 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 & Z c= ]. -1,1 .[ & f=exp_R*arccot/(f1+#Z 2)
 & (for x st x in Z holds f1.x=1) & Z = dom f & f|A is continuous
 implies integral(f,A)=(-exp_R*arccot).(upper_bound A)
                      -(-exp_R*arccot).(lower_bound A)
proof
   assume
A1:A c= Z & Z c= ]. -1,1 .[ & f=exp_R*arccot/(f1+#Z 2)
   & (for x st x in Z holds f1.x=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 (exp_R*arccot) /\ (dom (f1+#Z 2) \ (f1+#Z 2)"{0})
   by A1,RFUNCT_1:def 1;then
A3:Z c= dom (exp_R*arccot) & Z c= dom (f1+#Z 2) \ (f1+#Z 2)"{0} by XBOOLE_1:18;
then A4: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
A5:Z c= dom (f1+#Z 2) by A4;
A6:-exp_R*arccot is_differentiable_on Z by A1,A3,Th50;
A7:for x st x in Z holds f.x=exp_R.(arccot.x)/(1+x^2)
  proof
  let x;
  assume
A8:x in Z;
  then (exp_R*arccot/(f1+#Z 2)).x
  =(exp_R*arccot).x/(f1+#Z 2).x by A1,RFUNCT_1:def 1
 .=exp_R.(arccot.x)/(f1+#Z 2).x by A3,A8,FUNCT_1:12
 .=exp_R.(arccot.x)/(f1.x+( #Z 2).x) by A5,A8,VALUED_1:def 1
 .=exp_R.(arccot.x)/(1+( #Z 2).x) by A1,A8
 .=exp_R.(arccot.x)/(1+(x #Z 2)) by TAYLOR_1:def 1
 .=exp_R.(arccot.x)/(1+x^2) by FDIFF_7:1;
  hence thesis by A1;
  end;
A9:for x being Element of REAL
   st x in dom ((-exp_R*arccot)`|Z) holds ((-exp_R*arccot)`|Z).x=f.x
   proof
     let x be Element of REAL;
     assume x in dom ((-exp_R*arccot)`|Z);then
A10:  x in Z by A6,FDIFF_1:def 7;then
     ((-exp_R*arccot)`|Z).x=exp_R.(arccot.x)/(1+x^2) by A1,A3,Th50
                       .=f.x by A7,A10;
     hence thesis;
   end;
   dom ((-exp_R*arccot)`|Z)=dom f by A1,A6,FDIFF_1:def 7;
   then ((-exp_R*arccot)`|Z)= f by A9,PARTFUN1:5;
   hence thesis by A1,A2,A3,Th50,INTEGRA5:13;
end;
