let Z be open Subset of REAL; :: thesis: ( Z c= dom (ln ^) & ( for x being Real st x in Z holds
ln . x <> 0 ) implies ( ln ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) ) ) )
set f = ln ;
assume that
A1: Z c= dom (ln ^) and
A2: for x being Real st x in Z holds
ln . x <> 0 ; :: thesis: ( ln ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) ) )
dom (ln ^) c= dom ln by RFUNCT_1:1;
then A3: Z c= dom ln by A1;
then A4: ln is_differentiable_on Z by Th19;
then A5: ln ^ is_differentiable_on Z by A2, FDIFF_2:22;
for x being Real st x in Z holds
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2)))

proof

let x be Real; :: thesis: ( x in Z implies ((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) )
assume A6: x in Z ; :: thesis: ((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2)))
then A7: ( ln . x <> 0 & ln is_differentiable_in x ) by A2, A4, FDIFF_1:9;
((ln ^) `| Z) . x = diff ((ln ^),x) by A5, A6, FDIFF_1:def 7
.= - ((diff (ln,x)) / ((ln . x) ^2)) by A7, FDIFF_2:15
.= - (((ln `| Z) . x) / ((ln . x) ^2)) by A4, A6, FDIFF_1:def 7
.= - ((1 / x) / ((ln . x) ^2)) by A3, A6, Th19
.= - (1 / (x * ((ln . x) ^2))) by XCMPLX_1:78 ;
hence ((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) ; :: thesis: verum

end;

hence ( ln ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((ln ^) `| Z) . x = - (1 / (x * ((ln . x) ^2))) ) ) by A2, A4, FDIFF_2:22; :: thesis: verum