reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

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.||;
A6:   ||.seq.(m+k).|| <> 0 by A1,BHSP_1:26;
      ||.seq.(m+k+1).||/||.seq.(m+k).|| >= 1 & ||.seq.(m+k).|| >= 0 by A2,
BHSP_1:28,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 be Nat such that
A9: n = m + k by NAT_1:10;
    thus ||.seq.n.|| >= ||.seq.m.|| by A8,A9;
  end;
A10: ||.seq.m.|| <> 0 by A1,BHSP_1:26;
  ||.seq.m.|| >= 0 by BHSP_1:28;
  then not seq is convergent or lim seq <> 09(X) by A10,A3,Th29;
  hence thesis by Th9;
end;
