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 & ( for x being Real st x in Z holds
f . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2)) ) & Z c= dom (exp_R (#) cosec) & 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 & ( for x being Real st x in Z holds
f . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2)) ) & Z c= dom (exp_R (#) cosec) & 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 & ( for x being Real st x in Z holds
f . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2)) ) & Z c= dom (exp_R (#) cosec) & 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 & ( for x being Real st x in Z holds
f . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2)) ) & Z c= dom (exp_R (#) cosec) & 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;
A3: exp_R (#) cosec is_differentiable_on Z by A1, FDIFF_9:25;
A4: 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 A5: x in Z by A3, FDIFF_1:def 7;
then ((exp_R (#) cosec) `| Z) . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2)) by A1, FDIFF_9:25
.= f . x by A1, 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 7;
then (exp_R (#) cosec) `| Z = f by A4, PARTFUN1:5;
hence integral (f,A) = ((exp_R (#) cosec) . (upper_bound A)) - ((exp_R (#) cosec) . (lower_bound A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum