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

for x being Real st x in Z holds

((ln * (#Z n)) `| Z) . x = n / x

((ln * (#Z n)) `| Z) . x = n / x ) ) by A1, A3, FDIFF_1:9; :: thesis: verum

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

then A5:
ln * (#Z n) is_differentiable_on Z
by A1, FDIFF_1:9;
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: ln * (#Z n) is_differentiable_in x

then ( #Z n is_differentiable_in x & (#Z n) . x > 0 ) by A2, A4, PREPOWER:39, TAYLOR_1:2;

hence ln * (#Z n) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;A4: (#Z n) . x = x #Z n by TAYLOR_1:def 1;

assume x in Z ; :: thesis: ln * (#Z n) is_differentiable_in x

then ( #Z n is_differentiable_in x & (#Z n) . x > 0 ) by A2, A4, PREPOWER:39, TAYLOR_1:2;

hence ln * (#Z n) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

for x being Real st x in Z holds

((ln * (#Z n)) `| Z) . x = n / x

proof

hence
( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds
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 A2, A7, NEWTON:83;

A10: (#Z n) . x = x #Z n by TAYLOR_1:def 1;

then (#Z n) . x > 0 by A2, A7, PREPOWER:39;

then diff ((ln * (#Z n)),x) = (n * (x #Z (n - 1))) / (x #Z n) by A6, A10, TAYLOR_1:20

.= (n * (x #Z (n - 1))) / (x |^ n) by PREPOWER:36

.= (n * ((x |^ n) / (x |^ 1))) / (x |^ n) by A8, PREPOWER:43

.= 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 A9, XCMPLX_1:60

.= n * (1 / x)

.= (n * 1) / x by XCMPLX_1:74

.= n / x ;

hence ((ln * (#Z n)) `| Z) . x = n / x by A5, A7, FDIFF_1:def 7; :: thesis: verum

end;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 A2, A7, NEWTON:83;

A10: (#Z n) . x = x #Z n by TAYLOR_1:def 1;

then (#Z n) . x > 0 by A2, A7, PREPOWER:39;

then diff ((ln * (#Z n)),x) = (n * (x #Z (n - 1))) / (x #Z n) by A6, A10, TAYLOR_1:20

.= (n * (x #Z (n - 1))) / (x |^ n) by PREPOWER:36

.= (n * ((x |^ n) / (x |^ 1))) / (x |^ n) by A8, PREPOWER:43

.= 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 A9, XCMPLX_1:60

.= n * (1 / x)

.= (n * 1) / x by XCMPLX_1:74

.= n / x ;

hence ((ln * (#Z n)) `| Z) . x = n / x by A5, A7, FDIFF_1:def 7; :: thesis: verum

((ln * (#Z n)) `| Z) . x = n / x ) ) by A1, A3, FDIFF_1:9; :: thesis: verum