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

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

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

dom ((tan - (id Z)) `| Z) = dom f
by A4, A7, FDIFF_1:def 7;
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;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

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