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 x > 0) & dom (ln*( #Z n)) = Z & dom (
ln*( #Z n)) = dom f2 & (for x st x in Z holds f2.x = n/x ) & f2|A is continuous
implies integral(f2,A) = (ln*( #Z n)).(upper_bound A)-(ln*( #Z n)).(lower_bound
  A)
proof
  assume that
A1: A c= Z and
A2: for x st x in Z holds x > 0 and
A3: dom (ln*( #Z n)) = Z and
A4: dom (ln*( #Z n)) = dom f2 and
A5: for x st x in Z holds f2.x = n/x and
A6: f2|A is continuous;
A7: f2 is_integrable_on A by A1,A3,A4,A6,INTEGRA5:11;
A8: ln*( #Z n) is_differentiable_on Z by A2,A3,FDIFF_6:9;
A9: for x being Element of REAL
st x in dom ((ln*( #Z n))`|Z) holds ((ln*( #Z n))`|Z).x = f2.x
  proof
    let x be Element of REAL;
    assume x in dom ((ln*( #Z n))`|Z);
    then
A10: x in Z by A8,FDIFF_1:def 7;
    then ((ln*( #Z n))`|Z).x = n/x by A2,A3,FDIFF_6:9
      .= f2.x by A5,A10;
    hence thesis;
  end;
  dom ((ln*( #Z n))`|Z) = dom f2 by A3,A4,A8,FDIFF_1:def 7;
  then ((ln*( #Z n))`|Z) = f2 by A9,PARTFUN1:5;
  hence thesis by A1,A3,A4,A6,A7,A8,INTEGRA5:10,13;
end;
