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
& f.x>0) 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 & f.x>0;
  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_1:12;
  then
A5: Z c= dom (exp_R*f1) by XBOOLE_1:18;
A6: for x st x in Z holds f1.x=-x by A3;
  then
A7: exp_R*f1 is_differentiable_on Z by A5,Th14;
A8: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
  then
A9: f is_differentiable_on Z by A2,A4,A7,FDIFF_1:19;
A10: for x st x in Z holds ((exp_R-(exp_R*f1))`|Z).x =exp_R(x)+exp_R(-x)
  proof
    let x;
    assume
A11: x in Z;
    hence
    ((exp_R-(exp_R*f1))`|Z).x = diff(exp_R,x) - diff((exp_R*f1),x) by A4,A7,A8,
FDIFF_1:19
      .=exp_R.x- diff((exp_R*f1),x) by SIN_COS:65
      .=exp_R.x-((exp_R*f1)`|Z).x by A7,A11,FDIFF_1:def 7
      .=exp_R.x-(-exp_R(-x)) by A6,A5,A11,Th14
      .=exp_R.x+exp_R(-x)
      .=exp_R(x)+exp_R(-x) by SIN_COS:def 23;
  end;
A12: 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 A3,A9,FDIFF_1:9;
    hence thesis by TAYLOR_1:20;
  end;
  then
A13: 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
A14: x in Z;
    then
A15: f.x=exp_R.x - (exp_R*f1).x by A2,A4,VALUED_1:13
      .=exp_R.x -exp_R.(f1.x) by A5,A14,FUNCT_1:12
      .=exp_R.x -exp_R.(-x) by A3,A14
      .=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 A3,A9,A14,FDIFF_1:9;
    then diff((ln*f),x) =diff(f,x)/f.x by TAYLOR_1:20
      .=(f`|Z).x/f.x by A9,A14,FDIFF_1:def 7
      .=(exp_R(x)+exp_R(-x))/(exp_R(x) -exp_R(-x)) by A2,A10,A14,A15;
    hence thesis by A13,A14,FDIFF_1:def 7;
  end;
  hence thesis by A1,A12,FDIFF_1:9;
end;
