let A be non empty closed_interval Subset of REAL; 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; ( 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
; 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
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; verum