reserve x,a,b,c for Real,
  n for Nat,
  Z for open Subset of REAL,
  f, f1,f2 for PartFunc of REAL,REAL;

theorem
  Z c= dom (cosec*exp_R) implies cosec*exp_R is_differentiable_on Z &
for x st x in Z holds ((cosec*exp_R)`|Z).x = -exp_R.x*cos.(exp_R.x)/(sin.(exp_R
  .x))^2
proof
  assume
A1: Z c= dom (cosec*exp_R);
A2: for x st x in Z holds sin.(exp_R.x)<>0
  proof
    let x;
    assume x in Z;
    then exp_R.x in dom cosec by A1,FUNCT_1:11;
    hence thesis by RFUNCT_1:3;
  end;
A3: for x st x in Z holds cosec*exp_R is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then sin.(exp_R.x)<>0 by A2;
    then exp_R is_differentiable_in x & cosec is_differentiable_in exp_R.x by
Th2,SIN_COS:65;
    hence thesis by FDIFF_2:13;
  end;
  then
A4: cosec*exp_R is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((cosec*exp_R)`|Z).x = -exp_R.x*cos.(exp_R.x)/(
  sin.(exp_R.x))^2
  proof
    let x;
    assume
A5: x in Z;
    then
A6: sin.(exp_R.x)<>0 by A2;
    then exp_R is_differentiable_in x & cosec is_differentiable_in exp_R.x by
Th2,SIN_COS:65;
    then diff(cosec*exp_R,x) = diff(cosec, exp_R.x)*diff(exp_R,x) by FDIFF_2:13
      .=(-cos.(exp_R.x)/(sin.(exp_R.x))^2) * diff(exp_R,x) by A6,Th2
      .=exp_R.x*(-cos.(exp_R.x)/(sin.(exp_R.x))^2) by SIN_COS:65;
    hence thesis by A4,A5,FDIFF_1:def 7;
  end;
  hence thesis by A1,A3,FDIFF_1:9;
end;
