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: ( cot is_integrable_on A & cot | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: - (ln * cosec) is_differentiable_on Z by A1, Th5;
A4: 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 x in Z by A3, FDIFF_1:def 7;
then ((- (ln * cosec)) `| Z) . x = cot . x by A1, Th5;
hence ((- (ln * cosec)) `| Z) . x = cot . x ; :: thesis: verum

end;

dom ((- (ln * cosec)) `| Z) = dom cot by A1, A3, FDIFF_1:def 7;
then (- (ln * cosec)) `| Z = cot by A4, PARTFUN1:5;
hence integral (cot,A) = ((- (ln * cosec)) . (upper_bound A)) - ((- (ln * cosec)) . (lower_bound A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum