let Z be open Subset of REAL; ( 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)
; ( - (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
A5:
for x being Real st x in Z holds
sin . (ln . x) <> 0
A6:
for x being Real st x in Z holds
diff (ln,x) = 1 / x
A7:
cot * ln is_differentiable_on Z
by A1, FDIFF_8:15;
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
let x be
Real;
( x in Z implies ((- (cot * ln)) `| Z) . x = 1 / (x * ((sin . (ln . x)) ^2)) )
assume A9:
x in Z
;
((- (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))
;
verum
end;
hence
( - (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)) ) )
by A2, A7, FDIFF_1:20; verum