let Z be open Subset of REAL ; :: thesis: ( Z c= dom (sin * cosec ) implies ( sin * cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * cosec ) `| Z) . x = - (((cos . (cosec . x)) * (cos . x)) / ((sin . x) ^2 )) ) ) )
assume A1: Z c= dom (sin * cosec ) ; :: thesis: ( sin * cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * cosec ) `| Z) . x = - (((cos . (cosec . x)) * (cos . x)) / ((sin . x) ^2 )) ) )
dom (sin * cosec ) c= dom cosec by RELAT_1:44;
then A2: Z c= dom cosec by A1, XBOOLE_1:1;
A3: for x being Real st x in Z holds
sin * cosec is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sin * cosec is_differentiable_in x )
assume x in Z ; :: thesis: sin * cosec is_differentiable_in x
then sin . x <> 0 by A2, RFUNCT_1:13;
then A4: cosec is_differentiable_in x by FDIFF_9:2;
sin is_differentiable_in cosec . x by SIN_COS:69;
hence sin * cosec is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

then A5: sin * cosec is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((sin * cosec ) `| Z) . x = - (((cos . (cosec . x)) * (cos . x)) / ((sin . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((sin * cosec ) `| Z) . x = - (((cos . (cosec . x)) * (cos . x)) / ((sin . x) ^2 )) )
assume A6: x in Z ; :: thesis: ((sin * cosec ) `| Z) . x = - (((cos . (cosec . x)) * (cos . x)) / ((sin . x) ^2 ))
then A7: sin . x <> 0 by A2, RFUNCT_1:13;
then A8: cosec is_differentiable_in x by FDIFF_9:2;
sin is_differentiable_in cosec . x by SIN_COS:69;
then diff (sin * cosec ),x = (diff sin ,(cosec . x)) * (diff cosec ,x) by A8, FDIFF_2:13
.= (cos (cosec . x)) * (diff cosec ,x) by SIN_COS:69
.= (cos (cosec . x)) * (- ((cos . x) / ((sin . x) ^2 ))) by A7, FDIFF_9:2
.= - (((cos . (cosec . x)) * (cos . x)) / ((sin . x) ^2 )) ;
hence ((sin * cosec ) `| Z) . x = - (((cos . (cosec . x)) * (cos . x)) / ((sin . x) ^2 )) by A5, A6, FDIFF_1:def 8; :: thesis: verum

end;

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