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
  Z c= dom ((1/(4*a^2))(#)f) & f=ln*((f1+f2)/(f1-f2)) & f2=#Z 2 & (for x
  st x in Z holds f1.x=a^2 & (f1-f2).x>0 & a<>0) implies (1/(4*a^2))(#)f
is_differentiable_on Z & for x st x in Z holds (((1/(4*a^2))(#)f)`|Z).x =x/(a|^
  4-x|^4)
proof
  assume that
A1: Z c= dom ((1/(4*a^2))(#)f) and
A2: f=ln*((f1+f2)/(f1-f2)) & f2=#Z 2 and
A3: for x st x in Z holds f1.x=a^2 & (f1-f2).x>0 & a<>0;
A4: Z c= dom f by A1,VALUED_1:def 5;
  then
A5: f is_differentiable_on Z by A2,A3,Th5;
  for x st x in Z holds (((1/(4*a^2))(#)f)`|Z).x =x/(a|^4-x|^4)
  proof
    let x;
    assume
A6: x in Z;
    then a<>0 by A3;
    then a^2>0 by SQUARE_1:12;
    then
A7: 4*a^2>4*0 by XREAL_1:68;
    (((1/(4*a^2))(#)f)`|Z).x =(1/(4*a^2))*diff(f,x) by A1,A5,A6,FDIFF_1:20
      .=(1/(4*a^2))*(f`|Z).x by A5,A6,FDIFF_1:def 7
      .=(1/(4*a^2))*(4*a^2*x/(a|^4-x|^4)) by A2,A3,A4,A6,Th5
      .=(1/(4*a^2))*((4*a^2)*(x/(a|^4-x|^4))) by XCMPLX_1:74
      .=(x/(a|^4-x|^4))*(1/(4*a^2)*(4*a^2))
      .=x/(a|^4-x|^4) by A7,XCMPLX_1:108;
    hence thesis;
  end;
  hence thesis by A1,A5,FDIFF_1:20;
end;
