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 Th45:
  for p,q being Real holds
  Sum((p+q) rExpSeq) = (Sum(p rExpSeq))*(Sum(q rExpSeq))
proof
  let p,q be Real;
  reconsider p,q as Real;
 Sum((p+q) rExpSeq)=Re(Sum((p+q) ExpSeq)) by Th44
    .=Re((Sum(p ExpSeq)) *(Sum(q ExpSeq))) by Lm2
    .=Re((Re(Sum(p ExpSeq))+Im(Sum(p ExpSeq))*<i>)*(Sum(q ExpSeq)))
  by COMPLEX1:13
    .= Re((Re(Sum(p ExpSeq))+Im(Sum(p ExpSeq))*<i>)*
  (Re(Sum(q ExpSeq))+Im(Sum(q ExpSeq))*<i>)) by COMPLEX1:13
    .= Re((Sum(p rExpSeq)+Im(Sum(p ExpSeq))*<i>)*
  (Re(Sum(q ExpSeq))+Im(Sum(q ExpSeq))*<i>)) by Th44
    .= Re((Sum(p rExpSeq)+0*<i>)*
  (Re(Sum(q ExpSeq))+Im(Sum(q ExpSeq))*<i>)) by Th41
    .= Re(Sum(p rExpSeq)*(Sum(q rExpSeq)+Im(Sum(q ExpSeq))*<i>)) by Th44
    .= Re(Sum(p rExpSeq)*(Sum(q rExpSeq)+0*<i>)) by Th41
    .= Re((Sum(p rExpSeq))*(Sum(q rExpSeq))+0*<i>)
    .= (Sum(p rExpSeq))*(Sum(q rExpSeq)) by COMPLEX1:12;
  hence thesis;
end;
