 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 Th12:
  Z c= dom (cot(#)sec) implies (-cot(#)sec) is_differentiable_on Z &
  for x st x in Z holds
  ((-cot(#)sec)`|Z).x = 1/(sin.x)^2/cos.x-cot.x*sin.x/(cos.x)^2
proof
   assume
A1:Z c= dom (cot(#)sec);
then A2:Z c= dom (-cot(#)sec) by VALUED_1:8;
A3:cot(#)sec is_differentiable_on Z by A1,FDIFF_9:43;
then A4:(-1)(#)(cot(#)sec) is_differentiable_on Z by A2,FDIFF_1:20;
 for x st x in Z holds
   ((-cot(#)sec)`|Z).x = 1/(sin.x)^2/cos.x-cot.x*sin.x/(cos.x)^2
   proof
     let x;
     assume
A5:  x in Z;
     ((-cot(#)sec)`|Z).x = ((-1)(#)((cot(#)sec)`|Z)).x by A3,FDIFF_2:19
       .=(-1)*(((cot(#)sec)`|Z).x) by VALUED_1:6
       .=(-1)*(-1/(sin.x)^2/cos.x+cot.x*sin.x/(cos.x)^2) by A1,A5,FDIFF_9:43
       .=1/(sin.x)^2/cos.x-cot.x*sin.x/(cos.x)^2;
     hence thesis;
   end;
   hence thesis by A4;
end;
