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 & ( 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;
B1: Z c= dom ((id Z) ^ ) by FUNCT_1:171, A1;
B2: not 0 in Z

proof

assume K: 0 in Z ; :: thesis: contradiction
dom ((id Z) ^ ) = (dom (id Z)) \ ((id Z) " {0 }) by RFUNCT_1:def 8
.= (dom (id Z)) \ {0 } by Lm0, K ;
then not 0 in {0 } by XBOOLE_0:def 5, K, B1;
hence contradiction by TARSKI:def 1; :: thesis: verum

end;

then A3: tan * ((id Z) ^ ) is_differentiable_on Z by A1, FDIFF_8:8;
A4: for x being Real st x in dom ((tan * ((id Z) ^ )) `| Z) holds
((tan * ((id Z) ^ )) `| Z) . x = f . x

proof

let x be 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 A3, FDIFF_1:def 8;
then ((tan * ((id Z) ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) by A1, B2, FDIFF_8:8
.= f . x by A1, A5 ;
hence ((tan * ((id Z) ^ )) `| Z) . x = f . x ; :: thesis: verum

end;

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