let A be non empty closed_interval Subset of REAL; for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (tan / (id Z)) + (ln / (cos ^2)) & Z c= dom (ln (#) tan) & Z c= dom tan & Z = dom f & f | A is continuous holds
integral (f,A) = ((ln (#) tan) . (upper_bound A)) - ((ln (#) tan) . (lower_bound A))
let f be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & f = (tan / (id Z)) + (ln / (cos ^2)) & Z c= dom (ln (#) tan) & Z c= dom tan & Z = dom f & f | A is continuous holds
integral (f,A) = ((ln (#) tan) . (upper_bound A)) - ((ln (#) tan) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & f = (tan / (id Z)) + (ln / (cos ^2)) & Z c= dom (ln (#) tan) & Z c= dom tan & Z = dom f & f | A is continuous implies integral (f,A) = ((ln (#) tan) . (upper_bound A)) - ((ln (#) tan) . (lower_bound A)) )
assume A1:
( A c= Z & f = (tan / (id Z)) + (ln / (cos ^2)) & Z c= dom (ln (#) tan) & Z c= dom tan & Z = dom f & f | A is continuous )
; integral (f,A) = ((ln (#) tan) . (upper_bound A)) - ((ln (#) tan) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
A3:
ln (#) tan is_differentiable_on Z
by A1, FDIFF_8:32;
Z = (dom (tan / (id Z))) /\ (dom (ln / (cos ^2)))
by A1, VALUED_1:def 1;
then A4:
( Z c= dom (tan / (id Z)) & Z c= dom (ln / (cos ^2)) )
by XBOOLE_1:18;
A5:
for x being Real st x in Z holds
f . x = ((tan . x) / x) + ((ln . x) / ((cos . x) ^2))
A7:
for x being Element of REAL st x in dom ((ln (#) tan) `| Z) holds
((ln (#) tan) `| Z) . x = f . x
dom ((ln (#) tan) `| Z) = dom f
by A1, A3, FDIFF_1:def 7;
then
(ln (#) tan) `| Z = f
by A7, PARTFUN1:5;
hence
integral (f,A) = ((ln (#) tan) . (upper_bound A)) - ((ln (#) tan) . (lower_bound A))
by A1, A2, FDIFF_8:32, INTEGRA5:13; verum