reserve Z for open Subset of REAL;

theorem
  for n be Nat, f be PartFunc of REAL,REAL, x0,
      r be Real st
0 < r & ].x0-r,x0+r.[ c= dom f & f is_differentiable_on n+1, ].x0-r,x0+r.[
 for x be Real st x in ].x0-r, x0+r.[
  ex s be Real st 0 < s & s < 1 & |.f.x-
Partial_Sums(Taylor(f, ].x0-r,x0+r.[,x0,x)).n.|
    = |. (diff(f,].x0-r,x0+r.[).(n
  +1)).(x0+s*(x-x0)) * (x-x0) |^ (n+1) / ((n+1)!).|
proof
  let n be Nat;
  let f be PartFunc of REAL,REAL;
  let x0,r be Real such that
A1: 0 < r & ].x0-r,x0+r.[ c= dom f & f is_differentiable_on n+1, ].x0-r,
  x0+r .[;
  let x be Real;
  assume x in ].x0-r,x0+r.[;
  then ex s be Real
   st 0 < s & s < 1 & f.x=Partial_Sums(Taylor( f, ].x0-r,x0+r.[
,x0,x)).n + (diff(f,].x0-r,x0+r.[).(n+1)).(x0+s*(x-x0)) * (x-x0 ) |^ (n+1) /((n
  +1)!) by A1,TAYLOR_1:33;
  hence thesis;
end;
