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 Th30:
  ( for n holds seq.n <> 0.X ) & ( ex m st for n st n >= m holds
  ||.seq.(n+1).||/||.seq.n.|| >= 1 ) implies not seq is summable
proof
  assume
A1: for n holds seq.n <> 09(X);
  given m such that
A2: for n st n >= m holds ||.seq.(n+1).||/||.seq.n.|| >= 1;
A3: now
    defpred P[Nat] means ||.seq.(m+$1).|| >= ||.seq.m.||;
    let n;
A4: for k st P[k] holds P[k+1]
    proof
      let k;
      assume
A5:   ||.seq.(m+k).|| >= ||.seq.m.||;
      seq.(m+k) <> 09(X) by A1;
      then
A6:   ||.seq.(m+k).|| <> 0 by CSSPACE:42;
      ||.seq.(m+k+1).||/||.seq.(m+k).|| >= 1 & ||.seq.(m+k).|| >= 0 by A2,
CSSPACE:44,NAT_1:11;
      then ||.seq.(m+k+1).|| >= ||.seq.(m+k).|| by A6,XREAL_1:191;
      hence thesis by A5,XXREAL_0:2;
    end;
A7: P[0];
A8: for k holds P[k] from NAT_1:sch 2(A7,A4);
    assume n >= m;
    then consider k being Nat such that
A9: n = m + k by NAT_1:10;
    thus ||.seq.n.|| >= ||.seq.m.|| by A8,A9;
  end;
  seq.m <> 09(X) by A1;
  then ||.seq.m.|| <> 0 by CSSPACE:42;
  then ||.seq.m.|| > 0 by CSSPACE:44;
  then not seq is convergent or lim seq <> 09(X) by A3,Th29;
  hence thesis by Th9;
end;
