let A be non empty closed_interval Subset of REAL; :: thesis: for Z being open Subset of REAL st A c= Z & 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 & 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 that
A1: A c= Z and
A2: dom ln = Z and
A3: Z = dom ((id Z) ^) and
A4: ((id Z) ^) | A is continuous ; :: thesis: integral (((id Z) ^),A) = (ln . (upper_bound A)) - (ln . (lower_bound A))
A5: (id Z) ^ is_integrable_on A by A1, A3, A4, INTEGRA5:11;
A6: ln is_differentiable_on Z by A2, FDIFF_5:19;
A7: for x being Element of REAL st x in dom (ln `| Z) holds
(ln `| Z) . x = ((id Z) ^) . x

proof

let x be Element of 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 A8: x in Z by A6, FDIFF_1:def 7;
then (ln `| Z) . x = 1 / x by A2, FDIFF_5:19
.= x " by XCMPLX_1:215
.= ((id Z) . x) " by A8, FUNCT_1:18
.= ((id Z) ^) . x by A3, A8, RFUNCT_1:def 2 ;
hence (ln `| Z) . x = ((id Z) ^) . x ; :: thesis: verum

end;

dom (ln `| Z) = dom ((id Z) ^) by A3, A6, FDIFF_1:def 7;
then ln `| Z = (id Z) ^ by A7, PARTFUN1:5;
hence integral (((id Z) ^),A) = (ln . (upper_bound A)) - (ln . (lower_bound A)) by A1, A3, A4, A5, A6, INTEGRA5:10, INTEGRA5:13; :: thesis: verum