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_1:11;
then A3: Z c= dom ln by A1, XBOOLE_1:1;
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_1:16;
((ln ^ ) `| Z) . x = diff (ln ^ ),x by A5, A6, FDIFF_1:def 8
.= - ((diff ln ,x) / ((ln . x) ^2 )) by A7, FDIFF_2:15
.= - (((ln `| Z) . x) / ((ln . x) ^2 )) by A4, A6, FDIFF_1:def 8
.= - ((1 / x) / ((ln . x) ^2 )) by A3, A6, Th19
.= - (1 / (x * ((ln . x) ^2 ))) by XCMPLX_1:79 ;
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