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 Th35:
  Partial_Sums(||.seq.||) is non-decreasing
proof
  now
    let n;
    ||.seq.(n+1).|| >= 0 by CSSPACE:44;
    then ||.seq.||.(n+1) >= 0 by CLVECT_2:def 3;
    then
    0 + Partial_Sums(||.seq.||).n <= ||.seq.||.(n+1) + Partial_Sums(||.seq
    .||).n by XREAL_1:6;
    hence Partial_Sums(||.seq.||).n <= Partial_Sums(||.seq.||).(n+1) by
SERIES_1:def 1;
  end;
  hence thesis;
end;
