let A be closed-interval Subset of REAL ; :: thesis: for Z being open Subset of REAL st A c= Z & Z c= dom (ln * cosec ) & Z = dom cot & (- cot ) | A is continuous holds
integral (- cot ),A = ((ln * cosec ) . (upper_bound A)) - ((ln * cosec ) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & Z c= dom (ln * cosec ) & Z = dom cot & (- cot ) | A is continuous implies integral (- cot ),A = ((ln * cosec ) . (upper_bound A)) - ((ln * cosec ) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= dom (ln * cosec ) & Z = dom cot & (- cot ) | A is continuous ) ; :: thesis: integral (- cot ),A = ((ln * cosec ) . (upper_bound A)) - ((ln * cosec ) . (lower_bound A))
then B1: Z = dom (- cot ) by VALUED_1:8;
then A2: ( - cot is_integrable_on A & (- cot ) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A3: ln * cosec is_differentiable_on Z by A1, FDIFF_9:19;
AA: for x being Real st x in Z holds
sin . x <> 0

proof

let x be Real; :: thesis: ( x in Z implies sin . x <> 0 )
assume x in Z ; :: thesis: sin . x <> 0
then x in dom cosec by A1, FUNCT_1:21;
hence sin . x <> 0 by RFUNCT_1:13; :: thesis: verum

end;

A4: for x being Real st x in dom ((ln * cosec ) `| Z) holds
((ln * cosec ) `| Z) . x = (- cot ) . x

proof

let x be Real; :: thesis: ( x in dom ((ln * cosec ) `| Z) implies ((ln * cosec ) `| Z) . x = (- cot ) . x )
assume x in dom ((ln * cosec ) `| Z) ; :: thesis: ((ln * cosec ) `| Z) . x = (- cot ) . x
then A5: x in Z by A3, FDIFF_1:def 8;
then A6: sin . x <> 0 by AA;
((ln * cosec ) `| Z) . x = - (cot x) by A1, A5, FDIFF_9:19
.= - (cot . x) by A6, SIN_COS9:16
.= (- cot ) . x by VALUED_1:8 ;
hence ((ln * cosec ) `| Z) . x = (- cot ) . x ; :: thesis: verum

end;

dom ((ln * cosec ) `| Z) = dom (- cot ) by B1, A3, FDIFF_1:def 8;
then (ln * cosec ) `| Z = - cot by A4, PARTFUN1:34;
hence integral (- cot ),A = ((ln * cosec ) . (upper_bound A)) - ((ln * cosec ) . (lower_bound A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum