let Z be open Subset of REAL; ( Z c= dom (ln (#) cosec) implies ( ln (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) cosec) `| Z) . x = ((1 / (sin . x)) / x) - (((ln . x) * (cos . x)) / ((sin . x) ^2)) ) ) )
assume A1:
Z c= dom (ln (#) cosec)
; ( ln (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) cosec) `| Z) . x = ((1 / (sin . x)) / x) - (((ln . x) * (cos . x)) / ((sin . x) ^2)) ) )
then A2:
Z c= (dom ln) /\ (dom cosec)
by VALUED_1:def 4;
then A3:
Z c= dom cosec
by XBOOLE_1:18;
A4:
for x being Real st x in Z holds
( cosec is_differentiable_in x & diff (cosec,x) = - ((cos . x) / ((sin . x) ^2)) )
then
for x being Real st x in Z holds
cosec is_differentiable_in x
;
then A5:
cosec is_differentiable_on Z
by A3, FDIFF_1:9;
A6:
Z c= dom ln
by A2, XBOOLE_1:18;
A7:
for x being Real st x in Z holds
x > 0
then
for x being Real st x in Z holds
ln is_differentiable_in x
by TAYLOR_1:18;
then A8:
ln is_differentiable_on Z
by A6, FDIFF_1:9;
A9:
for x being Real st x in Z holds
diff (ln,x) = 1 / x
for x being Real st x in Z holds
((ln (#) cosec) `| Z) . x = ((1 / (sin . x)) / x) - (((ln . x) * (cos . x)) / ((sin . x) ^2))
proof
let x be
Real;
( x in Z implies ((ln (#) cosec) `| Z) . x = ((1 / (sin . x)) / x) - (((ln . x) * (cos . x)) / ((sin . x) ^2)) )
assume A10:
x in Z
;
((ln (#) cosec) `| Z) . x = ((1 / (sin . x)) / x) - (((ln . x) * (cos . x)) / ((sin . x) ^2))
then ((ln (#) cosec) `| Z) . x =
((cosec . x) * (diff (ln,x))) + ((ln . x) * (diff (cosec,x)))
by A1, A8, A5, FDIFF_1:21
.=
((cosec . x) * (1 / x)) + ((ln . x) * (diff (cosec,x)))
by A9, A10
.=
((cosec . x) * (1 / x)) + ((ln . x) * (- ((cos . x) / ((sin . x) ^2))))
by A4, A10
.=
((1 / (sin . x)) / x) - (((ln . x) * (cos . x)) / ((sin . x) ^2))
by A3, A10, RFUNCT_1:def 2
;
hence
((ln (#) cosec) `| Z) . x = ((1 / (sin . x)) / x) - (((ln . x) * (cos . x)) / ((sin . x) ^2))
;
verum
end;
hence
( ln (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) cosec) `| Z) . x = ((1 / (sin . x)) / x) - (((ln . x) * (cos . x)) / ((sin . x) ^2)) ) )
by A1, A8, A5, FDIFF_1:21; verum