reserve Z for open Subset of REAL;

theorem
  for r, e be Real st 0 < r & 0 < e
   ex n be Nat st for
  m be Nat st n <= m holds for x being Real st x in ].-r,r.[
  holds |.exp_R.x-Partial_Sums(Maclaurin(exp_R,].-r,r.[,x)).m.| < e
proof
  let r, e be Real such that
A1: r > 0 and
A2: e > 0;
  consider n be Nat such that
A3: for m be Nat st n <= m
    for x,s be Real st x in ].-r
,r.[ & 0 < s & s < 1 holds |. (diff(exp_R,].-r,r.[).m).(s*x) * x |^ m / (m!)
  .|< e by A1,A2,Th13;
  take n;
  let m be Nat such that
A4: n <= m;
  now
    m <= m+1 by NAT_1:11;
    then
A5: n <= m+1 by A4,XXREAL_0:2;
    let x be Real such that
A6: x in ].-r,r.[;
    ex s be Real
      st 0 < s & s < 1 & |.exp_R.x-Partial_Sums (Maclaurin(
exp_R,].-r,r.[,x)).m.| =|. (diff(exp_R,].-r,r.[).(m+1)).(s*x) * x |^ (m+1) / (
    (m+1)!).| by A1,A6,Th4,Th10,SIN_COS:47;
    hence
    |.exp_R.x-Partial_Sums(Maclaurin(exp_R,].-r,r.[,x)).m.| < e by A3,A6,A5;
  end;
  hence thesis;
end;
