reserve x,a,b,c for Real,
  n for Nat,
  Z for open Subset of REAL,
  f, f1,f2 for PartFunc of REAL,REAL;

theorem
  Z c= dom (f1/f2) & (for x st x in Z holds f1.x=x+a & f2.x=x-b) implies
(f1/f2) is_differentiable_on Z & for x st x in Z holds ((f1/f2)`|Z).x = (-a-b)/
  (x-b)^2
proof
  assume that
A1: Z c= dom (f1/f2) and
A2: for x st x in Z holds f1.x=x+a & f2.x=x-b;
  Z c= dom f1 /\ (dom f2 \ f2"{0}) by A1,RFUNCT_1:def 1;
  then
A3: Z c= dom f2 \ f2"{0} by XBOOLE_1:18;
  for x st x in Z holds f2.x<>0
  proof
    let x;
    assume x in Z;
    then x in dom f2 \ f2"{0} by A3;
    then x in dom (f2^) by RFUNCT_1:def 2;
    hence thesis by RFUNCT_1:3;
  end;
  then for x st x in Z holds f1.x=x-(-a) & f2.x=x-b & f2.x<>0 by A2;
  hence thesis by A1,FDIFF_5:3;
end;
