 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;
 reserve g for non zero Polynomial of INT.Ring;
 reserve f for Element of the carrier of Polynom-Ring INT.Ring;
 reserve z0 for non zero Element of F_Real;

theorem Th45:
  for M be non zero Nat, z0 be non zero Element of F_Real st z0 = number_e
  holds
  ex n1 be Nat st for n be Nat st n1 <= n holds
  |. (m|^(m+1))|^n/(n!) - 0 .| < 1/(2*(M*(z0 to_power m)))
    proof
      let M be non zero Nat, z0 be non zero Element of F_Real;
      assume
A1:   z0 = number_e;
A2:   ((m|^(m+1)) rExpSeq) is convergent & lim ((m|^(m+1)) rExpSeq) = 0
        by Th44; then
      consider n1 be Nat such that
A3:   for n be Nat st n1<=n holds
      |. ((m|^(m+1)) rExpSeq).n - lim ((m|^(m+1)) rExpSeq) .|
      < 1/(2*(M*(z0 to_power m))) by A1,TAYLOR_1:11,SEQ_2:def 7;
      take n1;
      let n be Nat;
      assume
A4:   n1 <= n;
      |. ((m|^(m+1)) rExpSeq).n - lim ((m|^(m+1)) rExpSeq) .|
        = |. (m|^(m+1))|^n/(n!) - lim ((m|^(m+1)) rExpSeq) .| by SIN_COS:def 5;
      hence thesis by A2,A4,A3;
    end;
