 reserve a,x for Real;
 reserve n for Nat;
 reserve A for non empty closed_interval Subset of REAL;
 reserve f,f1 for PartFunc of REAL,REAL;
 reserve Z for open Subset of REAL;

theorem Th9:
  Z c= dom (cosec*sin) implies -cosec*sin is_differentiable_on Z &
  for x st x in Z holds ((-cosec*sin)`|Z).x
  = cos.x*cos.(sin.x)/(sin.(sin.x))^2
proof
  assume
A1:Z c= dom (cosec*sin);
then A2:Z c= dom (-cosec*sin) by VALUED_1:8;
A3:cosec*sin is_differentiable_on Z by A1,FDIFF_9:36;
then A4:(-1)(#)(cosec*sin) is_differentiable_on Z by A2,FDIFF_1:20;
 for x st x in Z holds ((-cosec*sin)`|Z).x = cos.x*cos.(sin.x)/(sin.(sin.x))^2
    proof
       let x;
       assume
A5:x in Z;
  ((-cosec*sin)`|Z).x=((-1)(#)((cosec*sin)`|Z)).x by A3,FDIFF_2:19
   .=(-1)*(((cosec*sin)`|Z).x) by VALUED_1:6
   .=(-1)*(-cos.x*cos.(sin.x)/(sin.(sin.x))^2) by A1,A5,FDIFF_9:36
   .=cos.x*cos.(sin.x)/(sin.(sin.x))^2;
     hence thesis;
   end;
   hence thesis by A4;
end;
