let A be non empty closed_interval Subset of REAL; :: thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
sin . x > 0 ) & Z c= dom (ln * sin) & Z = dom cot & cot | A is continuous holds
integral (cot,A) = ((ln * sin) . (upper_bound A)) - ((ln * sin) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
sin . x > 0 ) & Z c= dom (ln * sin) & Z = dom cot & cot | A is continuous implies integral (cot,A) = ((ln * sin) . (upper_bound A)) - ((ln * sin) . (lower_bound A)) )
set f = cot ;
assume that
A1: A c= Z and
A2: for x being Real st x in Z holds
sin . x > 0 and
A3: Z c= dom (ln * sin) and
A4: Z = dom cot and
A5: cot | A is continuous ; :: thesis: integral (cot,A) = ((ln * sin) . (upper_bound A)) - ((ln * sin) . (lower_bound A))
A6: ln * sin is_differentiable_on Z by A2, A3, FDIFF_4:43;
A7: for x being Element of REAL st x in dom ((ln * sin) `| Z) holds
((ln * sin) `| Z) . x = cot . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((ln * sin) `| Z) implies ((ln * sin) `| Z) . x = cot . x )
assume x in dom ((ln * sin) `| Z) ; :: thesis: ((ln * sin) `| Z) . x = cot . x
then A8: x in Z by A6, FDIFF_1:def 7;
then A9: sin . x <> 0 by A2;
((ln * sin) `| Z) . x = cot x by A2, A3, A8, FDIFF_4:43
.= cot . x by A9, SIN_COS9:16 ;
hence ((ln * sin) `| Z) . x = cot . x ; :: thesis: verum

end;

dom ((ln * sin) `| Z) = dom cot by A4, A6, FDIFF_1:def 7;
then A10: (ln * sin) `| Z = cot by A7, PARTFUN1:5;
( cot is_integrable_on A & cot | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11;
hence integral (cot,A) = ((ln * sin) . (upper_bound A)) - ((ln * sin) . (lower_bound A)) by A1, A2, A3, A10, FDIFF_4:43, INTEGRA5:13; :: thesis: verum