let A be non empty closed_interval Subset of REAL; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) ) & Z c= dom (sec - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) ) & Z c= dom (sec - (id Z)) & Z = dom f & f | A is continuous holds
integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A))
let f be PartFunc of REAL,REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) ) & Z c= dom (sec - (id Z)) & Z = dom f & f | A is continuous implies integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A)) )
assume that
A1: A c= Z and
A2: for x being Real st x in Z holds
f . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) and
A3: Z c= dom (sec - (id Z)) and
A4: Z = dom f and
A5: f | A is continuous ; :: thesis: integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A))
A6: sec - (id Z) is_differentiable_on Z by A3, FDIFF_9:22;
A7: for x being Element of REAL st x in dom ((sec - (id Z)) `| Z) holds
((sec - (id Z)) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((sec - (id Z)) `| Z) implies ((sec - (id Z)) `| Z) . x = f . x )
assume x in dom ((sec - (id Z)) `| Z) ; :: thesis: ((sec - (id Z)) `| Z) . x = f . x
then A8: x in Z by A6, FDIFF_1:def 7;
then ((sec - (id Z)) `| Z) . x = ((sin . x) - ((cos . x) ^2)) / ((cos . x) ^2) by A3, FDIFF_9:22
.= f . x by A2, A8 ;
hence ((sec - (id Z)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((sec - (id Z)) `| Z) = dom f by A4, A6, FDIFF_1:def 7;
then A9: (sec - (id Z)) `| Z = f by A7, PARTFUN1:5;
( f is_integrable_on A & f | A is bounded ) by A1, A4, A5, INTEGRA5:10, INTEGRA5:11;
hence integral (f,A) = ((sec - (id Z)) . (upper_bound A)) - ((sec - (id Z)) . (lower_bound A)) by A1, A3, A9, FDIFF_9:22, INTEGRA5:13; :: thesis: verum