 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=-(n(#)((( #Z (n-1))*arccot)/(f1+#Z 2)))
 & Z c= ]. -1,1 .[ & Z c= dom (( #Z n)*arccot) & Z = dom f
 & f|A is continuous implies
 integral(f,A)=(( #Z n)*arccot).(upper_bound A)-
 (( #Z n)*arccot).(lower_bound A)
proof
  assume
A1:A c= Z & (for x st x in Z holds f1.x=1)
   & f=-(n(#)((( #Z (n-1))*arccot)/(f1+#Z 2)))
   & Z c= ]. -1,1 .[ & Z c= dom (( #Z n)*arccot) & Z = dom f
   & f|A is continuous;then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:( #Z n)*(arccot) is_differentiable_on Z by A1,SIN_COS9:92;
Z = dom (n(#)((( #Z (n-1))*arccot)/(f1+#Z 2))) by A1,VALUED_1:8;
then A4:Z = dom ((( #Z (n-1))*arccot)/(f1+#Z 2)) by VALUED_1:def 5;
   then Z c= dom (( #Z (n-1))*arccot) /\ (dom (f1+#Z 2) \ (f1+#Z 2)"{0})
   by RFUNCT_1:def 1;then
A5:Z c= dom (( #Z (n-1))*arccot) & Z c= dom (f1+#Z 2) \ (f1+#Z 2)"{0}
   by XBOOLE_1:18;
then A6: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
A7:Z c= dom (f1+#Z 2) by A6;
A8:for x st x in Z holds f.x=-n*(arccot.x) #Z (n-1) / (1+x^2)
   proof
   let x;
   assume
A9: x in Z;
   (-(n(#)((( #Z (n-1))*arccot)/(f1+#Z 2)))).x
   =-(n(#)((( #Z (n-1))*arccot)/(f1+#Z 2))).x by VALUED_1:8
  .=-n*(((( #Z (n-1))*arccot)/(f1+#Z 2))).x by VALUED_1:6
  .=-n*((( #Z (n-1))*arccot).x*((f1+#Z 2).x)") by A4,A9,RFUNCT_1:def 1
  .=-n*(( #Z (n-1))*arccot).x/(f1+#Z 2).x
  .=-n*(( #Z (n-1)).(arccot.x))/(f1+#Z 2).x by A5,A9,FUNCT_1:12
  .=-n*(arccot.x) #Z (n-1)/(f1+#Z 2).x by TAYLOR_1:def 1
  .=-n*(arccot.x) #Z (n-1)/(f1.x+( #Z 2).x) by A7,A9,VALUED_1:def 1
  .=-n*(arccot.x) #Z (n-1)/(1+( #Z 2).x) by A1,A9
  .=-n*(arccot.x) #Z (n-1)/(1+(x #Z 2)) by TAYLOR_1:def 1
  .=-n*(arccot.x) #Z (n-1) / (1+x^2) by FDIFF_7:1;
   hence thesis by A1;
   end;
A10:for x being Element of REAL st x in dom((( #Z n)*(arccot))`|Z) holds
   ((( #Z n)*(arccot))`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom((( #Z n)*(arccot))`|Z);then
A11:x in Z by A3,FDIFF_1:def 7;then
  ((( #Z n)*arccot)`|Z).x=-n*(arccot.x) #Z (n-1) / (1+x^2) by A1,SIN_COS9:92
  .=f.x by A11,A8;
  hence thesis;
  end;
  dom((( #Z n)*arccot)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
  then((( #Z n)*arccot)`|Z)= f by A10,PARTFUN1:5;
  hence thesis by A1,A2,INTEGRA5:13,SIN_COS9:92;
end;
