let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cosec * sin ) implies ( cosec * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * sin ) `| Z) . x = - (((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 )) ) ) )
assume A1: Z c= dom (cosec * sin ) ; :: thesis: ( cosec * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * sin ) `| Z) . x = - (((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 )) ) )
A2: for x being Real st x in Z holds
sin . (sin . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies sin . (sin . x) <> 0 )
assume x in Z ; :: thesis: sin . (sin . x) <> 0
then sin . x in dom cosec by A1, FUNCT_1:21;
hence sin . (sin . x) <> 0 by RFUNCT_1:13; :: thesis: verum

end;

A3: for x being Real st x in Z holds
cosec * sin is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cosec * sin is_differentiable_in x )
assume x in Z ; :: thesis: cosec * sin is_differentiable_in x
then sin . (sin . x) <> 0 by A2;
then ( sin is_differentiable_in x & cosec is_differentiable_in sin . x ) by Th2, SIN_COS:69;
hence cosec * sin is_differentiable_in x by FDIFF_2:13; :: thesis: verum

end;

then A4: cosec * sin is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((cosec * sin ) `| Z) . x = - (((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((cosec * sin ) `| Z) . x = - (((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 )) )
assume A5: x in Z ; :: thesis: ((cosec * sin ) `| Z) . x = - (((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 ))
then A6: sin . (sin . x) <> 0 by A2;
then ( sin is_differentiable_in x & cosec is_differentiable_in sin . x ) by Th2, SIN_COS:69;
then diff (cosec * sin ),x = (diff cosec ,(sin . x)) * (diff sin ,x) by FDIFF_2:13
.= (- ((cos . (sin . x)) / ((sin . (sin . x)) ^2 ))) * (diff sin ,x) by A6, Th2
.= (cos . x) * (- ((cos . (sin . x)) / ((sin . (sin . x)) ^2 ))) by SIN_COS:69 ;
hence ((cosec * sin ) `| Z) . x = - (((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 )) by A4, A5, FDIFF_1:def 8; :: thesis: verum

end;

hence ( cosec * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * sin ) `| Z) . x = - (((cos . x) * (cos . (sin . x))) / ((sin . (sin . x)) ^2 )) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum