 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: Z c= dom ((id Z)^) by A1,FUNCT_1:101;
A4:not 0 in Z
    proof
      assume A5: 0 in Z;
      dom ((id Z)^) = dom id Z \ (id Z)"{0} by RFUNCT_1:def 2
       .= dom id Z \ {0} by Lm1,A5; then
      not 0 in {0} by A5,A3,XBOOLE_0:def 5;
      hence thesis by TARSKI:def 1;
   end;
   then
A6:(tan*((id Z)^)) is_differentiable_on Z by A1,FDIFF_8:8;
A7: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
A8:x in Z by A6,FDIFF_1:def 7;then
  ((tan*((id Z)^))`|Z).x=-1/(x^2*(cos.(1/x))^2) by A1,A4,FDIFF_8:8
  .=f.x by A1,A8;
  hence thesis;
  end;
  dom ((tan*((id Z)^))`|Z)=dom f by A1,A6,FDIFF_1:def 7;
  then ((tan*((id Z)^))`|Z)= f by A7,PARTFUN1:5;
  hence thesis by A1,A2,A6,INTEGRA5:13;
end;
