reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

theorem
  seq is absolutely_summable implies z * seq is absolutely_summable
proof
A1: for n holds ||.z * seq.||.n >= 0 & ||.z * seq.||.n <= (|.z.| (#) ||.seq
  .||).n
  proof
    let n;
    ||.z * seq.||.n = ||.(z * seq).n.|| by CLVECT_2:def 3;
    hence ||.z * seq.||.n >= 0 by CSSPACE:44;
    ||.z * seq.||.n = ||.(z * seq).n.|| by CLVECT_2:def 3
      .= ||.z * seq.n.|| by CLVECT_1:def 14
      .= |.z.| * ||.seq.n.|| by CSSPACE:43
      .= |.z.| * ||.seq.||.n by CLVECT_2:def 3
      .= (|.z.| (#) ||.seq.||).n by SEQ_1:9;
    hence ||.z * seq.||.n <= (|.z.| (#) ||.seq.||).n;
  end;
  assume seq is absolutely_summable;
  then ||.seq.|| is summable;
  then |.z.| (#) ||.seq.|| is summable by SERIES_1:10;
  then ||.z * seq.|| is summable by A1,SERIES_1:20;
  hence thesis;
end;
