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;

theorem Th5:
  Partial_Sums(seq).k=Partial_Sums(Shift(seq)).k+seq.k
proof
  defpred X[Nat] means
  Partial_Sums(seq).$1=Partial_Sums(Shift(seq)).$1+seq.$1;
 Partial_Sums(seq).0=0c + seq.0 by SERIES_1:def 1
    .=(Shift(seq)).0 + seq.0 by Def8
    .=Partial_Sums(Shift(seq)).0+seq.0 by SERIES_1:def 1;
then A1: X[0];
A2: for k st X[k] holds X[k+1]
  proof
    let k such that
A3: Partial_Sums(seq).k=Partial_Sums(Shift(seq)).k+seq.k;
    thus
    Partial_Sums(seq).(k+1) = (Partial_Sums(Shift(seq)).k+seq.k) + seq.(k+1)
    by A3,SERIES_1:def 1
      .=(Partial_Sums(Shift(seq)).k+(Shift(seq)).(k+1)) + seq.(k+1) by Def8
      .=Partial_Sums(Shift(seq)).(k+1)+seq.(k+1) by SERIES_1:def 1;
  end;
 for k holds X[k] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
