reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th35:
  n!*Sum(eseq^\(n+1))>0
proof
A1: now
    let k;
    (eseq^\(n+1)).k = eseq.(n+1+k) by NAT_1:def 3
      .= 1/((n+1+k)!) by Def5;
    hence (eseq^\(n+1)).k>0;
  end;
  eseq^\(n+1) is summable by Th23,SERIES_1:12;
  then n!>0 & Sum(eseq^\(n+1))>0 by A1,Th34;
  hence thesis by XREAL_1:129;
end;
