let Z be open Subset of REAL ; :: thesis: ( Z c= dom (ln (#) sin ) implies ( ln (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) sin ) `| Z) . x = ((sin . x) / x) + ((ln . x) * (cos . x)) ) ) )
assume A1: Z c= dom (ln (#) sin ) ; :: thesis: ( ln (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) sin ) `| Z) . x = ((sin . x) / x) + ((ln . x) * (cos . x)) ) )
then A2: Z c= (dom ln ) /\ (dom sin ) by VALUED_1:def 4;
then A3: Z c= dom ln by XBOOLE_1:18;
A4: Z c= dom sin by A2, XBOOLE_1:18;
A5: for x being Real st x in Z holds
x > 0

proof

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 x in { g where g is Real : 0 < g } by XXREAL_1:230;
then ex g being Real st
( x = g & 0 < g ) ;
hence x > 0 ; :: thesis: verum

end;

for x being Real st x in Z holds
ln is_differentiable_in x by A5, TAYLOR_1:18;
then A6: ln is_differentiable_on Z by A3, FDIFF_1:16;
A7: for x being Real st x in Z holds
diff ln ,x = 1 / x

proof

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 A5;
then x in { g where g is Real : 0 < g } ;
then x in right_open_halfline 0 by XXREAL_1:230;
hence diff ln ,x = 1 / x by TAYLOR_1:18; :: thesis: verum

end;

for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:69;
then A8: sin is_differentiable_on Z by A4, FDIFF_1:16;
for x being Real st x in Z holds
((ln (#) sin ) `| Z) . x = ((sin . x) / x) + ((ln . x) * (cos . x))

proof

let x be Real; :: thesis: ( x in Z implies ((ln (#) sin ) `| Z) . x = ((sin . x) / x) + ((ln . x) * (cos . x)) )
assume A9: x in Z ; :: thesis: ((ln (#) sin ) `| Z) . x = ((sin . x) / x) + ((ln . x) * (cos . x))
then ((ln (#) sin ) `| Z) . x = ((sin . x) * (diff ln ,x)) + ((ln . x) * (diff sin ,x)) by A1, A6, A8, FDIFF_1:29
.= ((sin . x) * (1 / x)) + ((ln . x) * (diff sin ,x)) by A7, A9
.= ((sin . x) / x) + ((ln . x) * (cos . x)) by SIN_COS:69 ;
hence ((ln (#) sin ) `| Z) . x = ((sin . x) / x) + ((ln . x) * (cos . x)) ; :: thesis: verum

end;

hence ( ln (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln (#) sin ) `| Z) . x = ((sin . x) / x) + ((ln . x) * (cos . x)) ) ) by A1, A6, A8, FDIFF_1:29; :: thesis: verum