 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 Th6:
  Z c= dom (( #Z n)*cosec) & 1<=n
  implies -( #Z n)*cosec is_differentiable_on Z & for x st x in Z holds
  ((-( #Z n)*cosec)`|Z).x = n*cos.x/(sin.x) #Z (n+1)
proof
   assume
A1:Z c= dom (( #Z n)*cosec) & 1<=n;
then A2:Z c= dom (-( #Z n)*cosec) & 1<=n by VALUED_1:8;
A3:( #Z n)*cosec is_differentiable_on Z by A1,FDIFF_9:21;
then A4:(-1)(#)(( #Z n)*cosec) is_differentiable_on Z by A2,FDIFF_1:20;
 for x st x in Z holds ((-( #Z n)*cosec)`|Z).x = n*cos.x/(sin.x) #Z (n+1)
  proof
    let x;
    assume
A5:x in Z;
  ((-( #Z n)*cosec)`|Z).x=((-1)(#)((( #Z n)*cosec)`|Z)).x by A3,FDIFF_2:19
   .=(-1)*(((( #Z n)*cosec)`|Z).x) by VALUED_1:6
   .=(-1)*(-n*cos.x/(sin.x) #Z (n+1)) by A1,A5,FDIFF_9:21
   .=n*cos.x/(sin.x) #Z (n+1);
    hence thesis;
   end;
   hence thesis by A4;
end;
