let A be non empty closed_interval Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = ((id Z) (#) ((cos * ln) ^2)) ^ & Z c= dom (tan * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A))
let f be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & f = ((id Z) (#) ((cos * ln) ^2)) ^ & Z c= dom (tan * ln) & Z = dom f & f | A is continuous holds
integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & f = ((id Z) (#) ((cos * ln) ^2)) ^ & Z c= dom (tan * ln) & Z = dom f & f | A is continuous implies integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A)) )
assume A1: ( A c= Z & f = ((id Z) (#) ((cos * ln) ^2)) ^ & Z c= dom (tan * ln) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: tan * ln is_differentiable_on Z by A1, FDIFF_8:14;
Z c= dom ((id Z) (#) ((cos * ln) ^2)) by A1, RFUNCT_1:1;
then Z c= (dom (id Z)) /\ (dom ((cos * ln) ^2)) by VALUED_1:def 4;
then Z c= dom ((cos * ln) ^2) by XBOOLE_1:18;
then A4: Z c= dom (cos * ln) by VALUED_1:11;
A5: for x being Real st x in Z holds
f . x = 1 / (x * ((cos . (ln . x)) ^2))

proof

let x be Real; :: thesis: ( x in Z implies f . x = 1 / (x * ((cos . (ln . x)) ^2)) )
assume A6: x in Z ; :: thesis: f . x = 1 / (x * ((cos . (ln . x)) ^2))
then (((id Z) (#) ((cos * ln) ^2)) ^) . x = 1 / (((id Z) (#) ((cos * ln) ^2)) . x) by A1, RFUNCT_1:def 2
.= 1 / (((id Z) . x) * (((cos * ln) ^2) . x)) by VALUED_1:5
.= 1 / (x * (((cos * ln) ^2) . x)) by A6, FUNCT_1:18
.= 1 / (x * (((cos * ln) . x) ^2)) by VALUED_1:11
.= 1 / (x * ((cos . (ln . x)) ^2)) by A4, A6, FUNCT_1:12 ;
hence f . x = 1 / (x * ((cos . (ln . x)) ^2)) by A1; :: thesis: verum

end;

A7: for x being Element of REAL st x in dom ((tan * ln) `| Z) holds
((tan * ln) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((tan * ln) `| Z) implies ((tan * ln) `| Z) . x = f . x )
assume x in dom ((tan * ln) `| Z) ; :: thesis: ((tan * ln) `| Z) . x = f . x
then A8: x in Z by A3, FDIFF_1:def 7;
then ((tan * ln) `| Z) . x = 1 / (x * ((cos . (ln . x)) ^2)) by A1, FDIFF_8:14
.= f . x by A5, A8 ;
hence ((tan * ln) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((tan * ln) `| Z) = dom f by A1, A3, FDIFF_1:def 7;
then (tan * ln) `| Z = f by A7, PARTFUN1:5;
hence integral (f,A) = ((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A)) by A1, A2, FDIFF_8:14, INTEGRA5:13; :: thesis: verum