 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 Th14:
  Z c= dom (cos (#) cot) implies (-cos (#) cot) is_differentiable_on Z &
  for x st x in Z holds((-cos (#) cot)`|Z).x = cos.x+cos.x/(sin.x)^2
proof
   assume
A1:Z c= dom (cos (#) cot);
then A2:Z c= dom (-cos (#) cot) by VALUED_1:8;
A3:(cos (#) cot) is_differentiable_on Z by A1,FDIFF_10:11;
then A4:(-1)(#)(cos (#) cot) is_differentiable_on Z by A2,FDIFF_1:20;
 for x st x in Z holds ((-cos (#) cot)`|Z).x = cos.x+cos.x/(sin.x)^2
    proof
      let x;
      assume
A5:   x in Z;
     ((-cos (#) cot)`|Z).x = ((-1)(#)((cos (#) cot)`|Z)).x by A3,FDIFF_2:19
   .=(-1)*(((cos (#) cot)`|Z).x) by VALUED_1:6
   .=(-1)*(-cos.x-cos.x/(sin.x)^2) by A1,A5,FDIFF_10:11
   .=cos.x+cos.x/(sin.x)^2;
     hence thesis;
   end;
   hence thesis by A4;
end;
