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 (exp_R(#)cosec) implies (exp_R(#)cosec) is_differentiable_on
Z & for x st x in Z holds ((exp_R(#)cosec)`|Z).x = exp_R.x/sin.x-exp_R.x*cos.x/
  (sin.x)^2
proof
A1: for x st x in Z holds exp_R is_differentiable_in x by SIN_COS:65;
  assume
A2: Z c= dom (exp_R(#)cosec);
  then
A3: Z c= dom (exp_R) /\ dom cosec by VALUED_1:def 4;
  then
A4: Z c= dom cosec by XBOOLE_1:18;
  Z c= dom (exp_R) by A3,XBOOLE_1:18;
  then
A5: exp_R is_differentiable_on Z by A1,FDIFF_1:9;
A6: for x st x in Z holds cosec is_differentiable_in x & diff(cosec, x)=-cos
  .x/(sin.x)^2
  proof
    let x;
    assume x in Z;
    then sin.x<>0 by A4,RFUNCT_1:3;
    hence thesis by Th2;
  end;
  then for x st x in Z holds cosec is_differentiable_in x;
  then
A7: cosec is_differentiable_on Z by A4,FDIFF_1:9;
  for x st x in Z holds ((exp_R(#)cosec)`|Z).x = exp_R.x/sin.x-exp_R.x*
  cos.x/(sin.x)^2
  proof
    let x;
    assume
A8: x in Z;
    then ((exp_R(#)cosec)`|Z).x = (cosec.x)*diff(exp_R,x)+(exp_R.x)*diff(
    cosec,x) by A2,A5,A7,FDIFF_1:21
      .=(cosec.x)*exp_R.x+(exp_R.x)*diff(cosec,x) by SIN_COS:65
      .=(cosec.x)*exp_R.x+(exp_R.x)*(-cos.x/(sin.x)^2) by A6,A8
      .=exp_R.x/sin.x+(exp_R.x*(-cos.x))/(sin.x)^2 by A4,A8,RFUNCT_1:def 2;
    hence thesis;
  end;
  hence thesis by A2,A5,A7,FDIFF_1:21;
end;
