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 ((1/2)(#)(ln*f)) & f=(( #Z 2)*exp_R)-f1 & (for x st x in Z
holds f1.x=1 & f.x>0) implies (1/2)(#)(ln*f) is_differentiable_on Z & for x st
  x in Z holds (((1/2)(#)(ln*f))`|Z).x =exp_R(x)/(exp_R(x)-exp_R(-x))
proof
  assume that
A1: Z c= dom ((1/2)(#)(ln*f)) and
A2: f=(( #Z 2)*exp_R)-f1 and
A3: for x st x in Z holds f1.x=1 & f.x>0;
A4: Z c= dom (ln*f) by A1,VALUED_1:def 5;
  then for y being object st y in Z holds y in dom f by FUNCT_1:11;
  then
A5: Z c= dom ((( #Z 2)*exp_R)-f1) by A2,TARSKI:def 3;
A6: for x st x in Z holds f1.x=1 by A3;
  then
A7: (( #Z 2)*exp_R)-f1 is_differentiable_on Z by A5,Th25;
  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,A3,A7,FDIFF_1:9;
    hence thesis by TAYLOR_1:20;
  end;
  then
A8: ln*f is_differentiable_on Z by A4,FDIFF_1:9;
  Z c= dom (( #Z 2)*exp_R) /\ dom f1 by A5,VALUED_1:12;
  then
A9: Z c= dom (( #Z 2)*exp_R) by XBOOLE_1:18;
  for x st x in Z holds (((1/2)(#)(ln*f))`|Z).x =exp_R(x)/(exp_R(x)-exp_R (-x))
  proof
    let x;
A10: exp_R(x)>0 by SIN_COS:55;
    assume
A11: x in Z;
    then
A12: f is_differentiable_in x & f.x >0 by A2,A3,A7,FDIFF_1:9;
A13: ((( #Z 2)*exp_R)-f1).x =(( #Z 2)*exp_R).x-f1.x by A5,A11,VALUED_1:13
      .=( #Z 2).(exp_R.x)-f1.x by A9,A11,FUNCT_1:12
      .=(exp_R.x) #Z 2-f1.x by TAYLOR_1:def 1
      .=(exp_R.x) #Z 2-1 by A3,A11
      .=(exp_R(x)) #Z (1+1)-1 by SIN_COS:def 23
      .=(exp_R(x)) #Z 1 * (exp_R(x)) #Z 1 -1 by A10,PREPOWER:44
      .=exp_R(x) * (exp_R(x)) #Z 1 -1 by PREPOWER:35
      .=exp_R(x) * exp_R(x) -1 by PREPOWER:35;
    (((1/2)(#)(ln*f))`|Z).x =(1/2)*diff((ln*f),x) by A1,A8,A11,FDIFF_1:20
      .=(1/2)*(diff(f,x)/f.x) by A12,TAYLOR_1:20
      .=(1/2)*((((( #Z 2)*exp_R)-f1)`|Z).x/((( #Z 2)*exp_R)-f1).x) by A2,A7,A11
,FDIFF_1:def 7
      .=(1/2)*((2*exp_R(2*x))/(exp_R(x) * exp_R(x) -1)) by A6,A5,A11,A13,Th25
      .=(1/2)*(2*exp_R(2*x))/(exp_R(x) * exp_R(x) -1) by XCMPLX_1:74
      .=(exp_R(x+x)/exp_R(x))/((exp_R(x) * exp_R(x)-1)/exp_R(x)) by A10,
XCMPLX_1:55
      .=((exp_R(x)*exp_R(x))/exp_R(x))/((exp_R(x) * exp_R(x)-1)/exp_R(x)) by
SIN_COS:50
      .=((exp_R(x)*exp_R(x))/exp_R(x))/((exp_R(x)*exp_R(x))/exp_R(x)- 1/
    exp_R(x)) by XCMPLX_1:120
      .=exp_R(x)/(exp_R(x) * exp_R(x)/exp_R(x)-1/exp_R(x)) by A10,XCMPLX_1:89
      .=exp_R(x)/(exp_R(x)-1/exp_R(x)) by A10,XCMPLX_1:89
      .=exp_R(x)/(exp_R(x)-exp_R(-x)) by TAYLOR_1:4;
    hence thesis;
  end;
  hence thesis by A1,A8,FDIFF_1:20;
end;
