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
  number_e is irrational
proof
  assume number_e is rational;
  then consider n such that
A1: n>=2 and
A2: n!*number_e is integer by Th31;
  reconsider ne = n!*number_e as Integer by A2;
  set x = 1/(n+1);
  reconsider ne1 = n!*Partial_Sums(eseq).n as Integer by Th37;
  n!*number_e = n!*((Partial_Sums(eseq).n)+Sum(eseq^\(n+1))) by Th23,
SERIES_1:15
    .= n!*(Partial_Sums(eseq).n)+n!*Sum(eseq^\(n+1));
  then
A3: n!*Sum(eseq^\(n+1))=ne-ne1;
  x/(1-x)<1 & n!*Sum(eseq^\(n+1))<=x/(1-x) by A1,Th39,Th40;
  then n!*Sum(eseq^\(n+1))<0+1 by XXREAL_0:2;
  hence contradiction by A3,Th35,INT_1:7;
end;
