let Z be open Subset of REAL; :: thesis: ( Z c= dom (ln * ln) & ( for x being Real st x in Z holds
x > 0 ) implies ( ln * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * ln) `| Z) . x = 1 / ((ln . x) * x) ) ) )
assume that
A1: Z c= dom (ln * ln) and
A2: for x being Real st x in Z holds
x > 0 ; :: thesis: ( ln * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * ln) `| Z) . x = 1 / ((ln . x) * x) ) )
A3: for x being Real st x in Z holds
ln . x > 0

proof

let x be Real; :: thesis: ( x in Z implies ln . x > 0 )
assume x in Z ; :: thesis: ln . x > 0
then A4: ln . x in right_open_halfline 0 by A1, FUNCT_1:11, TAYLOR_1:18;
].0,+infty.[ = { g where g is Real : 0 < g } by XXREAL_1:230;
then ex g being Real st
( ln . x = g & 0 < g ) by A4;
hence ln . x > 0 ; :: thesis: verum

end;

A5: for x being Real st x in Z holds
ln * ln is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * ln is_differentiable_in x )
assume A6: x in Z ; :: thesis: ln * ln is_differentiable_in x
then A7: ln . x > 0 by A3;
ln is_differentiable_in x by A2, A6, TAYLOR_1:18;
hence ln * ln is_differentiable_in x by A7, TAYLOR_1:20; :: thesis: verum

end;

then A8: ln * ln is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * ln) `| Z) . x = 1 / ((ln . x) * x)

proof

let x be Real; :: thesis: ( x in Z implies ((ln * ln) `| Z) . x = 1 / ((ln . x) * x) )
A9: ].0,+infty.[ = { g where g is Real : 0 < g } by XXREAL_1:230;
assume A10: x in Z ; :: thesis: ((ln * ln) `| Z) . x = 1 / ((ln . x) * x)
then A11: ln . x > 0 by A3;
x > 0 by A2, A10;
then A12: x in right_open_halfline 0 by A9;
ln is_differentiable_in x by A2, A10, TAYLOR_1:18;
then diff ((ln * ln),x) = (diff (ln,x)) / (ln . x) by A11, TAYLOR_1:20
.= (1 / (ln . x)) * (1 / x) by A12, TAYLOR_1:18
.= 1 / ((ln . x) * x) by XCMPLX_1:102 ;
hence ((ln * ln) `| Z) . x = 1 / ((ln . x) * x) by A8, A10, FDIFF_1:def 7; :: thesis: verum

end;

hence ( ln * ln is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * ln) `| Z) . x = 1 / ((ln . x) * x) ) ) by A1, A5, FDIFF_1:9; :: thesis: verum