let Z be open Subset of REAL; :: thesis: ( Z c= dom (cot * ln) implies ( - (cot * ln) is_differentiable_on Z & ( for x being Real st x in Z holds

((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) ) ) )

assume A1: Z c= dom (cot * ln) ; :: thesis: ( - (cot * ln) is_differentiable_on Z & ( for x being Real st x in Z holds

((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) ) )

then A2: Z c= dom (- (cot * ln)) by VALUED_1:8;

dom (cot * ln) c= dom ln by RELAT_1:25;

then A3: Z c= dom ln by A1;

A4: for x being Real st x in Z holds

x > 0

sin . (ln . x) <> 0

diff (ln,x) = 1 / x

then A8: (- 1) (#) (cot * ln) is_differentiable_on Z by A2, FDIFF_1:20;

for x being Real st x in Z holds

((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2))

((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) ) ) by A2, A7, FDIFF_1:20; :: thesis: verum

((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) ) ) )

assume A1: Z c= dom (cot * ln) ; :: thesis: ( - (cot * ln) is_differentiable_on Z & ( for x being Real st x in Z holds

((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) ) )

then A2: Z c= dom (- (cot * ln)) by VALUED_1:8;

dom (cot * ln) c= dom ln by RELAT_1:25;

then A3: Z c= dom ln by A1;

A4: for x being Real st x in Z holds

x > 0

proof

A5:
for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies x > 0 )

assume x in Z ; :: thesis: x > 0

then x in right_open_halfline 0 by A3, TAYLOR_1:18;

then ex g being Real st

( x = g & 0 < g ) by Lm2;

hence x > 0 ; :: thesis: verum

end;assume x in Z ; :: thesis: x > 0

then x in right_open_halfline 0 by A3, TAYLOR_1:18;

then ex g being Real st

( x = g & 0 < g ) by Lm2;

hence x > 0 ; :: thesis: verum

sin . (ln . x) <> 0

proof

A6:
for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies sin . (ln . x) <> 0 )

assume x in Z ; :: thesis: sin . (ln . x) <> 0

then ln . x in dom (cos / sin) by A1, FUNCT_1:11;

hence sin . (ln . x) <> 0 by FDIFF_8:2; :: thesis: verum

end;assume x in Z ; :: thesis: sin . (ln . x) <> 0

then ln . x in dom (cos / sin) by A1, FUNCT_1:11;

hence sin . (ln . x) <> 0 by FDIFF_8:2; :: thesis: verum

diff (ln,x) = 1 / x

proof

A7:
cot * ln is_differentiable_on Z
by A1, FDIFF_8:15;
let x be Real; :: thesis: ( x in Z implies diff (ln,x) = 1 / x )

assume x in Z ; :: thesis: diff (ln,x) = 1 / x

then x > 0 by A4;

then x in right_open_halfline 0 by Lm2;

hence diff (ln,x) = 1 / x by TAYLOR_1:18; :: thesis: verum

end;assume x in Z ; :: thesis: diff (ln,x) = 1 / x

then x > 0 by A4;

then x in right_open_halfline 0 by Lm2;

hence diff (ln,x) = 1 / x by TAYLOR_1:18; :: thesis: verum

then A8: (- 1) (#) (cot * ln) is_differentiable_on Z by A2, FDIFF_1:20;

for x being Real st x in Z holds

((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2))

proof

hence
( - (cot * ln) is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) )

assume A9: x in Z ; :: thesis: ((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2))

then A10: ln is_differentiable_in x by A4, TAYLOR_1:18;

A11: ( x > 0 & sin . (ln . x) <> 0 ) by A4, A5, A9;

then A12: cot is_differentiable_in ln . x by FDIFF_7:47;

A13: cot * ln is_differentiable_in x by A7, A9, FDIFF_1:9;

((- (cot * ln)) `| Z) . x = diff ((- (cot * ln)),x) by A8, A9, FDIFF_1:def 7

.= (- 1) * (diff ((cot * ln),x)) by A13, FDIFF_1:15

.= (- 1) * ((diff (cot,(ln . x))) * (diff (ln,x))) by A10, A12, FDIFF_2:13

.= (- 1) * ((- (1 / ((sin . (ln . x)) ^2))) * (diff (ln,x))) by A11, FDIFF_7:47

.= (- 1) * (- ((diff (ln,x)) / ((sin . (ln . x)) ^2)))

.= (- 1) * (- ((1 / x) / ((sin . (ln . x)) ^2))) by A6, A9

.= 1 / (x * ((sin . (ln . x)) ^2)) by XCMPLX_1:78 ;

hence ((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) ; :: thesis: verum

end;assume A9: x in Z ; :: thesis: ((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2))

then A10: ln is_differentiable_in x by A4, TAYLOR_1:18;

A11: ( x > 0 & sin . (ln . x) <> 0 ) by A4, A5, A9;

then A12: cot is_differentiable_in ln . x by FDIFF_7:47;

A13: cot * ln is_differentiable_in x by A7, A9, FDIFF_1:9;

((- (cot * ln)) `| Z) . x = diff ((- (cot * ln)),x) by A8, A9, FDIFF_1:def 7

.= (- 1) * (diff ((cot * ln),x)) by A13, FDIFF_1:15

.= (- 1) * ((diff (cot,(ln . x))) * (diff (ln,x))) by A10, A12, FDIFF_2:13

.= (- 1) * ((- (1 / ((sin . (ln . x)) ^2))) * (diff (ln,x))) by A11, FDIFF_7:47

.= (- 1) * (- ((diff (ln,x)) / ((sin . (ln . x)) ^2)))

.= (- 1) * (- ((1 / x) / ((sin . (ln . x)) ^2))) by A6, A9

.= 1 / (x * ((sin . (ln . x)) ^2)) by XCMPLX_1:78 ;

hence ((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) ; :: thesis: verum

((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) ) ) by A2, A7, FDIFF_1:20; :: thesis: verum