reserve y for set,
  x,a,b for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2,g 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+a & f2.x<>
0) implies (f1/f2) is_differentiable_on Z & for x st x in Z holds ((f1/f2)`|Z).
  x = (2*a)/(x+a)^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+a & f2.x<>0;
A3: for x st x in Z holds f2.x = 1*x+a by A2;
A4: Z c= dom f1 /\ (dom f2 \ f2"{0}) by A1,RFUNCT_1:def 1;
  then
A5: Z c= dom f1 by XBOOLE_1:18;
A6: Z c= dom f2 by A4,XBOOLE_1:1;
  then
A7: f2 is_differentiable_on Z by A3,FDIFF_1:23;
A8: for x st x in Z holds f1.x = 1*x+-a
  proof
    let x;
A9: 1*x+-a=1*x-a;
    assume x in Z;
    hence thesis by A2,A9;
  end;
  then
A10: f1 is_differentiable_on Z by A5,FDIFF_1:23;
A11: for x st x in Z holds f2.x <> 0 by A2;
  then
A12: f1/f2 is_differentiable_on Z by A10,A7,FDIFF_2:21;
  for x st x in Z holds ((f1/f2)`|Z).x = (2*a)/(x+a)^2
  proof
    let x;
    assume
A13: x in Z;
    then
A14: f2.x <>0 by A2;
A15: f1.x=x-a & f2.x=x+a by A2,A13;
    f1 is_differentiable_in x & f2 is_differentiable_in x by A10,A7,A13,
FDIFF_1:9;
    then diff(f1/f2,x) =(diff(f1,x) * f2.x - diff(f2,x) * f1.x)/(f2.x)^2 by A14
,FDIFF_2:14
      .=((f1`|Z).x * f2.x-diff(f2,x) * f1.x)/(f2.x)^2 by A10,A13,FDIFF_1:def 7
      .= ((f1`|Z).x * f2.x-(f2`|Z).x * f1.x)/(f2.x)^2 by A7,A13,FDIFF_1:def 7
      .=(1*f2.x-(f2`|Z).x * f1.x)/(f2.x)^2 by A5,A8,A13,FDIFF_1:23
      .=(1*f2.x-1* f1.x)/(f2.x)^2 by A6,A3,A13,FDIFF_1:23
      .=(2*a)/(x+a)^2 by A15;
    hence thesis by A12,A13,FDIFF_1:def 7;
  end;
  hence thesis by A10,A7,A11,FDIFF_2:21;
end;
