 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=cot/(id Z)-ln/sin^2
 & Z c= dom (ln(#)cot) & Z c= dom 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=cot/(id Z)-ln/sin^2
& Z c= dom (ln(#)cot) & Z c= dom 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 (cot/(id Z)) /\ dom(ln/sin^2) by A1,VALUED_1:12;then
A4:Z c= dom (cot/(id Z)) & Z c= dom(ln/sin^2) by XBOOLE_1:18;
A5:for x st x in Z holds f.x=cot.x/x-ln.x/(sin.x)^2
   proof
   let x;
   assume
A6:x in Z;then
   (cot/(id Z)-ln/sin^2).x =(cot/(id Z)).x-(ln/sin^2).x by A1,VALUED_1:13
  .=cot.x/(id Z).x-(ln/sin^2).x by A6,A4,RFUNCT_1:def 1
  .=cot.x/x-(ln/sin^2).x by A6,FUNCT_1:18
  .=cot.x/x-ln.x/(sin^2).x by A6,A4,RFUNCT_1:def 1
  .=cot.x/x-ln.x/(sin.x)^2 by VALUED_1:11;
   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=cot(x)/x-ln.x/(sin.x)^2 by A1,FDIFF_8:33
  .=cot.x/x-ln.x/(sin.x)^2 by A1,A8,FDIFF_8:2,SIN_COS9:16
  .=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_8:33,INTEGRA5:13;
end;
