 reserve a,x for Real;
 reserve n for Element of NAT;
 reserve A for non empty closed_interval Subset of REAL;
 reserve f,h,f1,f2 for PartFunc of REAL,REAL;
 reserve Z for open Subset of REAL;

theorem Th23:
  Z c= dom (exp_R*cot) implies -exp_R*cot is_differentiable_on Z &
  for x st x in Z holds ((-exp_R*cot)`|Z).x = exp_R.(cot.x)/(sin.x)^2
proof
  assume
A1:Z c= dom (exp_R*cot);
then A2:Z c= dom (-exp_R*cot) by VALUED_1:8;
A3:for x st x in Z holds sin.x<>0
  proof
    let x;
    assume x in Z;
    then x in dom (cos/sin) by A1,FUNCT_1:11;
    hence thesis by FDIFF_8:2;
  end;
A4:exp_R*cot is_differentiable_on Z by A1,FDIFF_8:17;
then A5:(-1)(#)(exp_R*cot) is_differentiable_on Z by A2,FDIFF_1:20;
 for x st x in Z holds ((-exp_R*cot)`|Z).x = exp_R.(cot.x)/(sin.x)^2
   proof
     let x;
     assume
A6:x in Z; then
A7:sin.x<>0 by A3; then
A8:cot is_differentiable_in x by FDIFF_7:47;
A9:exp_R is_differentiable_in cot.x by SIN_COS:65;
A10:exp_R*cot is_differentiable_in x by A4,A6,FDIFF_1:9;
   ((-exp_R*cot)`|Z).x=diff(-exp_R*cot,x) by A5,A6,FDIFF_1:def 7
                     .=(-1)*(diff(exp_R*cot,x)) by A10,FDIFF_1:15
                     .=(-1)*(diff(exp_R,cot.x)*diff(cot,x))
   by A8,A9,FDIFF_2:13
                     .=(-1)*(diff(exp_R,cot.x)*(-1/(sin.x)^2))
   by A7,FDIFF_7:47
                     .=(-1)*(exp_R.(cot.x) * (-1/(sin.x)^2)) by SIN_COS:65
                     .=exp_R.(cot.x)/(sin.x)^2;
     hence thesis;
   end;
   hence thesis by A2,A4,FDIFF_1:20;
end;
