reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;
reserve a,b,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve f,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 = a+x & f.x<>0) & dom((-1)(#)(f^))
=Z & dom((-1)(#)(f^))=dom f2 & (for x st x in Z holds f2.x = 1/(a+x)^2 ) & f2|A
is continuous implies integral(f2,A) = ((-1)(#)(f^)).(upper_bound A)-((-1)(#)(f
  ^)).(lower_bound A)
proof
  assume that
A1: A c= Z and
A2: for x st x in Z holds f.x = a+x & f.x<>0 and
A3: dom((-1)(#)(f^))=Z and
A4: dom((-1)(#)(f^))=dom f2 and
A5: for x st x in Z holds f2.x = 1/(a+x)^2 and
A6: f2|A is continuous;
A7: f2 is_integrable_on A by A1,A3,A4,A6,INTEGRA5:11;
A8: ((-1)(#)(f^)) is_differentiable_on Z by A2,A3,FDIFF_4:15;
A9: for x being Element of REAL
st x in dom (((-1)(#)(f^))`|Z) holds (((-1)(#)(f^))`|Z).x = f2.x
  proof
    let x be Element of REAL;
    assume x in dom (((-1)(#)(f^))`|Z);
    then
A10: x in Z by A8,FDIFF_1:def 7;
    then (((-1)(#)(f^))`|Z).x = 1/(a+x)^2 by A2,A3,FDIFF_4:15
      .= f2.x by A5,A10;
    hence thesis;
  end;
  dom (((-1)(#)(f^))`|Z) = dom f2 by A3,A4,A8,FDIFF_1:def 7;
  then (((-1)(#)(f^))`|Z) = f2 by A9,PARTFUN1:5;
  hence thesis by A1,A3,A4,A6,A7,A8,INTEGRA5:10,13;
end;
