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 Th47:
  for th holds exp_R.th = Re(Sum(th ExpSeq))
proof
  let th;
 exp_R.th = Sum(th rExpSeq) by Def22;
  hence thesis by Th44;
end;
