let A be closed-interval Subset of REAL ; :: thesis: for f2 being PartFunc of REAL ,REAL
for Z being open Subset of REAL st not 0 in Z & A c= Z & dom (ln * ((id Z) ^ )) = Z & dom (ln * ((id Z) ^ )) = dom f2 & ( for x being Real st x in Z holds
f2 . x = - (1 / x) ) & f2 | A is continuous holds
integral f2,A = ((ln * ((id Z) ^ )) . (upper_bound A)) - ((ln * ((id Z) ^ )) . (lower_bound A))
let f2 be PartFunc of REAL ,REAL ; :: thesis: for Z being open Subset of REAL st not 0 in Z & A c= Z & dom (ln * ((id Z) ^ )) = Z & dom (ln * ((id Z) ^ )) = dom f2 & ( for x being Real st x in Z holds
f2 . x = - (1 / x) ) & f2 | A is continuous holds
integral f2,A = ((ln * ((id Z) ^ )) . (upper_bound A)) - ((ln * ((id Z) ^ )) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( not 0 in Z & A c= Z & dom (ln * ((id Z) ^ )) = Z & dom (ln * ((id Z) ^ )) = dom f2 & ( for x being Real st x in Z holds
f2 . x = - (1 / x) ) & f2 | A is continuous implies integral f2,A = ((ln * ((id Z) ^ )) . (upper_bound A)) - ((ln * ((id Z) ^ )) . (lower_bound A)) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: A c= Z and
A3: dom (ln * ((id Z) ^ )) = Z and
A4: dom (ln * ((id Z) ^ )) = dom f2 and
A5: for x being Real st x in Z holds
f2 . x = - (1 / x) and
A6: f2 | A is continuous ; :: thesis: integral f2,A = ((ln * ((id Z) ^ )) . (upper_bound A)) - ((ln * ((id Z) ^ )) . (lower_bound A))
A7: f2 is_integrable_on A by A2, A3, A4, A6, INTEGRA5:11;
A8: ln * ((id Z) ^ ) is_differentiable_on Z by A1, A3, FDIFF_8:5;
A9: for x being Real st x in dom ((ln * ((id Z) ^ )) `| Z) holds
((ln * ((id Z) ^ )) `| Z) . x = f2 . x

proof

let x be Real; :: thesis: ( x in dom ((ln * ((id Z) ^ )) `| Z) implies ((ln * ((id Z) ^ )) `| Z) . x = f2 . x )
assume x in dom ((ln * ((id Z) ^ )) `| Z) ; :: thesis: ((ln * ((id Z) ^ )) `| Z) . x = f2 . x
then A10: x in Z by A8, FDIFF_1:def 8;
then ((ln * ((id Z) ^ )) `| Z) . x = - (1 / x) by A1, A3, FDIFF_8:5
.= f2 . x by A5, A10 ;
hence ((ln * ((id Z) ^ )) `| Z) . x = f2 . x ; :: thesis: verum

end;

dom ((ln * ((id Z) ^ )) `| Z) = dom f2 by A3, A4, A8, FDIFF_1:def 8;
then (ln * ((id Z) ^ )) `| Z = f2 by A9, PARTFUN1:34;
hence integral f2,A = ((ln * ((id Z) ^ )) . (upper_bound A)) - ((ln * ((id Z) ^ )) . (lower_bound A)) by A2, A3, A4, A6, A7, A8, INTEGRA5:10, INTEGRA5:13; :: thesis: verum