 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 Th10:
  Z c= dom (sec*cot) implies -sec*cot is_differentiable_on Z &
  for x st x in Z holds ((-sec*cot)`|Z).x
  = sin.(cot.x)/(sin.x)^2/(cos.(cot.x))^2
proof
  assume
A1:Z c= dom (sec*cot);
then A2:Z c= dom (-sec*cot) by VALUED_1:8;
A3:sec*cot is_differentiable_on Z by A1,FDIFF_9:39;
then A4:(-1)(#)(sec*cot) is_differentiable_on Z by A2,FDIFF_1:20;
 for x st x in Z holds
   ((-sec*cot)`|Z).x = sin.(cot.x)/(sin.x)^2/(cos.(cot.x))^2
   proof
     let x;
     assume
A5:  x in Z;
     ((-sec*cot)`|Z).x=((-1)(#)((sec*cot)`|Z)).x by A3,FDIFF_2:19
     .=(-1)*(((sec*cot)`|Z).x) by VALUED_1:6
     .=(-1)*(-sin.(cot.x)/(sin.x)^2/(cos.(cot.x))^2) by A1,A5,FDIFF_9:39
     .=sin.(cot.x)/(sin.x)^2/(cos.(cot.x))^2;
     hence thesis;
   end;
   hence thesis by A4;
end;
