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))) ) )

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

then A2: Z c= dom ln by A1, XBOOLE_1:1;

A3: for x being Real st x in Z holds

x > 0

diff (ln,x) = 1 / x

sin . (ln . x) <> 0

cot * ln is_differentiable_in x

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 A1, A6, FDIFF_1:9; :: 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))) ) )

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

then A2: Z c= dom ln by A1, XBOOLE_1:1;

A3: for x being Real st x in Z holds

x > 0

proof

A4:
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 A2, TAYLOR_1:18;

then ex g being Real st

( x = g & 0 < g ) by Lm1;

hence x > 0 ; :: thesis: verum

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

then x in right_open_halfline 0 by A2, TAYLOR_1:18;

then ex g being Real st

( x = g & 0 < g ) by Lm1;

hence x > 0 ; :: thesis: verum

diff (ln,x) = 1 / x

proof

A5:
for x being Real st x in Z holds
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 A3;

then x in right_open_halfline 0 by Lm1;

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 A3;

then x in right_open_halfline 0 by Lm1;

hence diff (ln,x) = 1 / x by TAYLOR_1:18; :: 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 Th2; :: 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 Th2; :: thesis: verum

cot * ln is_differentiable_in x

proof

then A9:
cot * ln is_differentiable_on Z
by A1, FDIFF_1:9;
let x be Real; :: thesis: ( x in Z implies cot * ln is_differentiable_in x )

assume A7: x in Z ; :: thesis: cot * ln is_differentiable_in x

then sin . (ln . x) <> 0 by A5;

then A8: cot is_differentiable_in ln . x by FDIFF_7:47;

ln is_differentiable_in x by A3, A7, TAYLOR_1:18;

hence cot * ln is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

end;assume A7: x in Z ; :: thesis: cot * ln is_differentiable_in x

then sin . (ln . x) <> 0 by A5;

then A8: cot is_differentiable_in ln . x by FDIFF_7:47;

ln is_differentiable_in x by A3, A7, TAYLOR_1:18;

hence cot * ln is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

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 A10: x in Z ; :: thesis: ((cot * ln) `| Z) . x = - (1 / (x * ((sin . (ln . x)) ^2)))

then A11: ln is_differentiable_in x by A3, TAYLOR_1:18;

A12: sin . (ln . x) <> 0 by A5, A10;

then cot is_differentiable_in ln . x by FDIFF_7:47;

then diff ((cot * ln),x) = (diff (cot,(ln . x))) * (diff (ln,x)) by A11, FDIFF_2:13

.= (- (1 / ((sin . (ln . x)) ^2))) * (diff (ln,x)) by A12, FDIFF_7:47

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

.= - ((1 / x) / ((sin . (ln . x)) ^2)) by A4, A10

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

hence ((cot * ln) `| Z) . x = - (1 / (x * ((sin . (ln . x)) ^2))) by A9, A10, FDIFF_1:def 7; :: thesis: verum

end;assume A10: x in Z ; :: thesis: ((cot * ln) `| Z) . x = - (1 / (x * ((sin . (ln . x)) ^2)))

then A11: ln is_differentiable_in x by A3, TAYLOR_1:18;

A12: sin . (ln . x) <> 0 by A5, A10;

then cot is_differentiable_in ln . x by FDIFF_7:47;

then diff ((cot * ln),x) = (diff (cot,(ln . x))) * (diff (ln,x)) by A11, FDIFF_2:13

.= (- (1 / ((sin . (ln . x)) ^2))) * (diff (ln,x)) by A12, FDIFF_7:47

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

.= - ((1 / x) / ((sin . (ln . x)) ^2)) by A4, A10

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

hence ((cot * ln) `| Z) . x = - (1 / (x * ((sin . (ln . x)) ^2))) by A9, A10, FDIFF_1:def 7; :: thesis: verum

((cot * ln) `| Z) . x = - (1 / (x * ((sin . (ln . x)) ^2))) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum