let Z be open Subset of REAL; :: thesis: ( Z c= dom ((id Z) (#) ln) implies ( (id Z) (#) ln is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) ln) `| Z) . x = 1 + (ln . x) ) ) )
set f = ln ;
assume A1: Z c= dom ((id Z) (#) ln) ; :: thesis: ( (id Z) (#) ln is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) ln) `| Z) . x = 1 + (ln . x) ) )
then A2: Z c= (dom (id Z)) /\ (dom ln) by VALUED_1:def 4;
then A3: Z c= dom (id Z) by XBOOLE_1:18;
A4: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
then A5: id Z is_differentiable_on Z by A3, FDIFF_1:23;
A6: Z c= dom ln by A2, XBOOLE_1:18;
then A7: ln is_differentiable_on Z by Th19;
for x being Real st x in Z holds
(((id Z) (#) ln) `| Z) . x = 1 + (ln . x)

proof

let x be Real; :: thesis: ( x in Z implies (((id Z) (#) ln) `| Z) . x = 1 + (ln . x) )
assume A8: x in Z ; :: thesis: (((id Z) (#) ln) `| Z) . x = 1 + (ln . x)
then A9: x <> 0 by A6, TAYLOR_1:18, XXREAL_1:4;
(((id Z) (#) ln) `| Z) . x = (((id Z) . x) * (diff (ln,x))) + ((ln . x) * (diff ((id Z),x))) by A1, A5, A7, A8, FDIFF_1:21
.= (((id Z) . x) * ((ln `| Z) . x)) + ((ln . x) * (diff ((id Z),x))) by A7, A8, FDIFF_1:def 7
.= (((id Z) . x) * (1 / x)) + ((ln . x) * (diff ((id Z),x))) by A6, A8, Th19
.= (x * (1 / x)) + ((ln . x) * (diff ((id Z),x))) by A8, FUNCT_1:18
.= (x * (1 / x)) + ((ln . x) * (((id Z) `| Z) . x)) by A5, A8, FDIFF_1:def 7
.= (x * (1 / x)) + ((ln . x) * 1) by A3, A4, A8, FDIFF_1:23
.= 1 + (ln . x) by A9, XCMPLX_1:106 ;
hence (((id Z) (#) ln) `| Z) . x = 1 + (ln . x) ; :: thesis: verum

end;

hence ( (id Z) (#) ln is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) ln) `| Z) . x = 1 + (ln . x) ) ) by A1, A5, A7, FDIFF_1:21; :: thesis: verum