 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 & arctan.x>0) & f=((f1+#Z 2)(#)arctan)^
 & Z c= ]. -1,1 .[ & Z c= dom (ln*(arctan)) & Z = dom f
 & f|A is continuous implies
 integral(f,A)=(ln*(arctan)).(upper_bound A)-(ln* (arctan)).(lower_bound A)
proof
   assume
A1:A c= Z & (for x st x in Z holds f1.x=1 & arctan.x>0)
   & f=((f1+#Z 2)(#)arctan)^
   & Z c= ]. -1,1 .[ & Z c= dom (ln*(arctan)) & Z = dom f
   & f|A is continuous;then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:for x st x in Z holds arctan.x>0 by A1;
then A4:ln*(arctan) is_differentiable_on Z by A1,SIN_COS9:89;
   Z c= dom ((f1+#Z 2)(#)arctan) by A1,RFUNCT_1:1; then
   Z c= dom (f1+#Z 2) /\ dom arctan by VALUED_1:def 4;then
A5:Z c= dom (f1+#Z 2) by XBOOLE_1:18;
A6:for x st x in Z holds f.x=1/((1+x^2)*arctan.x)
  proof
  let x;
  assume
A7:x in Z;
    then
(((f1+#Z 2)(#)arctan)^).x =1/((f1+#Z 2)(#)arctan).x by A1,RFUNCT_1:def 2
  .=1/((f1+#Z 2).x*arctan.x) by VALUED_1:5
  .=1/((f1.x+( #Z 2).x)*arctan.x) by A5,A7,VALUED_1:def 1
  .=1/((f1.x+(x #Z 2))*arctan.x) by TAYLOR_1:def 1
  .=1/((1+(x #Z 2))*arctan.x) by A1,A7
  .=1/((1+x^2)*arctan.x) by FDIFF_7:1;
   hence thesis by A1;
   end;
A8:for x being Element of REAL
    st x in dom ((ln*(arctan))`|Z) holds ((ln*(arctan))`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((ln*(arctan))`|Z);then
A9:x in Z by A4,FDIFF_1:def 7;then
  ((ln*(arctan))`|Z).x=1/((1+x^2)*arctan.x) by A1,A3,SIN_COS9:89
  .=f.x by A6,A9;
  hence thesis;
  end;
  dom ((ln*(arctan))`|Z)=dom f by A1,A4,FDIFF_1:def 7;
  then ((ln*(arctan))`|Z)= f by A8,PARTFUN1:5;
  hence thesis by A1,A2,A4,INTEGRA5:13;
end;
