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 Th44:
  for th holds th rExpSeq is summable &
  Sum(th rExpSeq)=Re(Sum(th ExpSeq))
proof
  let th;
   Sum(th ExpSeq) =Sum(Re (th ExpSeq))+(Sum(Im (th ExpSeq)))*<i> & Sum(th
  rExpSeq)=Sum(Re (th ExpSeq)) by Th43,COMSEQ_3:53;
  hence thesis by Th43,COMPLEX1:12;
end;
