let f2 be PartFunc of REAL,REAL; for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL st A c= Z & dom tan = Z & dom tan = dom f2 & ( for x being Real st x in Z holds
( f2 . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) ) & f2 | A is continuous holds
integral (f2,A) = (tan . (upper_bound A)) - (tan . (lower_bound A))
let A be non empty closed_interval Subset of REAL; for Z being open Subset of REAL st A c= Z & dom tan = Z & dom tan = dom f2 & ( for x being Real st x in Z holds
( f2 . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) ) & f2 | A is continuous holds
integral (f2,A) = (tan . (upper_bound A)) - (tan . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & dom tan = Z & dom tan = dom f2 & ( for x being Real st x in Z holds
( f2 . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) ) & f2 | A is continuous implies integral (f2,A) = (tan . (upper_bound A)) - (tan . (lower_bound A)) )
assume that
A1:
A c= Z
and
A2:
dom tan = Z
and
A3:
dom tan = dom f2
and
A4:
for x being Real st x in Z holds
( f2 . x = 1 / ((cos . x) ^2) & cos . x <> 0 )
and
A5:
f2 | A is continuous
; integral (f2,A) = (tan . (upper_bound A)) - (tan . (lower_bound A))
A6:
f2 is_integrable_on A
by A1, A2, A3, A5, INTEGRA5:11;
A7:
tan is_differentiable_on Z
by A2, Th33;
A8:
for x being Element of REAL st x in dom (tan `| Z) holds
(tan `| Z) . x = f2 . x
dom (tan `| Z) = dom f2
by A2, A3, A7, FDIFF_1:def 7;
then
tan `| Z = f2
by A8, PARTFUN1:5;
hence
integral (f2,A) = (tan . (upper_bound A)) - (tan . (lower_bound A))
by A1, A2, A3, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; verum