 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 Th4:
  Z c= dom (exp_R*cosec) implies -exp_R*cosec is_differentiable_on Z
  & for x st x in Z holds
  ((-exp_R*cosec)`|Z).x = exp_R.(cosec.x)*cos.x/(sin.x)^2
proof
  assume
A1:Z c= dom (exp_R*cosec);
then A2:Z c= dom (-exp_R*cosec) by VALUED_1:8;
A3:exp_R*cosec is_differentiable_on Z by A1,FDIFF_9:17;
then A4:(-1)(#)(exp_R*cosec) is_differentiable_on Z by A2,FDIFF_1:20;
 for x st x in Z holds ((-exp_R*cosec)`|Z).x = exp_R.(cosec.x)*cos.x/(sin.x)^2
   proof
     let x;
     assume
A5:x in Z;
 ((-exp_R*cosec)`|Z).x=((-1)(#)((exp_R*cosec)`|Z)).x by A3,FDIFF_2:19
                 .=(-1)*(((exp_R*cosec)`|Z).x) by VALUED_1:6
                 .=(-1)*(-exp_R.(cosec.x)*cos.x/(sin.x)^2)
   by A1,A5,FDIFF_9:17
             .=exp_R.(cosec.x)*cos.x/(sin.x)^2;
     hence thesis;
   end;
  hence thesis by A4;
end;
