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
  seq1 is absolutely_summable & seq2 is absolutely_summable implies seq1
  + seq2 is absolutely_summable
proof
A1: for n holds ||.seq1 + seq2.||.n >= 0 & ||.seq1 + seq2.||.n <= (||.seq1
  .|| + ||.seq2.||).n
  proof
    let n;
    ||.seq1 + seq2.||.n = ||.(seq1 + seq2).n.|| by CLVECT_2:def 3;
    hence ||.seq1 + seq2.||.n >= 0 by CSSPACE:44;
    ||.seq1+seq2.||.n = ||.(seq1 + seq2).n.|| by CLVECT_2:def 3
      .= ||.(seq1).n + (seq2).n.|| by NORMSP_1:def 2;
    then ||.seq1+seq2.||.n <= ||.(seq1).n.|| + ||.(seq2).n.|| by CSSPACE:46;
    then ||.seq1+seq2.||.n <= ||.seq1.||.n + ||.(seq2).n.|| by CLVECT_2:def 3;
    then ||.seq1 + seq2.||.n <= ||.seq1.||.n + ||.seq2.||.n by CLVECT_2:def 3;
    hence ||.seq1 + seq2.||.n <= (||.seq1.|| + ||.seq2.||).n by SEQ_1:7;
  end;
  assume seq1 is absolutely_summable & seq2 is absolutely_summable;
  then ||.seq1.|| is summable & ||.seq2.|| is summable;
  then ||.seq1.|| + ||.seq2.|| is summable by SERIES_1:7;
  then ||.seq1 + seq2.|| is summable by A1,SERIES_1:20;
  hence thesis;
end;
