 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 & n>0 & (for x st x in Z holds f.x=cos.x/((sin.x) #Z (n+1))
 & sin.x<>0) & Z c= dom (( #Z n)*(sin^)) & Z =dom f
 & f|A is continuous implies
 integral(f,A)=((-1/n)(#)(( #Z n)*(sin^))).(upper_bound A)
              -((-1/n)(#)(( #Z n)*(sin^))).(lower_bound A)
proof
  assume
A1:A c= Z & n>0 & (for x st x in Z holds f.x=cos.x/((sin.x) #Z (n+1))
   & sin.x<>0) & Z c= dom (( #Z n)*(sin^)) & Z =dom f
   & f|A is continuous;
then A2:Z c= dom ((-1/n)(#)(( #Z n)*(sin^))) by VALUED_1:def 5;
A3:f is_integrable_on A & f|A is bounded by A1,INTEGRA5:10,11;
A4:for x st x in Z holds sin.x<>0 by A1;then
A5:(-1/n)(#)(( #Z n)*(sin^)) is_differentiable_on Z by A1,A2,FDIFF_7:30;
A6:for x being Element of REAL st x in dom (((-1/n)(#)(( #Z n)*(sin^)))`|Z)
holds
   (((-1/n)(#)(( #Z n)*(sin^)))`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom (((-1/n)(#)(( #Z n)*(sin^)))`|Z);then
A7:x in Z by A5,FDIFF_1:def 7;then
  (((-1/n)(#)(( #Z n)*(sin^)))`|Z).x=cos.x/((sin.x) #Z (n+1))
    by A1,A2,A4,FDIFF_7:30
   .=f.x by A1,A7;
   hence thesis;
   end;
   dom (((-1/n)(#)(( #Z n)*(sin^)))`|Z)=dom f by A1,A5,FDIFF_1:def 7;
   then(((-1/n)(#)(( #Z n)*(sin^)))`|Z)= f by A6,PARTFUN1:5;
   hence thesis by A1,A3,A5,INTEGRA5:13;
end;
