 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)(#)(cos*ln)^2)^
 & Z c= dom (tan*ln) & Z = dom f & f|A is continuous
 implies integral(f,A)=(tan*ln).(upper_bound A)-(tan*ln).(lower_bound A)
proof
  assume
A1:A c= Z & f=((id Z)(#)(cos*ln)^2)^
   & Z c= dom (tan*ln) & Z = dom f & f|A is continuous; then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:tan*ln is_differentiable_on Z by A1,FDIFF_8:14;
Z c= dom ((id Z)(#)(cos*ln)^2) by A1,RFUNCT_1:1;
   then Z c= dom (id Z) /\ dom ((cos*ln)^2) by VALUED_1:def 4;then
 Z c= dom ((cos*ln)^2) by XBOOLE_1:18;
then A4:Z c= dom (cos*ln) by VALUED_1:11;
A5:for x st x in Z holds f.x=1/(x*(cos.(ln.x))^2)
  proof
  let x;
  assume
A6:x in Z;
   then (((id Z)(#)(cos*ln)^2)^).x
  =1/(((id Z)(#)(cos*ln)^2).x) by A1,RFUNCT_1:def 2
 .=1/((id Z).x*((cos*ln)^2).x) by VALUED_1:5
 .=1/(x*((cos*ln)^2).x) by A6,FUNCT_1:18
 .=1/(x*((cos*ln).x)^2) by VALUED_1:11
 .=1/(x*(cos.(ln.x))^2) by A4,A6,FUNCT_1:12;
     hence thesis by A1;
     end;
A7:for x being Element of REAL
    st x in dom ((tan*ln)`|Z) holds ((tan*ln)`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((tan*ln)`|Z);then
A8:x in Z by A3,FDIFF_1:def 7;then
  ((tan*ln)`|Z).x=1/(x*(cos.(ln.x))^2) by A1,FDIFF_8:14
  .=f.x by A5,A8;
  hence thesis;
  end;
  dom ((tan*ln)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
  then ((tan*ln)`|Z)= f by A7,PARTFUN1:5;
  hence thesis by A1,A2,FDIFF_8:14,INTEGRA5:13;
end;
