 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=-((id Z)(#)(sin*ln)^2)^
 & Z c= dom (cot*ln) & Z = dom f & f|A is continuous
 implies integral(f,A)=(cot*ln).(upper_bound A)-(cot*ln).(lower_bound A)
proof
  assume
A1:A c= Z & f=-((id Z)(#)(sin*ln)^2)^
   & Z c= dom (cot*ln) & Z = dom f & f|A is continuous; then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:cot*ln is_differentiable_on Z by A1,FDIFF_8:15;
A4:Z = dom (((id Z)(#)(sin*ln)^2)^) by A1,VALUED_1:8;
then Z c= dom ((id Z)(#)(sin*ln)^2) by RFUNCT_1:1;
   then Z c= dom (id Z) /\ dom ((sin*ln)^2) by VALUED_1:def 4;then
 Z c= dom ((sin*ln)^2) by XBOOLE_1:18;
then A5:Z c= dom (sin*ln) by VALUED_1:11;
A6:for x st x in Z holds f.x=-1/(x*(sin.(ln.x))^2)
   proof
  let x;
  assume
A7:x in Z;
   (-((id Z)(#)(sin*ln)^2)^).x =-(((id Z)(#)(sin*ln)^2)^).x by VALUED_1:8
 .=-1/(((id Z)(#)(sin*ln)^2).x) by A4,A7,RFUNCT_1:def 2
 .=-1/((id Z).x*((sin*ln)^2).x) by VALUED_1:5
 .=-1/(x*((sin*ln)^2).x) by A7,FUNCT_1:18
 .=-1/(x*((sin*ln).x)^2) by VALUED_1:11
 .=-1/(x*(sin.(ln.x))^2) by A5,A7,FUNCT_1:12;
     hence thesis by A1;
     end;
A8:for x being Element of REAL
   st x in dom ((cot*ln)`|Z) holds ((cot*ln)`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((cot*ln)`|Z);then
A9:x in Z by A3,FDIFF_1:def 7;then
  ((cot*ln)`|Z).x= -1/(x*(sin.(ln.x))^2) by A1,FDIFF_8:15
  .=f.x by A6,A9;
  hence thesis;
  end;
  dom ((cot*ln)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
  then ((cot*ln)`|Z)= f by A8,PARTFUN1:5;
  hence thesis by A1,A2,FDIFF_8:15,INTEGRA5:13;
end;
