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=f1+2(#)sin & (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 =cos.x/(1+2*sin.x)
proof
  assume that
A1: Z c= dom ((1/2)(#)(ln*f)) and
A2: f=f1+2(#)sin 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 (f1+2(#)sin) by A2,TARSKI:def 3;
A6: for x st x in Z holds f1.x=1 by A3;
  then
A7: f is_differentiable_on Z by A2,A5,Lm6;
  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,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 f1 /\ dom (2(#)sin) by A5,VALUED_1:def 1;
  then
A9: Z c= dom (2(#)sin) by XBOOLE_1:18;
  for x st x in Z holds (((1/2)(#)(ln*f))`|Z).x =cos.x/(1+2*sin.x)
  proof
    let x;
    assume
A10: x in Z;
    then
A11: f.x=f1.x+(2(#)sin).x by A2,A5,VALUED_1:def 1
      .=1+(2(#)sin).x by A3,A10
      .=1+2*sin.x by A9,A10,VALUED_1:def 5;
A12: f is_differentiable_in x & f.x>0 by A3,A7,A10,FDIFF_1:9;
    (((1/2)(#)(ln*f))`|Z).x =(1/2)*diff((ln*f),x) by A1,A8,A10,FDIFF_1:20
      .=(1/2)*(diff(f,x)/(f.x)) by A12,TAYLOR_1:20
      .=(1/2)*((f`|Z).x/(f.x)) by A7,A10,FDIFF_1:def 7
      .=(1/2)*((2*cos.x)/(f.x)) by A2,A6,A5,A10,Lm6
      .=(1/2)*(2*cos.x)/(f.x) by XCMPLX_1:74
      .=cos.x/(1+2*sin.x) by A11;
    hence thesis;
  end;
  hence thesis by A1,A8,FDIFF_1:20;
end;
