 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 f.x=1/(x^2*(cos.(1/x))^2))
 & Z c= dom (tan*((id Z)^)) & Z = dom f & f|A is continuous
 implies integral(f,A)=(-tan*((id Z)^)).(upper_bound A) -
 (-tan*((id Z)^)).(lower_bound A)
proof
  assume
A1:A c= Z & (for x st x in Z holds f.x=1/(x^2*(cos.(1/x))^2))
 & Z c= dom (tan*((id Z)^)) & Z = dom f & f|A is continuous; then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:(-tan*((id Z)^)) is_differentiable_on Z by A1,Th34;
A4:for x being Element of REAL
    st x in dom ((-tan*((id Z)^))`|Z) holds ((-tan*((id Z)^))`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((-tan*((id Z)^))`|Z);then
A5:x in Z by A3,FDIFF_1:def 7;then
  ((-tan*((id Z)^))`|Z).x=1/(x^2*(cos.(1/x))^2) by A1,Th34
  .=f.x by A1,A5;
  hence thesis;
  end;
  dom ((-tan*((id Z)^))`|Z)=dom f by A1,A3,FDIFF_1:def 7;
  then ((-tan*((id Z)^))`|Z)= f by A4,PARTFUN1:5;
  hence thesis by A1,A2,A3,INTEGRA5:13;
end;
