let n be Element of NAT ; :: thesis: for Z being open Subset of REAL st Z c= dom (ln * (#Z n)) & ( for x being Real st x in Z holds
x > 0 ) holds
( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (#Z n)) `| Z) . x = n / x ) )

let Z be open Subset of REAL; :: thesis: ( Z c= dom (ln * (#Z n)) & ( for x being Real st x in Z holds
x > 0 ) implies ( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (#Z n)) `| Z) . x = n / x ) ) )

assume that
A1: Z c= dom (ln * (#Z n)) and
A2: for x being Real st x in Z holds
x > 0 ; :: thesis: ( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (#Z n)) `| Z) . x = n / x ) )

A3: for x being Real st x in Z holds
ln * (#Z n) is_differentiable_in x
proof
let x be Real; :: thesis: ( x in Z implies ln * (#Z n) is_differentiable_in x )
A4: (#Z n) . x = x #Z n by TAYLOR_1:def 1;
assume x in Z ; :: thesis:
then ( #Z n is_differentiable_in x & (#Z n) . x > 0 ) by ;
hence ln * (#Z n) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum
end;
then A5: ln * (#Z n) is_differentiable_on Z by ;
for x being Real st x in Z holds
((ln * (#Z n)) `| Z) . x = n / x
proof
let x be Real; :: thesis: ( x in Z implies ((ln * (#Z n)) `| Z) . x = n / x )
A6: ( #Z n is_differentiable_in x & diff ((#Z n),x) = n * (x #Z (n - 1)) ) by TAYLOR_1:2;
assume A7: x in Z ; :: thesis: ((ln * (#Z n)) `| Z) . x = n / x
then A8: x > 0 by A2;
A9: x |^ n > 0 by ;
A10: (#Z n) . x = x #Z n by TAYLOR_1:def 1;
then (#Z n) . x > 0 by ;
then diff ((ln * (#Z n)),x) = (n * (x #Z (n - 1))) / (x #Z n) by
.= (n * (x #Z (n - 1))) / (x |^ n) by PREPOWER:36
.= (n * ((x |^ n) / (x |^ 1))) / (x |^ n) by
.= n * (((x |^ n) / (x |^ 1)) / (x |^ n)) by XCMPLX_1:74
.= n * (((x |^ n) / (x |^ n)) / (x |^ 1)) by XCMPLX_1:48
.= n * (1 / (x |^ 1)) by
.= n * (1 / x)
.= (n * 1) / x by XCMPLX_1:74
.= n / x ;
hence ((ln * (#Z n)) `| Z) . x = n / x by ; :: thesis: verum
end;
hence ( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (#Z n)) `| Z) . x = n / x ) ) by ; :: thesis: verum