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)) & Rseq is convergent & lim
  Rseq > 1 implies not seq is summable
proof
  assume that
A1: for n holds Rseq.n = n-root (||.seq.||.n) and
A2: Rseq is convergent and
A3: lim Rseq > 1;
  lim Rseq - 1 > 0 by A3,XREAL_1:50;
  then consider m such that
A4: for n st n >= m holds |.Rseq.n - lim Rseq.| < lim Rseq - 1 by A2,
SEQ_2:def 7;
  now
    let n;
    assume
A5: n >= m + 1;
A6: Rseq.n = n-root (||.seq.||.n) by A1
      .= n-root ||.seq.n.|| by CLVECT_2:def 3;
    m + 1 >= m by NAT_1:11;
    then n >= m by A5,XXREAL_0:2;
    then |.n-root ||.seq.n.|| - lim Rseq.| < lim Rseq - 1 by A4,A6;
    then - (lim Rseq - 1) < n-root ||.seq.n.|| - lim Rseq by SEQ_2:1;
    then
    1 - lim Rseq + lim Rseq < n-root ||.seq.n.|| - lim Rseq + lim Rseq by
XREAL_1:6;
    then
A7: ||.seq.n.|| >= 0 & n-root ||.seq.n.|| |^ n >= 1 by CSSPACE:44,PREPOWER:11;
    m + 1 >= 1 by NAT_1:11;
    then n >= 1 by A5,XXREAL_0:2;
    hence ||.seq.n.|| >= 1 by A7,POWER:4;
  end;
  then not seq is convergent or lim seq <> 09(X) by Th29;
  hence thesis by Th9;
end;
