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 (exp_R*sin) implies exp_R*sin is_differentiable_on Z & for x
  st x in Z holds ((exp_R*sin)`|Z).x = exp_R.(sin.x) * cos.x
proof
A1: for x st x in Z holds exp_R*sin is_differentiable_in x
  proof
    let x;
    assume x in Z;
    sin is_differentiable_in x & exp_R is_differentiable_in sin.x by SIN_COS:64
,65;
    hence thesis by FDIFF_2:13;
  end;
  assume
A2: Z c= dom (exp_R*sin);
  then
A3: exp_R*sin is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((exp_R*sin)`|Z).x = exp_R.(sin.x) * cos.x
  proof
    let x;
    sin is_differentiable_in x & exp_R is_differentiable_in sin.x by SIN_COS:64
,65;
    then
A4: diff(exp_R*sin,x) = diff(exp_R,sin.x)*diff(sin,x) by FDIFF_2:13
      .=diff(exp_R,sin.x)*(cos.x ) by SIN_COS:64
      .=exp_R.(sin.x) * cos.x by SIN_COS:65;
    assume x in Z;
    hence thesis by A3,A4,FDIFF_1:def 7;
  end;
  hence thesis by A2,A1,FDIFF_1:9;
end;
