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)) . (upper_bound A)) - ((tan - (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
( 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)) . (upper_bound A)) - ((tan - (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
( 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)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) )
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)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A))
A6: f is_integrable_on A by A1, A4, A5, INTEGRA5:11;
A7: tan - (id Z) is_differentiable_on Z by A3, Th57;
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 A7, FDIFF_1:def 7;
then ((tan - (id Z)) `| Z) . x = (tan . x) ^2 by A3, Th57
.= f . x by A2, A9 ;
hence ((tan - (id Z)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((tan - (id Z)) `| Z) = dom f by A4, A7, FDIFF_1:def 7;
then (tan - (id Z)) `| Z = f by A8, PARTFUN1:5;
hence integral (f,A) = ((tan - (id Z)) . (upper_bound A)) - ((tan - (id Z)) . (lower_bound A)) by A1, A4, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; :: thesis: verum