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 Th58:
  for z,z1 being Complex holds (Sum((z1+z) ExpSeq))-(Sum(z1 ExpSeq))
  =(Sum(z1 ExpSeq))*z+z*(Sum(z P_dt))*(Sum(z1 ExpSeq))
proof
  let z,z1 be Complex;
 (Sum((z1+z) ExpSeq))-(Sum(z1 ExpSeq))
  =(Sum(z1 ExpSeq))*(Sum(z ExpSeq))-(Sum(z1 ExpSeq))*1r by Lm2
    .=(Sum(z1 ExpSeq))*(((Sum(z ExpSeq))-1r)-z+z)
    .=(Sum(z1 ExpSeq))* (z*(Sum(z P_dt))+z) by Th56
    .=(Sum(z1 ExpSeq))*z +z*(Sum(z P_dt))*(Sum(z1 ExpSeq));
  hence thesis;
end;
