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*(exp_R-f)) & (for x st x in Z holds f.x=1 & (exp_R-f).x>0
  ) implies ln*(exp_R-f) is_differentiable_on Z & for x st x in Z holds ((ln*(
  exp_R-f))`|Z).x = exp_R.x/(exp_R.x-1)
proof
  assume that
A1: Z c= dom (ln*(exp_R-f)) and
A2: for x st x in Z holds f.x=1 & (exp_R-f).x>0;
A3: for x st x in Z holds f.x=0*x+1 by A2;
  for y being object st y in Z holds y in dom (exp_R-f) by A1,FUNCT_1:11;
  then
A4: Z c= dom (exp_R-f) by TARSKI:def 3;
  then Z c= dom exp_R /\ dom f by VALUED_1:12;
  then
A5: Z c= dom f by XBOOLE_1:18;
  then
A6: f is_differentiable_on Z by A3,FDIFF_1:23;
A7: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
  then
A8: exp_R-f is_differentiable_on Z by A4,A6,FDIFF_1:19;
A9: for x st x in Z holds ((exp_R-f)`|Z).x =exp_R.x
  proof
    let x;
    assume
A10: x in Z;
    hence ((exp_R-f)`|Z).x = diff(exp_R,x) - diff(f,x) by A4,A6,A7,FDIFF_1:19
      .= exp_R.x- diff(f,x) by SIN_COS:65
      .= exp_R.x-(f`|Z).x by A6,A10,FDIFF_1:def 7
      .= exp_R.x-0 by A5,A3,A10,FDIFF_1:23
      .= exp_R.x;
  end;
A11: for x st x in Z holds ln*(exp_R-f) is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then exp_R-f is_differentiable_in x & (exp_R-f).x >0 by A2,A8,FDIFF_1:9;
    hence thesis by TAYLOR_1:20;
  end;
  then
A12: ln*(exp_R-f) is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((ln*(exp_R-f))`|Z).x = exp_R.x/(exp_R.x-1)
  proof
    let x;
    assume
A13: x in Z;
    then
A14: (exp_R-f).x=exp_R.x - f.x by A4,VALUED_1:13
      .=exp_R.x -1 by A2,A13;
    (exp_R-f) is_differentiable_in x & (exp_R-f).x >0 by A2,A8,A13,FDIFF_1:9;
    then diff(ln*(exp_R-f),x) =diff((exp_R-f),x)/((exp_R-f).x) by TAYLOR_1:20
      .=((exp_R-f)`|Z).x/((exp_R-f).x) by A8,A13,FDIFF_1:def 7
      .=exp_R.x/(exp_R.x -1) by A9,A13,A14;
    hence thesis by A12,A13,FDIFF_1:def 7;
  end;
  hence thesis by A1,A11,FDIFF_1:9;
end;
