reserve Z for open Subset of REAL;

theorem Th2:
  for n be Nat, f be PartFunc of REAL,REAL, r be Real st
  0 < r & ].-r,r.[ c= dom f & f is_differentiable_on n+1,].-r,r.[
 for x be Real st x in ].-r, r.[
  ex s be Real st 0 < s & s < 1 & f.x = Partial_Sums(
Maclaurin(f,].-r,r.[,x)).n + (diff(f,].-r,r.[).(n+1)).(s*x) * x |^ (n+1) / ((n+
  1)!)
proof
  let n be Nat, f be PartFunc of REAL,REAL,
      r be Real such that
A1: 0 < r & ].-r,r.[ c= dom f & f is_differentiable_on n+1, ].-r,r.[;
  let x be Real;
  assume x in ].-r, r.[;
  then consider s be Real such that
A2: 0 < s & s < 1 & f.x=Partial_Sums(Taylor(f, ].0-r,0+r.[,0,x)).n + (
  diff(f,].0-r,0+r.[).(n+1)).(0+s*(x-0)) * (x-0) |^ (n+1) / ((n+1)!) by A1,
TAYLOR_1:33;
  take s;
  thus thesis by A2;
end;
