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 & dom (- cosec) = Z & ( for x being Real st x in Z holds
f . x = (cos . x) / ((sin . x) ^2) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- cosec) . (upper_bound A)) - ((- cosec) . (lower_bound A))
let f be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & dom (- cosec) = Z & ( for x being Real st x in Z holds
f . x = (cos . x) / ((sin . x) ^2) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- cosec) . (upper_bound A)) - ((- cosec) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & dom (- cosec) = Z & ( for x being Real st x in Z holds
f . x = (cos . x) / ((sin . x) ^2) ) & Z = dom f & f | A is continuous implies integral (f,A) = ((- cosec) . (upper_bound A)) - ((- cosec) . (lower_bound A)) )
assume that
A1:
A c= Z
and
A2:
dom (- cosec) = Z
and
A3:
for x being Real st x in Z holds
f . x = (cos . x) / ((sin . x) ^2)
and
A4:
Z = dom f
and
A5:
f | A is continuous
; integral (f,A) = ((- cosec) . (upper_bound A)) - ((- cosec) . (lower_bound A))
A6:
- cosec is_differentiable_on Z
by A2, Th70;
A7:
for x being Element of REAL st x in dom ((- cosec) `| Z) holds
((- cosec) `| Z) . x = f . x
dom ((- cosec) `| Z) = dom f
by A4, A6, FDIFF_1:def 7;
then A9:
(- cosec) `| 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) . (upper_bound A)) - ((- cosec) . (lower_bound A))
by A1, A2, A9, Th70, INTEGRA5:13; verum