 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 & Z = dom (cos-sin) & (cos-sin)|A is continuous implies
 integral(cos-sin,A)=(sin+cos).(upper_bound A)-(sin+cos).(lower_bound A)
proof
  assume
A1:A c= Z & Z = dom (cos-sin) & (cos-sin)|A is continuous;then
A2:cos-sin is_integrable_on A & (cos-sin)|A is bounded by INTEGRA5:10,11;
   Z = dom cos /\ dom sin by A1,VALUED_1:12;then
A3:Z c= dom (sin+cos) by VALUED_1:def 1;
then A4:sin+cos is_differentiable_on Z by FDIFF_7:38;
A5:for x being Element of REAL
    st x in dom ((sin+cos)`|Z) holds ((sin+cos)`|Z).x=(cos-sin).x
   proof
   let x be Element of REAL;
   assume x in dom((sin+cos)`|Z);then
A6:x in Z by A4,FDIFF_1:def 7;
  then ((sin+cos)`|Z).x=cos.x-sin.x by A3,FDIFF_7:38
                 .=(cos-sin).x by A1,A6,VALUED_1:13;
   hence thesis;
   end;
  dom((sin+cos)`|Z)=dom (cos-sin) by A1,A4,FDIFF_1:def 7;
  then((sin+cos)`|Z)=(cos-sin) by A5,PARTFUN1:5;
  hence thesis by A1,A2,A3,FDIFF_7:38,INTEGRA5:13;
end;
