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 Th43:
  for th holds th rExpSeq=Re(th ExpSeq)
proof
  let th;
 for n being Element of NAT holds (th rExpSeq).n=Re(th ExpSeq).n
  proof
    let n be Element of NAT;
 Re(th ExpSeq).n= Re (th ExpSeq.n) by COMSEQ_3:def 5
      .=Re(th |^ n /(n!+0*<i>)) by Def4
      .=Re((th|^ n) /(n!)+0*<i>)
      .=(th|^ n) /(n!) by COMPLEX1:12
      .=(th rExpSeq).n by Def5;
    hence thesis;
  end;
  hence thesis;
end;
