reserve n for Nat,
  i for Integer,
  p, x, x0, y for Real,
  q for Rational,
  f for PartFunc of REAL,REAL;

theorem Th26:
  for n be Nat, f be PartFunc of REAL,REAL, Z be Subset
of REAL, b,l be Real
 ex g be Function of REAL,REAL st for x be Real holds
  g.x=f.b-Partial_Sums(Taylor(f,Z,x,b)).n -l*(b-x) |^ (n+1) /((n+1)!)
proof
  let n be Nat;
  let f be PartFunc of REAL,REAL;
  let Z be Subset of REAL;
  let b,l be Real;
  deffunc U(Real)
  = In(f.b-Partial_Sums(Taylor(f,Z,$1,b)).n -l*(b-$1) |^ (n+1) /((n+1)!),
       REAL);
  consider g being Function of REAL, REAL such that
A1: for d be Element of REAL holds g.d = U(d) from FUNCT_2:sch 4;
  take g;
  let x be Real;
   reconsider x as Element of REAL by XREAL_0:def 1;
   g.x= U(x) by A1;
  hence thesis;
end;
