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 & dom ((-id Z)(#)cos+sin) = Z & (for x st x in Z holds f.x =x*
sin.x) & Z = dom f & f|A is continuous implies integral(f,A) = ((-id Z)(#)cos+
  sin).(upper_bound A)-((-id Z)(#)cos+sin).(lower_bound A)
proof
  assume that
A1: A c= Z and
A2: dom ((-id Z)(#)cos+sin) = Z and
A3: for x st x in Z holds f.x =x*sin.x and
A4: Z = dom f and
A5: f|A is continuous;
A6: (-id Z)(#)cos+sin is_differentiable_on Z by A2,FDIFF_4:46;
A7: for x being Element of REAL
st x in dom (((-id Z)(#)cos+sin)`|Z) holds (((-id Z)(#)cos+sin)`|Z)
  .x = f.x
  proof
    let x be Element of REAL;
    assume x in dom (((-id Z)(#)cos+sin)`|Z);
    then
A8: x in Z by A6,FDIFF_1:def 7;
    then (((-id Z)(#)cos+sin)`|Z).x =x*sin.x by A2,FDIFF_4:46
      .= f.x by A3,A8;
    hence thesis;
  end;
  dom (((-id Z)(#)cos+sin)`|Z) = dom f by A4,A6,FDIFF_1:def 7;
  then
A9: (((-id Z)(#)cos+sin)`|Z) = f by A7,PARTFUN1:5;
  f is_integrable_on A & f|A is bounded by A1,A4,A5,INTEGRA5:10,11;
  hence thesis by A1,A2,A9,FDIFF_4:46,INTEGRA5:13;
end;
