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 Rseq.n = n-root (||.seq.||.n)) & (ex m st for n st n >= m
  holds Rseq.n >= 1) implies not seq is summable
proof
  assume
A1: for n holds Rseq.n = n-root (||.seq.||.n);
  given m such that
A2: for n st n >= m holds Rseq.n >= 1;
  now
    let n;
    assume
A3: n >= m+1;
    m + 1 >= 1 by NAT_1:11;
    then
A4: n >= 1 by A3,XXREAL_0:2;
    m+1 >= m by NAT_1:11;
    then
A5: n>=m by A3,XXREAL_0:2;
    Rseq.n = n-root (||.seq.||.n) by A1
      .= n-root ||.seq.n.|| by CLVECT_2:def 3;
    then ||.seq.n.|| >= 0 & n-root ||.seq.n.|| |^ n >= 1 by A2,A5,CSSPACE:44
,PREPOWER:11;
    hence ||.seq.n.|| >= 1 by A4,POWER:4;
  end;
  then not seq is convergent or lim seq <> 09(X) by Th29;
  hence thesis by Th9;
end;
