reserve Z for open Subset of REAL;

theorem Th4:
  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;
  let f be PartFunc of REAL,REAL;
  let 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 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)!) by A1,Th2;
  hence thesis;
end;
