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 (ln*f) & f=exp_R+(exp_R*f1) & (for x st x in Z holds f1.x=-x)
  implies ln*f is_differentiable_on Z & for x st x in Z holds ((ln*f)`|Z).x =(
  exp_R(x)-exp_R(-x))/(exp_R(x)+exp_R(-x))
proof
  assume that
A1: Z c= dom (ln*f) and
A2: f=exp_R+(exp_R*f1) and
A3: for x st x in Z holds f1.x=-x;
  for y being object st y in Z holds y in dom f by A1,FUNCT_1:11;
  then
A4: Z c= dom (exp_R+(exp_R*f1)) by A2,TARSKI:def 3;
  then Z c= dom exp_R /\ dom (exp_R*f1) by VALUED_1:def 1;
  then
A5: Z c= dom (exp_R*f1) by XBOOLE_1:18;
  then
A6: exp_R*f1 is_differentiable_on Z by A3,Th14;
A7: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
  then
A8: f is_differentiable_on Z by A2,A4,A6,FDIFF_1:18;
A9: for x st x in Z holds ((exp_R+(exp_R*f1))`|Z).x =exp_R(x)-exp_R(-x)
  proof
    let x;
    assume
A10: x in Z;
    hence
    ((exp_R+(exp_R*f1))`|Z).x = diff(exp_R,x) + diff((exp_R*f1),x) by A4,A6,A7,
FDIFF_1:18
      .=exp_R.x+ diff((exp_R*f1),x) by SIN_COS:65
      .=exp_R.x+((exp_R*f1)`|Z).x by A6,A10,FDIFF_1:def 7
      .=exp_R.x+(-exp_R(-x)) by A3,A5,A10,Th14
      .=exp_R(x)+(-exp_R(-x)) by SIN_COS:def 23
      .=exp_R(x)-exp_R(-x);
  end;
A11: for x st x in Z holds (exp_R+(exp_R*f1)).x>0
  proof
    let x;
A12: exp_R(x)>0 by SIN_COS:55;
    assume
A13: x in Z;
    then (exp_R+(exp_R*f1)).x=exp_R.x + (exp_R*f1).x by A4,VALUED_1:def 1
      .=exp_R.x +exp_R.(f1.x) by A5,A13,FUNCT_1:12
      .=exp_R.x +exp_R.(-x) by A3,A13
      .=exp_R(x) +exp_R.(-x) by SIN_COS:def 23
      .=exp_R(x) +exp_R(-x) by SIN_COS:def 23;
    hence thesis by A12,SIN_COS:55,XREAL_1:34;
  end;
A14: for x st x in Z holds ln*f is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then f is_differentiable_in x & f.x >0 by A2,A8,A11,FDIFF_1:9;
    hence thesis by TAYLOR_1:20;
  end;
  then
A15: ln*f is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((ln*f)`|Z).x =(exp_R(x)-exp_R(-x))/(exp_R(x)+
  exp_R(-x))
  proof
    let x;
    assume
A16: x in Z;
    then
A17: f.x=exp_R.x + (exp_R*f1).x by A2,A4,VALUED_1:def 1
      .=exp_R.x +exp_R.(f1.x) by A5,A16,FUNCT_1:12
      .=exp_R.x +exp_R.(-x) by A3,A16
      .=exp_R(x) +exp_R.(-x) by SIN_COS:def 23
      .=exp_R(x) +exp_R(-x) by SIN_COS:def 23;
    f is_differentiable_in x & f.x>0 by A2,A8,A11,A16,FDIFF_1:9;
    then diff((ln*f),x) =diff(f,x)/f.x by TAYLOR_1:20
      .=(f`|Z).x/f.x by A8,A16,FDIFF_1:def 7
      .=(exp_R(x)-exp_R(-x))/(exp_R(x) +exp_R(-x)) by A2,A9,A16,A17;
    hence thesis by A15,A16,FDIFF_1:def 7;
  end;
  hence thesis by A1,A14,FDIFF_1:9;
end;
