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
  not 0 in Z & A c= Z & dom (ln*((id Z)^)) = Z & dom (ln*((id Z)^)) =
  dom f2 & (for x st x in Z holds f2.x = -1/x) & f2|A is continuous implies
integral(f2,A) = (ln*((id Z)^)).(upper_bound A)-(ln*((id Z)^)).(lower_bound A)
proof
  set f = id Z;
  assume that
A1: not 0 in Z and
A2: A c= Z and
A3: dom (ln*(f^)) = Z and
A4: dom (ln*(f^)) = dom f2 and
A5: for x st x in Z holds f2.x = -1/x and
A6: f2|A is continuous;
A7: f2 is_integrable_on A by A2,A3,A4,A6,INTEGRA5:11;
A8: ln*(f^) is_differentiable_on Z by A1,A3,FDIFF_8:5;
A9: for x being Element of REAL
st x in dom ((ln*(f^))`|Z) holds ((ln*(f^))`|Z).x = f2.x
  proof
    let x be Element of REAL;
    assume x in dom ((ln*(f^))`|Z);
    then
A10: x in Z by A8,FDIFF_1:def 7;
    then ((ln*(f^))`|Z).x = -1/x by A1,A3,FDIFF_8:5
      .= f2.x by A5,A10;
    hence thesis;
  end;
  dom ((ln*(f^))`|Z) = dom f2 by A3,A4,A8,FDIFF_1:def 7;
  then ((ln*(f^))`|Z) = f2 by A9,PARTFUN1:5;
  hence thesis by A2,A3,A4,A6,A7,A8,INTEGRA5:10,13;
end;
