let A be non empty closed_interval Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f = (exp_R * cosec) (#) (cos / (sin ^2)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A))
let f be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & f = (exp_R * cosec) (#) (cos / (sin ^2)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & f = (exp_R * cosec) (#) (cos / (sin ^2)) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A)) )
assume A1: ( A c= Z & f = (exp_R * cosec) (#) (cos / (sin ^2)) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
Z = (dom (exp_R * cosec)) /\ (dom (cos / (sin ^2))) by A1, VALUED_1:def 4;
then A3: ( Z c= dom (exp_R * cosec) & Z c= dom (cos / (sin ^2)) ) by XBOOLE_1:18;
then A4: - (exp_R * cosec) is_differentiable_on Z by Th4;
A5: for x being Real st x in Z holds
f . x = ((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2)

proof

let x be Real; :: thesis: ( x in Z implies f . x = ((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2) )
assume A6: x in Z ; :: thesis: f . x = ((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2)
((exp_R * cosec) (#) (cos / (sin ^2))) . x = ((exp_R * cosec) . x) * ((cos / (sin ^2)) . x) by VALUED_1:5
.= (exp_R . (cosec . x)) * ((cos / (sin ^2)) . x) by A6, A3, FUNCT_1:12
.= (exp_R . (cosec . x)) * ((cos . x) / ((sin ^2) . x)) by A3, A6, RFUNCT_1:def 1
.= (exp_R . (cosec . x)) * ((cos . x) / ((sin . x) ^2)) by VALUED_1:11
.= ((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2) ;
hence f . x = ((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2) by A1; :: thesis: verum

end;

A7: for x being Element of REAL st x in dom ((- (exp_R * cosec)) `| Z) holds
((- (exp_R * cosec)) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((- (exp_R * cosec)) `| Z) implies ((- (exp_R * cosec)) `| Z) . x = f . x )
assume x in dom ((- (exp_R * cosec)) `| Z) ; :: thesis: ((- (exp_R * cosec)) `| Z) . x = f . x
then A8: x in Z by A4, FDIFF_1:def 7;
then ((- (exp_R * cosec)) `| Z) . x = ((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2) by A3, Th4
.= f . x by A5, A8 ;
hence ((- (exp_R * cosec)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- (exp_R * cosec)) `| Z) = dom f by A1, A4, FDIFF_1:def 7;
then (- (exp_R * cosec)) `| Z = f by A7, PARTFUN1:5;
hence integral (f,A) = ((- (exp_R * cosec)) . (upper_bound A)) - ((- (exp_R * cosec)) . (lower_bound A)) by A1, A2, A4, INTEGRA5:13; :: thesis: verum