((th*<i>) ExpSeq))).| < p;
  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
    let p be Real;
    assume p>0;
    then consider n such that
A11: for k st n <= k holds |.Partial_Sums(Im ((th*<i>) ExpSeq)).k-
    Im (Sum(((th*<i>) ExpSeq))).| < p by A5,SEQ_2:def 7;
    take n;
 now
      let k such that
A12:  n <= k;
A13:  |.Partial_Sums(th P_sin).k-Im (Sum(((th*<i>) ExpSeq))).|
      =|.Partial_Sums(Im ((th*<i>) ExpSeq)).(2*k+1)-
      Im (Sum(((th*<i>) ExpSeq))).| by Th34;
  2*k=k+k;
      then n <= 2*k by A12,NAT_1:12;
      then n<2*k+1 by NAT_1:13;
      hence |.Partial_Sums(th P_sin).k-Im (Sum(((th*<i>) ExpSeq))).|
      < p by A11,A13;
    end;
    hence for k st n <= k holds
    |.Partial_Sums(th P_sin).k-Im (Sum(((th*<i>) ExpSeq))).| < p;
  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 thesis by A6,A9,A10,SEQ_2:def 6,SERIES_1:def 3;
end;
