 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 exp_R.x < 1 & f1.x=1)
 & Z c= dom (arctan*exp_R)
 & Z = dom f & f=exp_R/(f1+(exp_R)^2)
 implies integral(f,A)=(arctan*exp_R).(upper_bound A)
                      -(arctan*exp_R).(lower_bound A)
proof
   assume
A1: A c= Z & (for x st x in Z holds exp_R.x < 1 & f1.x=1)
    & Z c= dom (arctan*exp_R)
    & Z = dom f & f=exp_R/(f1+(exp_R)^2);
then Z c= dom exp_R /\ (dom (f1+(exp_R)^2) \ (f1+(exp_R)^2)"{0})
   by RFUNCT_1:def 1;
then Z c= dom exp_R & Z c= dom (f1+(exp_R)^2) \ (f1+(exp_R)^2)"{0}
   by XBOOLE_1:18;
then A2:Z c= dom ((f1+(exp_R)^2)^) by RFUNCT_1:def 2;
   dom ((f1+(exp_R)^2)^) c= dom (f1+(exp_R)^2) by RFUNCT_1:1;then
A3:Z c= dom (f1+(exp_R)^2) by A2;
then A4:Z c= dom f1 /\ dom ((exp_R)^2) by VALUED_1:def 1;
then A5:Z c= dom f1 & Z c= dom ((exp_R)^2) by XBOOLE_1:18;
A6:Z c= dom ((exp_R)(#)(exp_R)) by A4,XBOOLE_1:18;
A7:exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
then A8:(exp_R)(#)(exp_R) is_differentiable_on Z by A6,FDIFF_1:21;
for x st x in Z holds f1.x=0*x+1 by A1;
then f1 is_differentiable_on Z by A5,FDIFF_1:23;
then A9:f1+(exp_R)^2 is_differentiable_on Z by A3,A8,FDIFF_1:18;
for x st x in Z holds (f1+(exp_R)^2).x<>0
   proof
   let x;
   assume x in Z;then
   x in dom exp_R /\ (dom (f1+(exp_R)^2) \ (f1+(exp_R)^2)"{0})
     by A1,RFUNCT_1:def 1;then
   x in dom (f1+(exp_R)^2) \ (f1+(exp_R)^2)"{0} by XBOOLE_0:def 4; then
   x in dom ((f1+(exp_R)^2)^) by RFUNCT_1:def 2;
      hence thesis by RFUNCT_1:3;
   end;
then f is_differentiable_on Z by A1,A7,A9,FDIFF_2:21;
then f|Z is continuous by FDIFF_1:25;
then f|A is continuous by A1,FCONT_1:16;
then A10:f is_integrable_on A & f|A is bounded by A1,INTEGRA5:10,11;
A11:for x st x in Z holds exp_R.x < 1 by A1;then
A12:arctan*exp_R is_differentiable_on Z by A1,SIN_COS9:115;
A13:for x st x in Z holds f.x=exp_R.x/(1+(exp_R.x)^2)
   proof
   let x;
   assume
A14:x in Z;
    then (exp_R/(f1+(exp_R)^2)).x=(exp_R.x)*((f1+(exp_R)^2).x)"
                                 by A1,RFUNCT_1:def 1
                           .=(exp_R.x)*(f1.x+((exp_R)^2).x)"
                                by A14,A3,VALUED_1:def 1
                           .=(exp_R.x)*(f1.x+(exp_R.x)^2)"
                                by VALUED_1:11
                           .=exp_R.x/(1+(exp_R.x)^2) by A1,A14;
        hence thesis by A1;
      end;
A15:for x being Element of REAL
     st x in dom ((arctan*exp_R)`|Z) holds ((arctan*exp_R)`|Z).x=f.x
    proof
       let x be Element of REAL;
       assume x in dom ((arctan*exp_R)`|Z);then
A16:x in Z by A12,FDIFF_1:def 7;then
   ((arctan*exp_R)`|Z).x=exp_R.x/(1+(exp_R.x)^2) by A1,A11,SIN_COS9:115
                       .=f.x by A16,A13;
   hence thesis;
   end;
   dom ((arctan*exp_R)`|Z)=dom f by A1,A12,FDIFF_1:def 7;
   then ((arctan*exp_R)`|Z)= f by A15,PARTFUN1:5;
   hence thesis by A1,A10,A12,INTEGRA5:13;
end;
