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 = 1 / ((x ^2) * ((cos . (1 / x)) ^2)) ) & Z c= dom (tan * ((id Z) ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (tan * ((id Z) ^))) . ()) - ((- (tan * ((id Z) ^))) . ())

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 = 1 / ((x ^2) * ((cos . (1 / x)) ^2)) ) & Z c= dom (tan * ((id Z) ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (tan * ((id Z) ^))) . ()) - ((- (tan * ((id Z) ^))) . ())

let Z be open Subset of REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
f . x = 1 / ((x ^2) * ((cos . (1 / x)) ^2)) ) & Z c= dom (tan * ((id Z) ^)) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (tan * ((id Z) ^))) . ()) - ((- (tan * ((id Z) ^))) . ()) )

assume A1: ( A c= Z & ( for x being Real st x in Z holds
f . x = 1 / ((x ^2) * ((cos . (1 / x)) ^2)) ) & Z c= dom (tan * ((id Z) ^)) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((- (tan * ((id Z) ^))) . ()) - ((- (tan * ((id Z) ^))) . ())
then A2: ( f is_integrable_on A & f | A is bounded ) by ;
A3: - (tan * ((id Z) ^)) is_differentiable_on Z by ;
A4: for x being Element of REAL st x in dom ((- (tan * ((id Z) ^))) `| Z) holds
((- (tan * ((id Z) ^))) `| Z) . x = f . x
proof
let x be Element of REAL ; :: thesis: ( x in dom ((- (tan * ((id Z) ^))) `| Z) implies ((- (tan * ((id Z) ^))) `| Z) . x = f . x )
assume x in dom ((- (tan * ((id Z) ^))) `| Z) ; :: thesis: ((- (tan * ((id Z) ^))) `| Z) . x = f . x
then A5: x in Z by ;
then ((- (tan * ((id Z) ^))) `| Z) . x = 1 / ((x ^2) * ((cos . (1 / x)) ^2)) by
.= f . x by A1, A5 ;
hence ((- (tan * ((id Z) ^))) `| Z) . x = f . x ; :: thesis: verum
end;
dom ((- (tan * ((id Z) ^))) `| Z) = dom f by ;
then (- (tan * ((id Z) ^))) `| Z = f by ;
hence integral (f,A) = ((- (tan * ((id Z) ^))) . ()) - ((- (tan * ((id Z) ^))) . ()) by ; :: thesis: verum