let f2 be PartFunc of REAL,REAL; :: thesis: for A being non empty closed_interval Subset of REAL
for Z being open Subset of REAL st A c= Z & dom cot = Z & dom cot = dom f2 & ( for x being Real st x in Z holds
( f2 . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & f2 | A is continuous holds
integral (f2,A) = (cot . (upper_bound A)) - (cot . (lower_bound A))
let A be non empty closed_interval Subset of REAL; :: thesis: for Z being open Subset of REAL st A c= Z & dom cot = Z & dom cot = dom f2 & ( for x being Real st x in Z holds
( f2 . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & f2 | A is continuous holds
integral (f2,A) = (cot . (upper_bound A)) - (cot . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & dom cot = Z & dom cot = dom f2 & ( for x being Real st x in Z holds
( f2 . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & f2 | A is continuous implies integral (f2,A) = (cot . (upper_bound A)) - (cot . (lower_bound A)) )
assume that
A1: A c= Z and
A2: dom cot = Z and
A3: dom cot = dom f2 and
A4: for x being Real st x in Z holds
( f2 . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) and
A5: f2 | A is continuous ; :: thesis: integral (f2,A) = (cot . (upper_bound A)) - (cot . (lower_bound A))
A6: f2 is_integrable_on A by A1, A2, A3, A5, INTEGRA5:11;
A7: cot is_differentiable_on Z by A2, Th34;
A8: for x being Element of REAL st x in dom (cot `| Z) holds
(cot `| Z) . x = f2 . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (cot `| Z) implies (cot `| Z) . x = f2 . x )
assume x in dom (cot `| Z) ; :: thesis: (cot `| Z) . x = f2 . x
then A9: x in Z by A7, FDIFF_1:def 7;
then (cot `| Z) . x = - (1 / ((sin . x) ^2)) by A2, Th34
.= f2 . x by A4, A9 ;
hence (cot `| Z) . x = f2 . x ; :: thesis: verum

end;

dom (cot `| Z) = dom f2 by A2, A3, A7, FDIFF_1:def 7;
then cot `| Z = f2 by A8, PARTFUN1:5;
hence integral (f2,A) = (cot . (upper_bound A)) - (cot . (lower_bound A)) by A1, A2, A3, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; :: thesis: verum