reserve y for set,
  x,a for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2 for PartFunc of REAL,REAL;

theorem Th4:
  Z c= dom ((f1+f2)/(f1-f2)) & f2=#Z 2 & (for x st x in Z holds f1.
x=a^2 & (f1-f2).x<>0) implies (f1+f2)/(f1-f2) is_differentiable_on Z & for x st
  x in Z holds (((f1+f2)/(f1-f2))`|Z).x = 4*a^2*x/(a^2-x|^2)^2
proof
  assume that
A1: Z c= dom ((f1+f2)/(f1-f2)) and
A2: f2=#Z 2 and
A3: for x st x in Z holds f1.x=a^2 & (f1-f2).x<>0;
A4: for x st x in Z holds f1.x=a^2 by A3;
A5: Z c=dom (f1+f2) /\ (dom (f1-f2) \ (f1-f2)"{0}) by A1,RFUNCT_1:def 1;
  then
A6: Z c= dom (f1+f2) by XBOOLE_1:18;
  then
A7: f1+f2 is_differentiable_on Z by A2,A4,FDIFF_4:17;
A8: Z c= dom (f1-f2) by A5,XBOOLE_1:1;
  then
A9: f1-f2 is_differentiable_on Z by A2,A4,Th3;
A10: for x st x in Z holds (f1-f2).x <> 0 by A3;
  then
A11: (f1+f2)/(f1-f2) is_differentiable_on Z by A7,A9,FDIFF_2:21;
  for x st x in Z holds (((f1+f2)/(f1-f2))`|Z).x=4*a^2*x/(a^2-x|^2)^2
  proof
    let x;
A12: f2.x=x #Z 2 by A2,TAYLOR_1:def 1
      .=x |^2 by PREPOWER:36;
    assume
A13: x in Z;
    then
A14: (f1-f2).x<>0 by A3;
A15: (f1-f2).x=f1.x-f2.x by A8,A13,VALUED_1:13
      .=a^2-x |^2 by A3,A13,A12;
A16: (f1+f2).x=f1.x+f2.x by A6,A13,VALUED_1:def 1
      .=a^2+x |^2 by A3,A13,A12;
    f1+f2 is_differentiable_in x & f1-f2 is_differentiable_in x by A7,A9,A13,
FDIFF_1:9;
    then
    diff((f1+f2)/(f1-f2),x) =(diff(f1+f2,x)*(f1-f2).x - diff(f1-f2,x)*(f1
    +f2).x)/((f1-f2).x)^2 by A14,FDIFF_2:14
      .=(((f1+f2)`|Z).x * (f1-f2).x-diff(f1-f2,x) * (f1+f2).x)/((f1-f2).x)^2
    by A7,A13,FDIFF_1:def 7
      .= (((f1+f2)`|Z).x * (f1-f2).x-((f1-f2)`|Z).x * (f1+f2).x)/((f1-f2).x)
    ^2 by A9,A13,FDIFF_1:def 7
      .=((2*x)*(f1-f2).x-((f1-f2)`|Z).x * (f1+f2).x)/((f1-f2).x)^2 by A2,A6,A4
,A13,FDIFF_4:17
      .=((2*x)*(f1-f2).x-(-2*x) * (f1+f2).x)/((f1-f2).x)^2 by A2,A8,A4,A13,Th3
      .=4*a^2*x/(a^2-x|^2)^2 by A16,A15;
    hence thesis by A11,A13,FDIFF_1:def 7;
  end;
  hence thesis by A7,A9,A10,FDIFF_2:21;
end;
