 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;
A3:Z = dom (exp_R*arccot/(f1+#Z 2)) by A1,VALUED_1:8;
   then Z = dom (exp_R*arccot) /\ (dom (f1+#Z 2) \ (f1+#Z 2)"{0})
   by RFUNCT_1:def 1;then
A4:Z c= dom (exp_R*arccot) & 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:exp_R*arccot is_differentiable_on Z by A1,A4,SIN_COS9:120;
A8:for x st x in Z holds f.x=-exp_R.(arccot.x)/(1+x^2)
  proof
  let x;
  assume
A9:x in Z;
   (-exp_R*arccot/(f1+#Z 2)).x =-(exp_R*arccot/(f1+#Z 2)).x by VALUED_1:8
 .=-(exp_R*arccot).x/(f1+#Z 2).x by A9,A3,RFUNCT_1:def 1
 .=-exp_R.(arccot.x)/(f1+#Z 2).x by A4,A9,FUNCT_1:12
 .=-exp_R.(arccot.x)/(f1.x+( #Z 2).x) by A6,A9,VALUED_1:def 1
 .=-exp_R.(arccot.x)/(1+( #Z 2).x) by A1,A9
 .=-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;
A10: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
A11:  x in Z by A7,FDIFF_1:def 7;then
     ((exp_R*arccot)`|Z).x=-exp_R.(arccot.x)/(1+x^2) by A1,A4,SIN_COS9:120
                       .=f.x by A11,A8;
     hence thesis;
   end;
  dom ((exp_R*arccot)`|Z)=dom f by A1,A7,FDIFF_1:def 7;
  then ((exp_R*arccot)`|Z)= f by A10,PARTFUN1:5;
  hence thesis by A1,A2,A4,INTEGRA5:13,SIN_COS9:120;
end;
