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) ^)) . (upper_bound A)) - ((tan * ((id Z) ^)) . (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 = - (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) ^)) . (upper_bound A)) - ((tan * ((id Z) ^)) . (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 = - (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) ^)) . (upper_bound A)) - ((tan * ((id Z) ^)) . (lower_bound A)) )
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) ^)) . (upper_bound A)) - ((tan * ((id Z) ^)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: Z c= dom ((id Z) ^) by A1, FUNCT_1:101;
A4: not 0 in Z

proof

assume A5: 0 in Z ; :: thesis: contradiction
dom ((id Z) ^) = (dom (id Z)) \ ((id Z) " {0}) by RFUNCT_1:def 2
.= (dom (id Z)) \ {0} by Lm1, A5 ;
then not 0 in {0} by A5, A3, XBOOLE_0:def 5;
hence contradiction by TARSKI:def 1; :: thesis: verum

end;

then A6: tan * ((id Z) ^) is_differentiable_on Z by A1, FDIFF_8:8;
A7: 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 A8: x in Z by A6, FDIFF_1:def 7;
then ((tan * ((id Z) ^)) `| Z) . x = - (1 / ((x ^2) * ((cos . (1 / x)) ^2))) by A1, A4, FDIFF_8:8
.= f . x by A1, A8 ;
hence ((tan * ((id Z) ^)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((tan * ((id Z) ^)) `| Z) = dom f by A1, A6, FDIFF_1:def 7;
then (tan * ((id Z) ^)) `| Z = f by A7, PARTFUN1:5;
hence integral (f,A) = ((tan * ((id Z) ^)) . (upper_bound A)) - ((tan * ((id Z) ^)) . (lower_bound A)) by A1, A2, A6, INTEGRA5:13; :: thesis: verum