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
( cos . x > 0 & f . x = (tan . 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: for f being PartFunc of REAL,REAL st A c= Z & ( for x being Real st x in Z holds
( cos . x > 0 & f . x = (tan . 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: ( A c= Z & ( for x being Real st x in Z holds
( cos . x > 0 & f . x = (tan . 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 that
A1: A c= Z and
A2: for x being Real st x in Z holds
( cos . x > 0 & f . x = (tan . x) ^2 ) and
A3: Z c= dom (tan - (id Z)) and
A4: Z = dom f and
A5: f | A is continuous ; :: thesis: integral (f,A) = ((tan - (id Z)) . ()) - ((tan - (id Z)) . ())
A6: f is_integrable_on A by ;
A7: tan - (id Z) is_differentiable_on Z by ;
A8: 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 A9: x in Z by ;
then ((tan - (id Z)) `| Z) . x = (tan . x) ^2 by
.= f . x by A2, A9 ;
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