 reserve a,b,x,r for Real;
 reserve y for set;
 reserve n for Element of NAT;
 reserve A for non empty closed_interval Subset of REAL;
 reserve f,g,f1,f2,g1,g2 for PartFunc of REAL,REAL;
 reserve Z for open Subset of REAL;

theorem
 A c= Z & Z = dom f & f=(exp_R*sin)(#)cos implies
 integral(f,A)=(exp_R*sin).(upper_bound A)-(exp_R*sin).(lower_bound A)
proof
  assume
A1:A c= Z & Z = dom f & f=(exp_R*sin)(#)cos;
then Z = dom (exp_R*sin) /\ dom cos by VALUED_1:def 4;
then A2:Z c= dom (exp_R*sin) by XBOOLE_1:18;
then A3:exp_R*sin is_differentiable_on Z by FDIFF_7:37;
cos is_differentiable_on Z by FDIFF_1:26,SIN_COS:67;
   then f|Z is continuous by A1,A3,FDIFF_1:21,25;then
f|A is continuous by A1,FCONT_1:16;
then A4:f is_integrable_on A & f|A is bounded by A1,INTEGRA5:10,11;
A5:for x st x in Z holds f.x=exp_R.(sin.x) * cos.x
   proof
   let x;
   assume
A6:x in Z;
   then ((exp_R*sin)(#)cos).x=(exp_R*sin).x*cos.x by A1,VALUED_1:def 4
                       .=exp_R.(sin.x) * cos.x by A2,A6,FUNCT_1:12;
   hence thesis by A1;
   end;
A7:for x being Element of REAL
    st x in dom((exp_R*sin)`|Z) holds ((exp_R*sin)`|Z).x=f.x
   proof
     let x be Element of REAL;
     assume x in dom((exp_R*sin)`|Z);then
A8: x in Z by A3,FDIFF_1:def 7; then
  ((exp_R*sin)`|Z).x=exp_R.(sin.x) * cos.x by A2,FDIFF_7:37
   .=f.x by A5,A8;
   hence thesis;
   end;
  dom((exp_R*sin)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
  then((exp_R*sin)`|Z)= f by A7,PARTFUN1:5;
  hence thesis by A1,A2,A4,FDIFF_7:37,INTEGRA5:13;
end;
