 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 & f=exp_R(#)(cos-sin)
& Z c= dom (exp_R(#)cos) & Z = dom f & f|A is continuous implies
integral(f,A)=(exp_R(#)cos).(upper_bound A)-(exp_R(#)cos).(lower_bound A)
proof
   assume
A1:A c= Z & f=exp_R(#)(cos-sin)
   & Z c= dom (exp_R(#)cos) & Z = dom f & f|A is continuous;then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:exp_R(#)cos is_differentiable_on Z by A1,FDIFF_7:45;
    dom f = dom exp_R /\ dom (cos-sin) by A1,VALUED_1:def 4;then
A4:Z c= dom (cos-sin) by A1,XBOOLE_1:18;
A5:for x st x in Z holds f.x=exp_R.x*(cos.x-sin.x)
    proof
    let x;
    assume
A6:x in Z;
    (exp_R(#)(cos-sin)).x=exp_R.x*((cos-sin).x) by VALUED_1:5
                               .=exp_R.x*(cos.x-sin.x) by A4,A6,VALUED_1:13;
    hence thesis by A1;
    end;
A7:for x being Element of REAL
     st x in dom ((exp_R(#)cos)`|Z) holds ((exp_R(#)cos)`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((exp_R(#)cos)`|Z);then
A8:x in Z by A3,FDIFF_1:def 7;then
   ((exp_R(#)cos)`|Z).x=exp_R.x*(cos.x-sin.x) by A1,FDIFF_7:45
   .=f.x by A5,A8;
   hence thesis;
   end;
  dom ((exp_R(#)cos)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
  then ((exp_R(#)cos)`|Z)= f by A7,PARTFUN1:5;
  hence thesis by A1,A2,FDIFF_7:45,INTEGRA5:13;
end;
