let A be closed-interval Subset of REAL ; :: thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
x > 0 ) & dom ln = Z & Z = dom ((id Z) ^ ) & ((id Z) ^ ) | A is continuous holds
integral ((id Z) ^ ),A = (ln . (upper_bound A)) - (ln . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
x > 0 ) & dom ln = Z & Z = dom ((id Z) ^ ) & ((id Z) ^ ) | A is continuous implies integral ((id Z) ^ ),A = (ln . (upper_bound A)) - (ln . (lower_bound A)) )
set f2 = (id Z) ^ ;
assume A1: ( A c= Z & ( for x being Real st x in Z holds
x > 0 ) & dom ln = Z & Z = dom ((id Z) ^ ) & ((id Z) ^ ) | A is continuous ) ; :: thesis: integral ((id Z) ^ ),A = (ln . (upper_bound A)) - (ln . (lower_bound A))
then A2: ( (id Z) ^ is_integrable_on A & ((id Z) ^ ) | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: ln is_differentiable_on Z by A1, FDIFF_5:19;
A4: for x being Real st x in dom (ln `| Z) holds
(ln `| Z) . x = ((id Z) ^ ) . x

proof

let x be Real; :: thesis: ( x in dom (ln `| Z) implies (ln `| Z) . x = ((id Z) ^ ) . x )
assume x in dom (ln `| Z) ; :: thesis: (ln `| Z) . x = ((id Z) ^ ) . x
then A5: x in Z by A3, FDIFF_1:def 8;
then (ln `| Z) . x = 1 / x by A1, FDIFF_5:19
.= x " by XCMPLX_1:217
.= ((id Z) . x) " by A5, FUNCT_1:35
.= ((id Z) ^ ) . x by A1, A5, RFUNCT_1:def 8 ;
hence (ln `| Z) . x = ((id Z) ^ ) . x ; :: thesis: verum

end;

dom (ln `| Z) = dom ((id Z) ^ ) by A1, A3, FDIFF_1:def 8;
then ln `| Z = (id Z) ^ by A4, PARTFUN1:34;
hence integral ((id Z) ^ ),A = (ln . (upper_bound A)) - (ln . (lower_bound A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum