 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/( #Z 2)-((id Z)(#)(f1+#Z 2))^
 & Z c= dom ((id Z)^(#)arctan) & 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/( #Z 2)-((id Z)(#)(f1+#Z 2))^
   & Z c= dom ((id Z)^(#)arctan) & 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;
A3:(-(id Z)^(#)arctan) is_differentiable_on Z by A1,Th54;
A4:Z=dom (arctan/( #Z 2)) /\ dom (((id Z)(#)(f1+#Z 2))^) by A1,VALUED_1:12;then
A5:Z c= dom (arctan/( #Z 2)) by XBOOLE_1:18;
A6:Z c= dom (((id Z)(#)(f1+#Z 2))^) by A4,XBOOLE_1:18;
   dom (((id Z)(#)(f1+#Z 2))^) c= dom ((id Z)(#)(f1+#Z 2)) by RFUNCT_1:1;then
Z c= dom ((id Z)(#)(f1+#Z 2)) by A6;
   then Z c= dom (id Z) /\ dom (f1+#Z 2) by VALUED_1:def 4;then
A7:Z c= dom (f1+#Z 2) by XBOOLE_1:18;
A8:for x st x in Z holds f.x=arctan.x/(x^2)-1/(x*(1+x^2))
   proof
   let x;
   assume
A9:x in Z;then
A10:x in dom (((id Z)(#)(f1+#Z 2))^) by A6;
  (arctan/( #Z 2)-((id Z)(#)(f1+#Z 2))^).x
 =(arctan/( #Z 2)).x-(((id Z)(#)(f1+#Z 2))^).x by A1,A9,VALUED_1:13
.=(arctan/( #Z 2)).x-1/(((id Z)(#)(f1+#Z 2)).x) by A10,RFUNCT_1:def 2
.=(arctan/( #Z 2)).x-1/((id Z).x*(f1+#Z 2).x) by VALUED_1:5
.=(arctan/( #Z 2)).x-1/((id Z).x*(f1.x+( #Z 2).x)) by A7,A9,VALUED_1:def 1
.=(arctan/( #Z 2)).x-1/(x*(f1.x+( #Z 2).x)) by A9,FUNCT_1:18
.=(arctan/( #Z 2)).x-1/(x*(1+( #Z 2).x)) by A1,A9
.=arctan.x/( #Z 2).x-1/(x*(1+( #Z 2).x)) by A9,A5,RFUNCT_1:def 1
.=arctan.x/(x #Z 2)-1/(x*(1+( #Z 2).x)) by TAYLOR_1:def 1
.=arctan.x/(x #Z 2)-1/(x*(1+( x #Z 2))) by TAYLOR_1:def 1
.=arctan.x/(x^2)-1/(x*(1+( x #Z 2))) by FDIFF_7:1
.=arctan.x/(x^2)-1/(x*(1+x^2)) by FDIFF_7:1;
   hence thesis by A1;
   end;
A11: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
A12:  x in Z by A3,FDIFF_1:def 7;then
     ((-(id Z)^(#)arctan)`|Z).x=arctan.x/(x^2)-1/(x*(1+x^2)) by A1,Th54
       .=f.x by A8,A12;
     hence thesis;
   end;
   dom((-(id Z)^(#)arctan)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
   then((-(id Z)^(#)arctan)`|Z)= f by A11,PARTFUN1:5;
   hence thesis by A1,A2,A3,INTEGRA5:13;
end;
