let a be Real; :: thesis: for A being closed-interval Subset of REAL
for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A))
let A be closed-interval Subset of REAL; :: thesis: for f1, f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A))
let f1, f be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f holds
integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f implies integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= dom (tan * f1) & f = ((sin * f1) ^2) / ((cos * f1) ^2) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f ) ; :: thesis: integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A))
then A2: Z c= dom ((1 / a) (#) (tan * f1)) by VALUED_1:def 5;
dom (id Z) = Z by RELAT_1:71;
then Z c= (dom ((1 / a) (#) (tan * f1))) /\ (dom (id Z)) by XBOOLE_1:19, A2;
then A4: Z c= dom (((1 / a) (#) (tan * f1)) - (id Z)) by VALUED_1:12;
A5: for x being Real st x in Z holds
f1 . x = (a * x) + 0 by A1;
Z = (dom ((sin * f1) ^2)) /\ ((dom ((cos * f1) ^2)) \ (((cos * f1) ^2) " {0})) by A1, RFUNCT_1:def 4;
then A7: ( Z c= dom ((sin * f1) ^2) & Z c= (dom ((cos * f1) ^2)) \ (((cos * f1) ^2) " {0}) ) by XBOOLE_1:18;
then A8: Z c= dom (sin * f1) by VALUED_1:11;
A9: Z c= dom (((cos * f1) ^2) ^) by RFUNCT_1:def 8, A7;
dom (((cos * f1) ^2) ^) c= dom ((cos * f1) ^2) by RFUNCT_1:11;
then Z c= dom ((cos * f1) ^2) by XBOOLE_1:1, A9;
then A11: Z c= dom (cos * f1) by VALUED_1:11;
A12: sin * f1 is_differentiable_on Z by FDIFF_4:37, A8, A5;
A13: cos * f1 is_differentiable_on Z by FDIFF_4:38, A5, A11;
A15: (sin * f1) ^2 is_differentiable_on Z by FDIFF_2:20, A12;
A17: (cos * f1) ^2 is_differentiable_on Z by FDIFF_2:20, A13;
for x being Real st x in Z holds
((cos * f1) ^2) . x <> 0

proof

let x be Real; :: thesis: ( x in Z implies ((cos * f1) ^2) . x <> 0 )
assume x in Z ; :: thesis: ((cos * f1) ^2) . x <> 0
then x in (dom ((sin * f1) ^2)) /\ ((dom ((cos * f1) ^2)) \ (((cos * f1) ^2) " {0})) by A1, RFUNCT_1:def 4;
then x in (dom ((cos * f1) ^2)) \ (((cos * f1) ^2) " {0}) by XBOOLE_0:def 4;
then x in dom (((cos * f1) ^2) ^) by RFUNCT_1:def 8;
hence ((cos * f1) ^2) . x <> 0 by RFUNCT_1:13; :: thesis: verum

end;

then f is_differentiable_on Z by A1, A15, A17, FDIFF_2:21;
then f | Z is continuous by FDIFF_1:33;
then f | A is continuous by A1, FCONT_1:17;
then A21: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A22: ((1 / a) (#) (tan * f1)) - (id Z) is_differentiable_on Z by A1, A4, FDIFF_8:26;
B1: for x being Real st x in Z holds
f . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2)

proof

let x be Real; :: thesis: ( x in Z implies f . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2) )
assume B2: x in Z ; :: thesis: f . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2)
then (((sin * f1) ^2) / ((cos * f1) ^2)) . x = (((sin * f1) ^2) . x) / (((cos * f1) ^2) . x) by RFUNCT_1:def 4, A1
.= (((sin * f1) . x) ^2) / (((cos * f1) ^2) . x) by VALUED_1:11
.= (((sin * f1) . x) ^2) / (((cos * f1) . x) ^2) by VALUED_1:11
.= ((sin . (f1 . x)) ^2) / (((cos * f1) . x) ^2) by FUNCT_1:22, A8, B2
.= ((sin . (f1 . x)) ^2) / ((cos . (f1 . x)) ^2) by FUNCT_1:22, A11, B2
.= ((sin . (a * x)) ^2) / ((cos . (f1 . x)) ^2) by B2, A1
.= ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2) by B2, A1 ;
hence f . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2) by A1; :: thesis: verum

end;

A23: for x being Real st x in dom ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) holds
((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) implies ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = f . x )
assume x in dom ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) ; :: thesis: ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = f . x
then A24: x in Z by A22, FDIFF_1:def 8;
then ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = ((sin . (a * x)) ^2) / ((cos . (a * x)) ^2) by A1, A4, FDIFF_8:26
.= f . x by B1, A24 ;
hence ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((((1 / a) (#) (tan * f1)) - (id Z)) `| Z) = dom f by A1, A22, FDIFF_1:def 8;
then (((1 / a) (#) (tan * f1)) - (id Z)) `| Z = f by A23, PARTFUN1:34;
hence integral (f,A) = ((((1 / a) (#) (tan * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (tan * f1)) - (id Z)) . (lower_bound A)) by A1, A21, A22, INTEGRA5:13; :: thesis: verum