reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th29:
  for z being Complex holds Partial_Sums(z (#) seq) = z (#)
  Partial_Sums(seq)
proof
  let z be Complex;
  defpred P[Nat] means Partial_Sums(z (#) seq).$1= (z (#)
  Partial_Sums(seq)).$1;
A1: now
    let n be Nat;
    assume
A2: P[n];
    Partial_Sums(z (#) seq).(n+1) =Partial_Sums(z (#) seq).n + (z (#) seq)
    .(n+1) by SERIES_1:def 1
      .=(z * Partial_Sums(seq).n )+ (z (#) seq).(n+1) by A2,VALUED_1:6
      .=(z * Partial_Sums(seq).n )+ (z * seq.(n+1)) by VALUED_1:6
      .= z * ( Partial_Sums(seq).n + seq.(n+1))
      .= z * ( Partial_Sums(seq).(n+1)) by SERIES_1:def 1
      .= (z (#) Partial_Sums(seq)).(n+1) by VALUED_1:6;
    hence P[n+1];
  end;
  Partial_Sums(z(#)seq).0=(z (#) seq).0 by SERIES_1:def 1
    .=z * seq.0 by VALUED_1:6
    .=z * Partial_Sums(seq).0 by SERIES_1:def 1
    .=(z (#) Partial_Sums(seq)).0 by VALUED_1:6;
  then
A3: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
