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
  ( for n holds ||.seq.||.n <= Rseq.n ) & Rseq is summable implies seq
  is absolutely_summable
proof
A1: for n holds ||.seq.||.n >= 0
  proof
    let n;
    ||.seq.||.n = ||.seq.n.|| by CLVECT_2:def 3;
    hence thesis by CSSPACE:44;
  end;
  assume ( for n holds ||.seq.||.n <= Rseq.n)& Rseq is summable;
  then ||.seq.|| is summable by A1,SERIES_1:20;
  hence thesis;
end;
