 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*cot/sin^2 & Z = dom f & f|A is continuous
 implies integral(f,A)=(exp_R*cot).(upper_bound A)-(exp_R*cot).(lower_bound A)
proof
  assume
A1:A c= Z & f=-exp_R*cot/sin^2 & Z = dom f & f|A is continuous;then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:Z = dom (exp_R*cot/sin^2) by A1,VALUED_1:8;
   then Z c= dom (exp_R*cot) /\ (dom (sin^2) \ (sin^2)"{0})
   by RFUNCT_1:def 1;then
A4:Z c= dom (exp_R*cot) by XBOOLE_1:18;
then A5:exp_R*cot is_differentiable_on Z by FDIFF_8:17;
A6:for x st x in Z holds f.x=-exp_R.(cot.x)/(sin.x)^2
   proof
    let x;
    assume
A7:x in Z;
   (-exp_R*cot/sin^2).x =-(exp_R*cot/sin^2).x by VALUED_1:8
  .=-(exp_R*cot).x/(sin^2).x by A7,A3,RFUNCT_1:def 1
  .=-exp_R.(cot.x)/(sin^2).x by A4,A7,FUNCT_1:12
  .=-exp_R.(cot.x)/(sin.x)^2 by VALUED_1:11;
   hence thesis by A1;
   end;
A8:for x being Element of REAL
     st x in dom ((exp_R*cot)`|Z) holds ((exp_R*cot)`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((exp_R*cot)`|Z);then
A9:x in Z by A5,FDIFF_1:def 7;then
  ((exp_R*cot)`|Z).x=-exp_R.(cot.x)/(sin.x)^2 by A4,FDIFF_8:17
  .=f.x by A6,A9;
  hence thesis;
  end;
  dom ((exp_R*cot)`|Z)=dom f by A1,A5,FDIFF_1:def 7;
  then ((exp_R*cot)`|Z)= f by A8,PARTFUN1:5;
  hence thesis by A1,A2,A4,FDIFF_8:17,INTEGRA5:13;
end;
