 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=cos/sin/(id Z)-ln/sin^2
 & 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=cos/sin/(id Z)-ln/sin^2
& 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_8:33;
   Z = dom (cos/sin/(id Z)) /\ dom (ln/sin^2) by A1,VALUED_1:12;
   then
A4:Z c= dom (cos/sin/(id Z)) & Z c= dom (ln/sin^2) by XBOOLE_1:18;
   dom (cos/sin/(id Z)) c= dom (cos/sin) /\ (dom (id Z) \ (id Z)"{0})
   by RFUNCT_1:def 1;then
   dom (cos/sin/(id Z)) c= dom (cos/sin) by XBOOLE_1:18;then
A5:Z c= dom (cos/sin) by A4;
A6:for x st x in Z holds f.x=cos.x/sin.x/x-ln.x/(sin.x)^2
   proof
   let x;
   assume
A7:x in Z;then
   (cos/sin/(id Z)-ln/sin^2).x=(cos/sin/(id Z)).x-(ln/sin^2).x
    by A1,VALUED_1:13
   .=(cos/sin).x/(id Z).x-(ln/sin^2).x by A7,A4,RFUNCT_1:def 1
   .=cos.x/sin.x/(id Z).x-(ln/sin^2).x by A5,A7,RFUNCT_1:def 1
   .=cos.x/sin.x/x-(ln/sin^2).x by A7,FUNCT_1:18
   .=cos.x/sin.x/x-ln.x/(sin^2).x by A7,A4,RFUNCT_1:def 1
   .=cos.x/sin.x/x-ln.x/(sin.x)^2 by VALUED_1:11;
    hence thesis by A1;
    end;
A8: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
A9:x in Z by A3,FDIFF_1:def 7;then
  ((ln(#)cot)`|Z).x=cos.x/sin.x/x-ln.x/(sin.x)^2 by A1,FDIFF_8:33
  .=f.x by A9,A6;
  hence thesis;
  end;
  dom ((ln(#)cot)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
  then ((ln(#)cot)`|Z)= f by A8,PARTFUN1:5;
  hence thesis by A1,A2,FDIFF_8:33,INTEGRA5:13;
end;
