reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem
  ( for n being Nat holds ||.seq.||.n <= Rseq.n ) &
    Rseq is summable implies seq
  is absolutely_summable
proof
A1: for n being Nat holds ||.seq.||.n >= 0
  proof
    let n be Nat;
    ||.seq.||.n = ||.seq.n.|| by BHSP_2:def 3;
    hence thesis by BHSP_1:28;
  end;
  assume ( for n being Nat  holds ||.seq.||.n <= Rseq.n)& Rseq is summable;
  then ||.seq.|| is summable by A1,SERIES_1:20;
  hence thesis;
end;
