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 Th23:
  eseq is summable & Sum(eseq)=exp_R(1)
proof
  now
    let k be Element of NAT;
    thus eseq.k = 1/(k!) by Def5
      .= (1 |^ k)/(k!)
      .= (1 rExpSeq).k by SIN_COS:def 5;
  end;
  then
A1: eseq=(1 rExpSeq) by FUNCT_2:63;
  hence eseq is summable by SIN_COS:45;
  thus exp_R(1) = exp_R.1 by SIN_COS:def 23
    .= Sum(eseq) by A1,SIN_COS:def 22;
end;
