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 = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & dom cot = Z & Z = dom f & f | A is continuous holds
integral (f,A) = (cot . (upper_bound A)) - (cot . (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 = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & dom cot = Z & Z = dom f & f | A is continuous holds
integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A))
let f be PartFunc of REAL,REAL; ( A c= Z & ( for x being Real st x in Z holds
( f . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & dom cot = Z & Z = dom f & f | A is continuous implies integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A)) )
assume that
A1:
A c= Z
and
A2:
for x being Real st x in Z holds
( f . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 )
and
A3:
dom cot = Z
and
A4:
Z = dom f
and
A5:
f | A is continuous
; integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A))
A6:
f is_integrable_on A
by A1, A4, A5, INTEGRA5:11;
A7:
cot is_differentiable_on Z
by A3, INTEGRA8:34;
A8:
for x being Element of REAL st x in dom (cot `| Z) holds
(cot `| Z) . x = f . x
dom (cot `| Z) = dom f
by A4, A7, FDIFF_1:def 7;
then
cot `| Z = f
by A8, PARTFUN1:5;
hence
integral (f,A) = (cot . (upper_bound A)) - (cot . (lower_bound A))
by A1, A4, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; verum