 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/(f1+#Z 2)
 & Z c= ]. -1,1 .[ & Z c= dom (( #Z 2)*(arctan))
 & Z = dom f & f|A is continuous implies
 integral(f,A)=((1/2)(#)(( #Z 2)*(arctan))).(upper_bound A)
              -((1/2)(#)(( #Z 2)*(arctan))).(lower_bound A)
proof
  assume
A1:A c= Z & (for x st x in Z holds f1.x=1) & f=arctan/(f1+#Z 2)
   & Z c= ]. -1,1 .[ & Z c= dom (( #Z 2)*(arctan))
   & Z = dom f & f|A is continuous;then
A2:Z c= dom((1/2)(#)(( #Z 2)*(arctan))) by VALUED_1:def 5;
A3:f is_integrable_on A & f|A is bounded by A1,INTEGRA5:10,11;
A4:(1/2)(#)(( #Z 2)*arctan) is_differentiable_on Z by A1,A2,SIN_COS9:93;
   Z c= dom (arctan) /\ (dom (f1+#Z 2) \ (f1+#Z 2)"{0})
   by A1,RFUNCT_1:def 1;then
Z c= dom (f1+#Z 2) \ (f1+#Z 2)"{0} by XBOOLE_1:18;
then A5:Z c= dom ((f1+#Z 2)^) by RFUNCT_1:def 2;
   dom ((f1+#Z 2)^) c= dom (f1+#Z 2) by RFUNCT_1:1;then
A6:Z c= dom (f1+#Z 2) by A5;
A7:for x st x in Z holds f.x=arctan.x / (1+x^2)
   proof
   let x;
   assume
A8:x in Z;then
   (arctan/(f1+#Z 2)).x =arctan.x/(f1+#Z 2).x by A1,RFUNCT_1:def 1
   .=arctan.x/(f1.x+(( #Z 2).x)) by A6,A8,VALUED_1:def 1
   .=arctan.x/(f1.x+(x #Z 2)) by TAYLOR_1:def 1
   .=arctan.x/(1+(x #Z 2)) by A1,A8
   .=arctan.x / (1+x^2) by FDIFF_7:1;
    hence thesis by A1;
    end;
A9:for x being Element of REAL st x in dom(((1/2)(#)(( #Z 2)*(arctan)))`|Z)
    holds
   (((1/2)(#)(( #Z 2)*arctan))`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom(((1/2)(#)(( #Z 2)*arctan))`|Z);then
A10:x in Z by A4,FDIFF_1:def 7;then
  (((1/2)(#)(( #Z 2)*arctan))`|Z).x=arctan.x / (1+x^2) by A1,A2,SIN_COS9:93
  .=f.x by A7,A10;
  hence thesis;
  end;
  dom(((1/2)(#)(( #Z 2)*arctan))`|Z)=dom f by A1,A4,FDIFF_1:def 7;
  then(((1/2)(#)(( #Z 2)*arctan))`|Z)= f by A9,PARTFUN1:5;
  hence thesis by A1,A2,A3,INTEGRA5:13,SIN_COS9:93;
end;
