reserve Z for open Subset of REAL;

theorem Th13:
  for r, e be Real st 0 < r & 0 < e
  ex n be Nat st 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
proof
  let r, e be Real such that
A1: 0 < r and
A2: 0 < e;
  consider M, L be Real such that
A3: M >= 0 & L >= 0 and
A4: for n be Nat holds
   for x,s be Real st x in ].-r,r.[ & 0 <
s & s < 1 holds |.(diff(exp_R,].-r,r.[).n).(s*x) * x |^ n / (n!).| <= M*L |^ n
  / (n!) by A1,Th11;
  consider n be Nat such that
A5: for m be Nat st n <= m holds (M*L |^ m / (m!)) < e by A2,A3,Th12;
  take n;
  let m be Nat;
  assume n <= m;
  then
A6: (M*L |^ m / (m!)) < e by A5;
  let x,s be Real;
  assume x in ].-r,r.[ & 0 < s & s < 1;
  then
  |.(diff(exp_R,].-r,r.[).m).(s*x) * x |^ m /(m!).| <= M*L |^ m /(m!) by A4;
  hence thesis by A6,XXREAL_0:2;
end;
