let Z be open Subset of REAL ; :: thesis: ( Z c= dom (tan (#) cosec ) implies ( tan (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) cosec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (sin . x)) - (((tan . x) * (cos . x)) / ((sin . x) ^2 )) ) ) )
assume A1: Z c= dom (tan (#) cosec ) ; :: thesis: ( tan (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) cosec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (sin . x)) - (((tan . x) * (cos . x)) / ((sin . x) ^2 )) ) )
then A2: Z c= (dom tan ) /\ (dom cosec ) by VALUED_1:def 4;
then A3: Z c= dom tan by XBOOLE_1:18;
for x being Real st x in Z holds
tan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies tan is_differentiable_in x )
assume x in Z ; :: thesis: tan is_differentiable_in x
then cos . x <> 0 by A3, FDIFF_8:1;
hence tan is_differentiable_in x by FDIFF_7:46; :: thesis: verum

end;

then A4: tan is_differentiable_on Z by A3, FDIFF_1:16;
A5: Z c= dom cosec by A2, XBOOLE_1:18;
A6: for x being Real st x in Z holds
( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) )

proof

let x be Real; :: thesis: ( x in Z implies ( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) ) )
assume x in Z ; :: thesis: ( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) )
then sin . x <> 0 by A5, RFUNCT_1:13;
hence ( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) ) by Th2; :: thesis: verum

end;

then for x being Real st x in Z holds
cosec is_differentiable_in x ;
then A7: cosec is_differentiable_on Z by A5, FDIFF_1:16;
for x being Real st x in Z holds
((tan (#) cosec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (sin . x)) - (((tan . x) * (cos . x)) / ((sin . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((tan (#) cosec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (sin . x)) - (((tan . x) * (cos . x)) / ((sin . x) ^2 )) )
assume A8: x in Z ; :: thesis: ((tan (#) cosec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (sin . x)) - (((tan . x) * (cos . x)) / ((sin . x) ^2 ))
then A9: cos . x <> 0 by A3, FDIFF_8:1;
((tan (#) cosec ) `| Z) . x = ((cosec . x) * (diff tan ,x)) + ((tan . x) * (diff cosec ,x)) by A1, A4, A7, A8, FDIFF_1:29
.= ((cosec . x) * (1 / ((cos . x) ^2 ))) + ((tan . x) * (diff cosec ,x)) by A9, FDIFF_7:46
.= ((cosec . x) * (1 / ((cos . x) ^2 ))) + ((tan . x) * (- ((cos . x) / ((sin . x) ^2 )))) by A6, A8
.= ((1 / ((cos . x) ^2 )) / (sin . x)) + ((tan . x) * (- ((cos . x) / ((sin . x) ^2 )))) by A5, A8, RFUNCT_1:def 8 ;
hence ((tan (#) cosec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (sin . x)) - (((tan . x) * (cos . x)) / ((sin . x) ^2 )) ; :: thesis: verum

end;

hence ( tan (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan (#) cosec ) `| Z) . x = ((1 / ((cos . x) ^2 )) / (sin . x)) - (((tan . x) * (cos . x)) / ((sin . x) ^2 )) ) ) by A1, A4, A7, FDIFF_1:29; :: thesis: verum