 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=-(sin(#)cos)^ & Z c= dom (ln*cot) & Z = dom f & f|A is continuous
 implies integral(f,A)=(ln*cot).(upper_bound A)-(ln*cot).(lower_bound A)
proof
   assume
A1:A c= Z & f=-(sin(#)cos)^
   & Z c= dom (ln*cot) & Z = dom f & f|A is continuous;then
A2: f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3: (ln*cot) is_differentiable_on Z by A1,FDIFF_8:19;
A4:Z = dom ((sin(#)cos)^) by A1,VALUED_1:8;
A5:for x st x in Z holds f.x=-1/(sin.x*cos.x)
   proof
   let x;
   assume
A6:x in Z;
   (-(sin(#)cos)^).x=-(((sin(#)cos)^).x) by VALUED_1:8
                   .=-1/((sin(#)cos).x) by A4,A6,RFUNCT_1:def 2
                   .=-1/(sin.x*cos.x) by VALUED_1:5;
    hence thesis by A1;
    end;
A7: for x being Element of REAL
st x in dom ((ln*cot)`|Z) holds ((ln*cot)`|Z).x = f.x
    proof
      let x be Element of REAL;
      assume x in dom ((ln*cot)`|Z);then
A8:   x in Z by A3,FDIFF_1:def 7; then
      ((ln*cot)`|Z).x = -1/(sin.x*cos.x) by A1,FDIFF_8:19
        .=f.x by A5,A8;
      hence thesis;
    end;
    dom ((ln*cot)`|Z) = dom f by A1,A3,FDIFF_1:def 7;then
    ((ln*cot)`|Z)=f by A7,PARTFUN1:5;
    hence thesis by A1,A2,FDIFF_8:19,INTEGRA5:13;
end;
