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;
