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 * tan ) / (cos ^2 ) & Z = dom f & f | A is continuous holds
integral f,A = ((exp_R * tan ) . (upper_bound A)) - ((exp_R * tan ) . (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 * tan ) / (cos ^2 ) & Z = dom f & f | A is continuous holds
integral f,A = ((exp_R * tan ) . (upper_bound A)) - ((exp_R * tan ) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & f = (exp_R * tan ) / (cos ^2 ) & Z = dom f & f | A is continuous implies integral f,A = ((exp_R * tan ) . (upper_bound A)) - ((exp_R * tan ) . (lower_bound A)) )
assume A1: ( A c= Z & f = (exp_R * tan ) / (cos ^2 ) & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = ((exp_R * tan ) . (upper_bound A)) - ((exp_R * tan ) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
Z = (dom (exp_R * tan )) /\ ((dom (cos ^2 )) \ ((cos ^2 ) " {0 })) by RFUNCT_1:def 4, A1;
then A3: Z c= dom (exp_R * tan ) by XBOOLE_1:18;
then A4: exp_R * tan is_differentiable_on Z by FDIFF_8:16;
A5: for x being Real st x in Z holds
f . x = (exp_R . (tan . x)) / ((cos . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies f . x = (exp_R . (tan . x)) / ((cos . x) ^2 ) )
assume A6: x in Z ; :: thesis: f . x = (exp_R . (tan . x)) / ((cos . x) ^2 )
then ((exp_R * tan ) / (cos ^2 )) . x = ((exp_R * tan ) . x) / ((cos ^2 ) . x) by RFUNCT_1:def 4, A1
.= (exp_R . (tan . x)) / ((cos ^2 ) . x) by FUNCT_1:22, A3, A6
.= (exp_R . (tan . x)) / ((cos . x) ^2 ) by VALUED_1:11 ;
hence f . x = (exp_R . (tan . x)) / ((cos . x) ^2 ) by A1; :: thesis: verum

end;

A7: for x being Real st x in dom ((exp_R * tan ) `| Z) holds
((exp_R * tan ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((exp_R * tan ) `| Z) implies ((exp_R * tan ) `| Z) . x = f . x )
assume x in dom ((exp_R * tan ) `| Z) ; :: thesis: ((exp_R * tan ) `| Z) . x = f . x
then A8: x in Z by A4, FDIFF_1:def 8;
then ((exp_R * tan ) `| Z) . x = (exp_R . (tan . x)) / ((cos . x) ^2 ) by A3, FDIFF_8:16
.= f . x by A5, A8 ;
hence ((exp_R * tan ) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((exp_R * tan ) `| Z) = dom f by A1, A4, FDIFF_1:def 8;
then (exp_R * tan ) `| Z = f by A7, PARTFUN1:34;
hence integral f,A = ((exp_R * tan ) . (upper_bound A)) - ((exp_R * tan ) . (lower_bound A)) by A1, A2, A3, FDIFF_8:16, INTEGRA5:13; :: thesis: verum