 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/(id Z)+ln/(f1+( #Z 2))
 & Z c= ]. -1,1 .[ & Z = dom f & f|A is continuous implies
 integral(f,A)=(ln(#)arctan).(upper_bound A)-(ln(#)arctan).(lower_bound A)
proof
   assume
A1:A c= Z & (for x st x in Z holds f1.x=1) & f=arctan/(id Z)+ln/(f1+( #Z 2))
   & Z c= ]. -1,1 .[ & Z = dom f & f|A is continuous; then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
   Z = dom (arctan/(id Z)) /\ dom (ln/(f1+( #Z 2))) by A1,VALUED_1:def 1;
    then
A3:Z c= dom (arctan/(id Z)) & Z c= dom (ln/(f1+( #Z 2))) by XBOOLE_1:18;
   then
Z c= dom (arctan) /\ (dom (id Z) \ (id Z)"{0}) by RFUNCT_1:def 1;then
A4:Z c= dom (arctan) by XBOOLE_1:18;
   Z c= dom ln /\ (dom (f1+( #Z 2)) \ (f1+( #Z 2))"{0}) by A3,RFUNCT_1:def 1;
   then
A5:Z c= dom ln & Z c= (dom (f1+( #Z 2)) \ (f1+( #Z 2))"{0}) by XBOOLE_1:18;
   then Z c= dom (arctan) /\ dom ln by A4,XBOOLE_1:19;then
A6:Z c= dom (ln(#)arctan) by VALUED_1:def 4;
then A7:ln(#)arctan is_differentiable_on Z by A1,SIN_COS9:127;
A8:Z c= dom ((f1+#Z 2)^) by A5,RFUNCT_1:def 2;
   dom ((f1+#Z 2)^) c= dom (f1+#Z 2) by RFUNCT_1:1;then
A9:Z c= dom (f1+#Z 2) by A8;
A10:for x st x in Z holds f.x=arctan.x/x+ln.x/(1+x^2)
    proof
    let x;
    assume
A11:x in Z;
    then (arctan/(id Z)+ln/(f1+( #Z 2))).x
    =(arctan/(id Z)).x+(ln/(f1+( #Z 2))).x by A1,VALUED_1:def 1
   .=arctan.x*((id Z).x)" +(ln/(f1+( #Z 2))).x by A3,A11,RFUNCT_1:def 1
   .=arctan.x*((id Z).x)"+(ln.x*((f1+( #Z 2)).x)") by A3,A11,RFUNCT_1:def 1
   .=arctan.x/x+(ln.x/(f1+( #Z 2)).x) by A11,FUNCT_1:18
   .=arctan.x/x+ln.x/(f1.x+( #Z 2).x) by A9,A11,VALUED_1:def 1
   .=arctan.x/x+ln.x/(1+( #Z 2).x) by A1,A11
   .=arctan.x/x+ln.x/(1+(x #Z 2)) by TAYLOR_1:def 1
   .=arctan.x/x+ln.x/(1+x^2) by FDIFF_7:1;
   hence thesis by A1;
   end;
A12:for x being Element of REAL
    st x in dom((ln(#)arctan)`|Z) holds ((ln(#)arctan)`|Z).x=f.x
   proof
     let x be Element of REAL;
     assume x in dom((ln(#)arctan)`|Z);then
A13:  x in Z by A7,FDIFF_1:def 7;then
     ((ln(#)arctan)`|Z).x=arctan.x/x+ln.x/(1+x^2) by A1,A6,SIN_COS9:127
                       .=f.x by A13,A10;
     hence thesis;
   end;
   dom((ln(#)arctan)`|Z)=dom f by A1,A7,FDIFF_1:def 7;
   then ((ln(#)arctan)`|Z)= f by A12,PARTFUN1:5;
   hence thesis by A1,A2,A6,INTEGRA5:13,SIN_COS9:127;
end;
