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 (sin*ln) implies sin*ln is_differentiable_on Z & for x st x
  in Z holds ((sin*ln)`|Z).x =cos.(log(number_e,x))/x
proof
  assume
A1: Z c= dom (sin*ln);
  then for y being object st y in Z holds y in dom ln by FUNCT_1:11;
  then
A2: Z c= dom ln by TARSKI:def 3;
  then
A3: ln is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:18;
A4: for x st x in Z holds sin*ln is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then
A5: ln is_differentiable_in x by A3,FDIFF_1:9;
    sin is_differentiable_in ln.x by SIN_COS:64;
    hence thesis by A5,FDIFF_2:13;
  end;
  then
A6: sin*ln is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((sin*ln)`|Z).x =cos.(log(number_e,x))/x
  proof
    let x;
A7: sin is_differentiable_in ln.x by SIN_COS:64;
    assume
A8: x in Z;
    then
A9: x in right_open_halfline(0) by A1,FUNCT_1:11,TAYLOR_1:18;
    ln is_differentiable_in x by A3,A8,FDIFF_1:9;
    then diff(sin*ln,x) = diff(sin,ln.x)*diff(ln,x) by A7,FDIFF_2:13
      .=cos.(ln.x)*diff(ln,x) by SIN_COS:64
      .=cos.(log(number_e,x))*diff(ln,x) by A9,TAYLOR_1:def 2
      .=cos.(log(number_e,x))*(1/x) by A2,A8,TAYLOR_1:18
      .=cos.(log(number_e,x))/x by XCMPLX_1:99;
    hence thesis by A6,A8,FDIFF_1:def 7;
  end;
  hence thesis by A1,A4,FDIFF_1:9;
end;
