 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
 A c= Z & f=-exp_R/(sin*exp_R)^2
 & Z c= dom (cot*exp_R) & Z = dom f & f|A is continuous
 implies integral(f,A)=(cot*exp_R).(upper_bound A)-(cot*exp_R).(lower_bound A)
proof
  assume
A1:A c= Z & f=-exp_R/(sin*exp_R)^2
   & Z c= dom (cot*exp_R) & Z = dom f & f|A is continuous;then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:cot*exp_R is_differentiable_on Z by A1,FDIFF_8:13;
A4:Z = dom (exp_R/(sin*exp_R)^2) by A1,VALUED_1:8;
   then Z c= dom (exp_R) /\ (dom ((sin*exp_R)^2) \ ((sin*exp_R)^2)"{0})
   by RFUNCT_1:def 1;then
   Z c= dom ((sin*exp_R)^2) \ ((sin*exp_R)^2)"{0} by XBOOLE_1:18;
   then
A5:Z c= dom (((sin*exp_R)^2)^) by RFUNCT_1:def 2;
   dom (((sin*exp_R)^2)^) c= dom ((sin*exp_R)^2) by RFUNCT_1:1;then
Z c= dom ((sin*exp_R)^2) by A5;
then A6:Z c= dom (sin*exp_R) by VALUED_1:11;
A7:for x st x in Z holds f.x=-exp_R.x/(sin.(exp_R.x))^2
   proof
   let x;
   assume
A8:x in Z;
   (-exp_R/(sin*exp_R)^2).x =-(exp_R/(sin*exp_R)^2).x by VALUED_1:8
  .=-exp_R.x/((sin*exp_R)^2).x by A4,A8,RFUNCT_1:def 1
  .=-exp_R.x/((sin*exp_R).x)^2 by VALUED_1:11
  .=-exp_R.x/(sin.(exp_R.x))^2 by A6,A8,FUNCT_1:12;
   hence thesis by A1;
   end;
A9:for x being Element of REAL
    st x in dom ((cot*exp_R)`|Z) holds ((cot*exp_R)`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((cot*exp_R)`|Z);then
A10:x in Z by A3,FDIFF_1:def 7;then
  ((cot*exp_R)`|Z).x=-exp_R.x/(sin.(exp_R.x))^2 by A1,FDIFF_8:13
  .=f.x by A7,A10;
  hence thesis;
  end;
  dom ((cot*exp_R)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
  then ((cot*exp_R)`|Z)= f by A9,PARTFUN1:5;
  hence thesis by A1,A2,FDIFF_8:13,INTEGRA5:13;
end;
