let A be non empty 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 A2: Z = dom (- cot) by VALUED_1:8;
then A3: ( - cot is_integrable_on A & (- cot) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A4: ln * cosec is_differentiable_on Z by A1, FDIFF_9:19;
A5: 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_1:11;
hence sin . x <> 0 by RFUNCT_1:3; :: thesis: verum

end;

A6: for x being Element of REAL st x in dom ((ln * cosec) `| Z) holds
((ln * cosec) `| Z) . x = (- cot) . x

proof

let x be Element of 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 A7: x in Z by A4, FDIFF_1:def 7;
then A8: sin . x <> 0 by A5;
((ln * cosec) `| Z) . x = - (cot x) by A1, A7, FDIFF_9:19
.= - (cot . x) by A8, 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 A2, A4, FDIFF_1:def 7;
then (ln * cosec) `| Z = - cot by A6, PARTFUN1:5;
hence integral ((- cot),A) = ((ln * cosec) . (upper_bound A)) - ((ln * cosec) . (lower_bound A)) by A1, A3, A4, INTEGRA5:13; :: thesis: verum