let A be 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 B1: ( Z c= dom (exp_R * cosec ) & Z c= dom (cos / (sin ^2 )) ) by XBOOLE_1:18;
then A3: - (exp_R * cosec ) is_differentiable_on Z by Th4;
B2: 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 B3: 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 B3, FUNCT_1:22, B1
.= (exp_R . (cosec . x)) * ((cos . x) / ((sin ^2 ) . x)) by RFUNCT_1:def 4, B1, B3
.= (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;

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

proof

let x be 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 A5: x in Z by A3, FDIFF_1:def 8;
then ((- (exp_R * cosec )) `| Z) . x = ((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2 ) by B1, Th4
.= f . x by B2, A5 ;
hence ((- (exp_R * cosec )) `| Z) . x = f . x ; :: thesis: verum

end;

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