reserve y for set,
  x,r,a,b for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2,f3 for PartFunc of REAL,REAL;

theorem
  Z c= dom (sin*ln) & (for x st x in Z holds x>0) implies sin*ln
  is_differentiable_on Z & for x st x in Z holds ((sin*ln)`|Z).x =cos.(ln.x)/x
proof
  assume that
A1: Z c= dom (sin*ln) and
A2: for x st x in Z holds x>0;
A3: for x st x in Z holds sin*ln is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then
A4: ln is_differentiable_in x by A2,TAYLOR_1:18;
    sin is_differentiable_in ln.x by SIN_COS:64;
    hence thesis by A4,FDIFF_2:13;
  end;
  then
A5: sin*ln is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((sin*ln)`|Z).x =cos.(ln.x)/x
  proof
    let x;
A6: sin is_differentiable_in ln.x by SIN_COS:64;
    assume
A7: x in Z;
    then x>0 by A2;
    then
A8: x in right_open_halfline(0) by Lm3;
    ln is_differentiable_in x by A2,A7,TAYLOR_1:18;
    then diff(sin*ln,x) = diff(sin,ln.x)*diff(ln,x) by A6,FDIFF_2:13
      .=cos.(ln.x)*diff(ln,x) by SIN_COS:64
      .=cos.(ln.x)*(1/x) by A8,TAYLOR_1:18
      .=cos.(ln.x)/x by XCMPLX_1:99;
    hence thesis by A5,A7,FDIFF_1:def 7;
  end;
  hence thesis by A1,A3,FDIFF_1:9;
end;
