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 (-(cos*ln)) implies -(cos*ln) is_differentiable_on Z & for x
  st x in Z holds ((-(cos*ln))`|Z).x =sin.(log(number_e,x))/x
proof
  assume
A1: Z c= dom (-(cos*ln));
  then
A2: Z c= dom (cos*ln) by VALUED_1:8;
  then for y being object st y in Z holds y in dom ln by FUNCT_1:11;
  then
A3: Z c= dom ln by TARSKI:def 3;
  then
A4: ln is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:18;
  for x st x in Z holds cos*ln is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then
A5: ln is_differentiable_in x by A4,FDIFF_1:9;
    cos is_differentiable_in ln.x by SIN_COS:63;
    hence thesis by A5,FDIFF_2:13;
  end;
  then
A6: cos*ln is_differentiable_on Z by A2,FDIFF_1:9;
A7: for x st x in Z holds ((-(cos*ln))`|Z).x =sin.(log(number_e,x))/x
  proof
    let x;
A8: cos is_differentiable_in ln.x by SIN_COS:63;
    assume
A9: x in Z;
    then
A10: x in right_open_halfline(0) by A2,FUNCT_1:11,TAYLOR_1:18;
A11: ln is_differentiable_in x by A4,A9,FDIFF_1:9;
    ((-(cos*ln))`|Z).x =(-1)*diff((cos*ln),x) by A1,A6,A9,FDIFF_1:20
      .=(-1)*(diff(cos,ln.x)*diff(ln,x)) by A11,A8,FDIFF_2:13
      .=(-1)*((-sin.(ln.x))*diff(ln,x)) by SIN_COS:63
      .=(-1)*(-sin.(ln.x))*diff(ln,x)
      .=(-1)*(-sin.(log(number_e,x)))*diff(ln,x) by A10,TAYLOR_1:def 2
      .=(-1)*(-sin.(log(number_e,x)))*(1/x) by A3,A9,TAYLOR_1:18
      .=sin.(log(number_e,x))/x by XCMPLX_1:99;
    hence thesis;
  end;
  Z c= dom ((-1)(#)(cos*ln)) by A1;
  hence thesis by A6,A7,FDIFF_1:20;
end;
