 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 & (for x st x in Z holds f1.x=1) & f=arctan+(id Z)/(f1+#Z 2)
 & Z c= ]. -1,1 .[ & Z = dom f & f|A is continuous implies
 integral(f,A)=((id Z)(#)(arctan)).(upper_bound A)-
 ((id Z)(#)(arctan)).(lower_bound A)
proof
  assume
A1:A c= Z & (for x st x in Z holds f1.x=1) & f=arctan+(id Z)/(f1+#Z 2)
   & Z c= ]. -1,1 .[ & Z = dom f & f|A is continuous;then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
   Z = dom arctan /\ dom ((id Z)/(f1+#Z 2)) by A1,VALUED_1:def 1;
   then
A3:Z c= dom arctan & Z c= dom ((id Z)/(f1+#Z 2)) by XBOOLE_1:18;
then
Z c= dom (id Z) /\ (dom (f1+#Z 2) \ (f1+#Z 2)"{0}) by RFUNCT_1:def 1;
then A4: Z c= dom (f1+#Z 2) \ (f1+#Z 2)"{0} by XBOOLE_1:18;
A5:(id Z)(#)(arctan) is_differentiable_on Z by A1,SIN_COS9:95;
A6:Z c= dom ((f1+#Z 2)^) by A4,RFUNCT_1:def 2;
   dom ((f1+#Z 2)^) c= dom (f1+#Z 2) by RFUNCT_1:1;then
A7:Z c= dom (f1+#Z 2) by A6;
A8:for x st x in Z holds f.x=arctan.x + x/(1+x^2)
  proof
  let x;
  assume
A9: x in Z;
   then (arctan+(id Z)/(f1+#Z 2)).x
   =arctan.x+((id Z)/(f1+#Z 2)).x by A1,VALUED_1:def 1
  .=arctan.x+(id Z).x/(f1+#Z 2).x by A3,A9,RFUNCT_1:def 1
  .=arctan.x+x/(f1+#Z 2).x by A9,FUNCT_1:18
  .=arctan.x+x/(f1.x+( #Z 2).x) by A7,A9,VALUED_1:def 1
  .=arctan.x+x/(1+( #Z 2).x) by A1,A9
  .=arctan.x+x/(1+(x #Z 2)) by TAYLOR_1:def 1
  .=arctan.x + x/(1+x^2) by FDIFF_7:1;
    hence thesis by A1;
    end;
A10:for x being Element of REAL st x in dom(((id Z)(#)(arctan))`|Z) holds
   (((id Z)(#)(arctan))`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom(((id Z)(#)(arctan))`|Z);then
A11:x in Z by A5,FDIFF_1:def 7;then
  (((id Z)(#)(arctan))`|Z).x=arctan.x + x/(1+x^2) by A1,SIN_COS9:95
  .=f.x by A8,A11;
  hence thesis;
  end;
  dom(((id Z)(#)(arctan))`|Z)=dom f by A1,A5,FDIFF_1:def 7;
  then(((id Z)(#)(arctan))`|Z)= f by A10,PARTFUN1:5;
  hence thesis by A1,A2,INTEGRA5:13,SIN_COS9:95;
end;
