 reserve a,b,x,r for Real;
 reserve y for set;
 reserve n for Element of NAT;
 reserve A for non empty closed_interval Subset of REAL;
 reserve f,g,f1,f2,g1,g2 for PartFunc of REAL,REAL;
 reserve Z for open Subset of REAL;

theorem
 A c= Z & (for x st x in Z holds x>0)
 & Z = dom f & f=(cos*ln)(#)((id Z)^) implies
 integral(f,A)=(sin*ln).(upper_bound A)-(sin*ln).(lower_bound A)
 proof
   assume
A1:A c= Z & (for x st x in Z holds x>0) & Z = dom f & f=(cos*ln)(#)((id Z)^);
then A2:Z = dom ((cos*ln)/(id Z)) by RFUNCT_1:31;
Z = dom (cos*ln) /\ dom ((id Z)^) by A1,VALUED_1:def 4;
then A3:Z c= dom (cos*ln) by XBOOLE_1:18;
for y being object st y in Z holds y in dom (sin*ln)
   proof
   let y be object;
   assume y in Z;then
   y in dom ln & ln.y in dom sin by A3,FUNCT_1:11,SIN_COS:24;
   hence thesis by FUNCT_1:11;
   end;
then A4:Z c= dom (sin*ln);
A5:cos*ln is_differentiable_on Z by A3,A1,FDIFF_7:33;
not 0 in Z by A1;
then (id Z)^ is_differentiable_on Z by FDIFF_5:4;
   then f|Z is continuous by A1,A5,FDIFF_1:21,25;then
f|A is continuous by A1,FCONT_1:16;
then A6:f is_integrable_on A & f|A is bounded by A1,INTEGRA5:10,11;
A7:sin*ln is_differentiable_on Z by A4,A1,FDIFF_7:32;
A8:for x st x in Z holds f.x=cos.(ln.x)/x
   proof
   let x;
   assume
A9:x in Z;
   ((cos*ln)(#)((id Z)^)).x=((cos*ln)/(id Z)).x by RFUNCT_1:31
                          .=(cos*ln).x*((id Z).x)" by A2,A9,RFUNCT_1:def 1
                          .=(cos*ln).x/x by A9,FUNCT_1:18
                          .=cos.(ln.x)/x by A3,A9,FUNCT_1:12;
   hence thesis by A1;
   end;
A10:for x being Element of REAL st x in dom((sin*ln)`|Z)
  holds ((sin*ln)`|Z).x=f.x
   proof
     let x be Element of REAL;
     assume x in dom((sin*ln)`|Z);then
A11: x in Z by A7,FDIFF_1:def 7; then
  ((sin*ln)`|Z).x=cos.(ln.x)/x by A4,A1,FDIFF_7:32
   .=f.x by A11,A8;
   hence thesis;
   end;
  dom((sin*ln)`|Z)=dom f by A1,A7,FDIFF_1:def 7;
  then ((sin*ln)`|Z)= f by A10,PARTFUN1:5;
  hence thesis by A6,A4,A1,FDIFF_7:32,INTEGRA5:13;
end;
