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 Th41:
  for p being Real holds Im(Sum(p ExpSeq))=0
proof
  let p be Real;
A1: for n holds (Partial_Sums(Im(p ExpSeq))).n=0
  proof
    let n;
    defpred X[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: X[0];
A3: for k st X[k] holds X[k+1]
    proof
      let k;
      assume (Partial_Sums(Im(p ExpSeq))).k=0; 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 thesis;
    end;
 for n holds X[n] from NAT_1:sch 2(A2,A3);
    hence thesis;
  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;
    m in NAT & (Partial_Sums(Im(p ExpSeq))).n=0 by A1,ORDINAL1:def 12;
    hence thesis by A1;
  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 thesis by A4,SERIES_1:def 3;
end;
