let A be non empty closed_interval Subset of REAL; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds
f . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2) ) & Z c= dom ((- cosec) - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A))
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds
f . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2) ) & Z c= dom ((- cosec) - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A))
let f be PartFunc of REAL,REAL; ( A c= Z & ( for x being Real st x in Z holds
f . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2) ) & Z c= dom ((- cosec) - (id Z)) & Z = dom f & f | A is continuous implies integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A)) )
assume that
A1:
A c= Z
and
A2:
for x being Real st x in Z holds
f . x = ((cos . x) - ((sin . x) ^2)) / ((sin . x) ^2)
and
A3:
Z c= dom ((- cosec) - (id Z))
and
A4:
Z = dom f
and
A5:
f | A is continuous
; integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A))
A6:
(- cosec) - (id Z) is_differentiable_on Z
by A3, FDIFF_9:23;
A7:
for x being Element of REAL st x in dom (((- cosec) - (id Z)) `| Z) holds
(((- cosec) - (id Z)) `| Z) . x = f . x
dom (((- cosec) - (id Z)) `| Z) = dom f
by A4, A6, FDIFF_1:def 7;
then A9:
((- cosec) - (id Z)) `| Z = f
by A7, PARTFUN1:5;
( f is_integrable_on A & f | A is bounded )
by A1, A4, A5, INTEGRA5:10, INTEGRA5:11;
hence
integral (f,A) = (((- cosec) - (id Z)) . (upper_bound A)) - (((- cosec) - (id Z)) . (lower_bound A))
by A1, A3, A9, FDIFF_9:23, INTEGRA5:13; verum