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 * cot ) / (sin ^2 ) & Z = dom f & f | A is continuous holds
integral f,A = ((- (exp_R * cot )) . (upper_bound A)) - ((- (exp_R * cot )) . (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 * cot ) / (sin ^2 ) & Z = dom f & f | A is continuous holds
integral f,A = ((- (exp_R * cot )) . (upper_bound A)) - ((- (exp_R * cot )) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & f = (exp_R * cot ) / (sin ^2 ) & Z = dom f & f | A is continuous implies integral f,A = ((- (exp_R * cot )) . (upper_bound A)) - ((- (exp_R * cot )) . (lower_bound A)) )
assume A1: ( A c= Z & f = (exp_R * cot ) / (sin ^2 ) & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = ((- (exp_R * cot )) . (upper_bound A)) - ((- (exp_R * cot )) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
Z = (dom (exp_R * cot )) /\ ((dom (sin ^2 )) \ ((sin ^2 ) " {0 })) by RFUNCT_1:def 4, A1;
then A4: Z c= dom (exp_R * cot ) by XBOOLE_1:18;
then A3: - (exp_R * cot ) is_differentiable_on Z by Th3;
A6: for x being Real st x in Z holds
f . x = (exp_R . (cot . x)) / ((sin . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies f . x = (exp_R . (cot . x)) / ((sin . x) ^2 ) )
assume A7: x in Z ; :: thesis: f . x = (exp_R . (cot . x)) / ((sin . x) ^2 )
then ((exp_R * cot ) / (sin ^2 )) . x = ((exp_R * cot ) . x) / ((sin ^2 ) . x) by RFUNCT_1:def 4, A1
.= (exp_R . (cot . x)) / ((sin ^2 ) . x) by FUNCT_1:22, A4, A7
.= (exp_R . (cot . x)) / ((sin . x) ^2 ) by VALUED_1:11 ;
hence f . x = (exp_R . (cot . x)) / ((sin . x) ^2 ) by A1; :: thesis: verum

end;

A8: for x being Real st x in dom ((- (exp_R * cot )) `| Z) holds
((- (exp_R * cot )) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((- (exp_R * cot )) `| Z) implies ((- (exp_R * cot )) `| Z) . x = f . x )
assume x in dom ((- (exp_R * cot )) `| Z) ; :: thesis: ((- (exp_R * cot )) `| Z) . x = f . x
then A9: x in Z by A3, FDIFF_1:def 8;
then ((- (exp_R * cot )) `| Z) . x = (exp_R . (cot . x)) / ((sin . x) ^2 ) by Th3, A4
.= f . x by A6, A9 ;
hence ((- (exp_R * cot )) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- (exp_R * cot )) `| Z) = dom f by A1, A3, FDIFF_1:def 8;
then (- (exp_R * cot )) `| Z = f by A8, PARTFUN1:34;
hence integral f,A = ((- (exp_R * cot )) . (upper_bound A)) - ((- (exp_R * cot )) . (lower_bound A)) by A1, A2, Th3, A4, INTEGRA5:13; :: thesis: verum