let Z be open Subset of REAL; :: thesis: ( Z c= dom (ln * sin) & ( for x being Real st x in Z holds
sin . x > 0 ) implies ( ln * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * sin) `| Z) . x = cot x ) ) )
assume that
A1: Z c= dom (ln * sin) and
A2: for x being Real st x in Z holds
sin . x > 0 ; :: thesis: ( ln * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * sin) `| Z) . x = cot x ) )
A3: sin is_differentiable_on Z by FDIFF_1:26, SIN_COS:68;
A4: for x being Real st x in Z holds
ln * sin is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * sin is_differentiable_in x )
assume x in Z ; :: thesis: ln * sin is_differentiable_in x
then ( sin is_differentiable_in x & sin . x > 0 ) by A2, A3, FDIFF_1:9;
hence ln * sin is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;

then A5: ln * sin is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * sin) `| Z) . x = cot x

proof

let x be Real; :: thesis: ( x in Z implies ((ln * sin) `| Z) . x = cot x )
assume A6: x in Z ; :: thesis: ((ln * sin) `| Z) . x = cot x
then ( sin is_differentiable_in x & sin . x > 0 ) by A2, A3, FDIFF_1:9;
then diff ((ln * sin),x) = (diff (sin,x)) / (sin . x) by TAYLOR_1:20
.= (cos . x) / (sin . x) by SIN_COS:64
.= (cos x) / (sin . x) by SIN_COS:def 19
.= (cos x) / (sin x) by SIN_COS:def 17
.= cot x by SIN_COS4:def 2 ;
hence ((ln * sin) `| Z) . x = cot x by A5, A6, FDIFF_1:def 7; :: thesis: verum

end;

hence ( ln * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * sin) `| Z) . x = cot x ) ) by A1, A4, FDIFF_1:9; :: thesis: verum