let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL , REAL st Z c= dom (ln * (f ^ )) & ( for x being Real st x in Z holds
f . x = x ) holds
( ln * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f ^ )) `| Z) . x = - (1 / x) ) )
let f be PartFunc of REAL , REAL ; :: thesis: ( Z c= dom (ln * (f ^ )) & ( for x being Real st x in Z holds
f . x = x ) implies ( ln * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f ^ )) `| Z) . x = - (1 / x) ) ) )
assume that
A1: Z c= dom (ln * (f ^ )) and
A2: for x being Real st x in Z holds
f . x = x ; :: thesis: ( ln * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f ^ )) `| Z) . x = - (1 / x) ) )
dom (ln * (f ^ )) c= dom (f ^ ) by RELAT_1:44;
then A3: Z c= dom (f ^ ) by A1, XBOOLE_1:1;
dom (f ^ ) c= dom f by RFUNCT_1:11;
then A4: Z c= dom f by A3, XBOOLE_1:1;
A5: for x being Real st x in Z holds
(f ^ ) . x > 0

proof

let x be Real; :: thesis: ( x in Z implies (f ^ ) . x > 0 )
assume A6: x in Z ; :: thesis: (f ^ ) . x > 0
then A7: (f ^ ) . x in right_open_halfline 0 by A1, FUNCT_1:21, TAYLOR_1:18;
A8: (f ^ ) . x = (f . x) " by A3, A6, RFUNCT_1:def 8
.= 1 / x by A2, A6 ;
then ex g being Real st
( 1 / x = g & 0 < g ) by A7, Lm1;
hence (f ^ ) . x > 0 by A8; :: thesis: verum

end;

A9: for x being Real st x in Z holds
f . x > 0

proof

let x be Real; :: thesis: ( x in Z implies f . x > 0 )
assume A10: x in Z ; :: thesis: f . x > 0
then (f ^ ) . x > 0 by A5;
then (f . x) " > 0 by A3, A10, RFUNCT_1:def 8;
hence f . x > 0 by XREAL_1:124; :: thesis: verum

end;

then A11: for x being Real st x in Z holds
( f . x = x & f . x <> 0 ) by A2;
then A12: ( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((f ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by A4, FDIFF_5:4;
A13: for x being Real st x in Z holds
( ln * (f ^ ) is_differentiable_in x & diff (ln * (f ^ )),x = (diff (f ^ ),x) / ((f ^ ) . x) )

proof

let x be Real; :: thesis: ( x in Z implies ( ln * (f ^ ) is_differentiable_in x & diff (ln * (f ^ )),x = (diff (f ^ ),x) / ((f ^ ) . x) ) )
assume A14: x in Z ; :: thesis: ( ln * (f ^ ) is_differentiable_in x & diff (ln * (f ^ )),x = (diff (f ^ ),x) / ((f ^ ) . x) )
then A15: f ^ is_differentiable_in x by A12, FDIFF_1:16;
(f ^ ) . x > 0 by A5, A14;
hence ( ln * (f ^ ) is_differentiable_in x & diff (ln * (f ^ )),x = (diff (f ^ ),x) / ((f ^ ) . x) ) by A15, TAYLOR_1:20; :: thesis: verum

end;

then A16: for x being Real st x in Z holds
ln * (f ^ ) is_differentiable_in x ;
then A17: ln * (f ^ ) is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((ln * (f ^ )) `| Z) . x = - (1 / x)

proof

let x be Real; :: thesis: ( x in Z implies ((ln * (f ^ )) `| Z) . x = - (1 / x) )
assume A18: x in Z ; :: thesis: ((ln * (f ^ )) `| Z) . x = - (1 / x)
then f . x <> 0 by A9;
then A19: x <> 0 by A2, A18;
diff (ln * (f ^ )),x = (diff (f ^ ),x) / ((f ^ ) . x) by A13, A18
.= (((f ^ ) `| Z) . x) / ((f ^ ) . x) by A12, A18, FDIFF_1:def 8
.= (((f ^ ) `| Z) . x) / ((f . x) " ) by A3, A18, RFUNCT_1:def 8
.= (((f ^ ) `| Z) . x) / (1 * (x " )) by A2, A18
.= (- (1 / (x ^2 ))) / (1 * (x " )) by A4, A11, A18, FDIFF_5:4
.= - (x / (x ^2 ))
.= - ((x / x) / x) by XCMPLX_1:79
.= - (1 / x) by A19, XCMPLX_1:60 ;
hence ((ln * (f ^ )) `| Z) . x = - (1 / x) by A17, A18, FDIFF_1:def 8; :: thesis: verum

end;

hence ( ln * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f ^ )) `| Z) . x = - (1 / x) ) ) by A1, A16, FDIFF_1:16; :: thesis: verum