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 Th56:
  seq is summable implies for z being Complex holds z (#)
  seq is summable & Sum(z (#) seq)= z * Sum(seq)
proof
  assume
A1: seq is summable;
  let z be Complex;
A2: Partial_Sums(z (#) seq)=(z (#) Partial_Sums(seq)) by Th29;
  Sum(z (#) seq)=lim((z (#) Partial_Sums(seq))) by Th29
    .=z * Sum(seq) by A1,COMSEQ_2:18;
  hence thesis by A1,A2;
end;
