 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 & f=(n(#)(( #Z (n-1))*sin))/(( #Z (n+1))*cos) &
 1<=n & Z c= dom (( #Z n)*tan) & Z = dom f
 implies
 integral(f,A)=(( #Z n)*tan).(upper_bound A)-(( #Z n)*tan).(lower_bound A)
proof
  assume
A1:A c= Z & f=n(#)(( #Z (n-1))*sin)/(( #Z (n+1))*cos)
 & 1<=n & Z c= dom (( #Z n)*tan) & Z = dom f;
then Z = dom (n(#)(( #Z (n-1))*sin)) /\
                (dom (( #Z (n+1))*cos) \ (( #Z (n+1))*cos)"{0})
   by RFUNCT_1:def 1;
then A2:Z c= dom (n(#)(( #Z (n-1))*sin)) &
   Z c= dom (( #Z (n+1))*cos) \ (( #Z (n+1))*cos)"{0} by XBOOLE_1:18;
then A3:Z c= dom ((( #Z (n+1))*cos)^) by RFUNCT_1:def 2;
    dom ((( #Z (n+1))*cos)^) c= dom (( #Z (n+1))*cos) by RFUNCT_1:1;then
A4:Z c= dom (( #Z (n+1))*cos) by A3;
A5: x in Z implies (( #Z (n+1))*cos).x <> 0
   proof
   assume x in Z;
   then x in dom (n(#)(( #Z (n-1))*sin)) /\
                (dom (( #Z (n+1))*cos) \ (( #Z (n+1))*cos)"{0})
        by A1,RFUNCT_1:def 1;
   then x in dom (( #Z (n+1))*cos) \ (( #Z (n+1))*cos)"{0}
        by XBOOLE_0:def 4; then
   x in dom ((( #Z (n+1))*cos)^) by RFUNCT_1:def 2;
     hence thesis by RFUNCT_1:3;
   end;
A6:Z c= dom (( #Z (n-1))*sin) by A2,VALUED_1:def 5;
A7:#Z (n-1)*sin is_differentiable_in x
   proof
   consider m being Nat such that
A8: n = m + 1 by A1,NAT_1:6;
    sin is_differentiable_in x by SIN_COS:64;
   hence thesis by A8,TAYLOR_1:3;
   end;
 #Z (n-1)*sin is_differentiable_on Z
   proof
   for x st x in Z holds #Z (n-1)*sin is_differentiable_in x by A7;
   hence thesis by A6,FDIFF_1:9;
   end;
then A9:(n(#)(( #Z (n-1))*sin)) is_differentiable_on Z by A2,FDIFF_1:20;
A10: #Z (n+1)*cos is_differentiable_in x
    proof
    cos is_differentiable_in x by SIN_COS:63;
    hence thesis by TAYLOR_1:3;
    end;
(( #Z (n+1))*cos) is_differentiable_on Z
   proof
   for x st x in Z holds ( #Z (n+1))*cos is_differentiable_in x by A10;
   hence thesis by A4,FDIFF_1:9;
   end;
then f|Z is continuous by A1,A5,A9,FDIFF_1:25,FDIFF_2:21;
then f|A is continuous by A1,FCONT_1:16;
then A11:f is_integrable_on A & f|A is bounded by A1,INTEGRA5:10,11;
A12:(( #Z n)*tan) is_differentiable_on Z by A1,FDIFF_8:20;
A13:for x st x in Z holds f.x = n*(sin.x) #Z (n-1)/(cos.x) #Z (n+1)
   proof
   let x;
   assume
A14:x in Z;
    then ((n(#)(( #Z (n-1))*sin))/(( #Z (n+1))*cos)).x
    =(n(#)(( #Z (n-1))*sin)).x/(( #Z (n+1))*cos).x by A1,RFUNCT_1:def 1
   .=n*(( #Z (n-1))*sin).x/(( #Z (n+1))*cos).x by VALUED_1:6
   .=n*(( #Z (n-1)).(sin.x))/(( #Z (n+1))*cos).x by A6,A14,FUNCT_1:12
   .=n*(sin.x) #Z (n-1)/(( #Z (n+1))*cos).x by TAYLOR_1:def 1
   .=n*(sin.x) #Z (n-1)/(( #Z (n+1)).(cos.x)) by A4,A14,FUNCT_1:12
   .=n*(sin.x) #Z (n-1)/(cos.x) #Z (n+1) by TAYLOR_1:def 1;
   hence thesis by A1;
   end;
A15:for x being Element of REAL
    st x in dom ((( #Z n)*tan)`|Z) holds ((( #Z n)*tan)`|Z).x = f.x
   proof
      let x be Element of REAL;
      assume x in dom ((( #Z n)*tan)`|Z);then
A16: x in Z by A12,FDIFF_1:def 7; then
    ((( #Z n)*tan)`|Z).x = n*(sin.x) #Z (n-1)/(cos.x) #Z (n+1) by A1,FDIFF_8:20
    .=f.x by A13,A16;
    hence thesis;
   end;
  dom ((( #Z n)*tan)`|Z) = dom f by A1,A12,FDIFF_1:def 7;
  then ((( #Z n)*tan)`|Z)=f by A15,PARTFUN1:5;
  hence thesis by A1,A11,FDIFF_8:20,INTEGRA5:13;
end;
