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

proof

let x be Real; :: thesis: ( x in Z implies cosec is_differentiable_in x )
assume x in Z ; :: thesis: cosec is_differentiable_in x
then sin . x <> 0 by A3, RFUNCT_1:13;
hence cosec is_differentiable_in x by FDIFF_9:2; :: thesis: verum

end;

then A4: cosec is_differentiable_on Z by A3, FDIFF_1:16;
A5: arctan is_differentiable_on Z by A2, SIN_COS9:81;
for x being Real st x in Z holds
((cosec (#) arctan ) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2 ))) + (1 / ((sin . x) * (1 + (x ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((cosec (#) arctan ) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2 ))) + (1 / ((sin . x) * (1 + (x ^2 )))) )
assume A6: x in Z ; :: thesis: ((cosec (#) arctan ) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2 ))) + (1 / ((sin . x) * (1 + (x ^2 ))))
then A7: sin . x <> 0 by A3, RFUNCT_1:13;
((cosec (#) arctan ) `| Z) . x = ((arctan . x) * (diff cosec ,x)) + ((cosec . x) * (diff arctan ,x)) by A1, A4, A5, A6, FDIFF_1:29
.= ((arctan . x) * (- ((cos . x) / ((sin . x) ^2 )))) + ((cosec . x) * (diff arctan ,x)) by A7, FDIFF_9:2
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2 ))) + ((cosec . x) * ((arctan `| Z) . x)) by A5, A6, FDIFF_1:def 8
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2 ))) + ((cosec . x) * (1 / (1 + (x ^2 )))) by A2, A6, SIN_COS9:81
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2 ))) + ((1 / (sin . x)) * (1 / (1 + (x ^2 )))) by A3, A6, RFUNCT_1:def 8
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2 ))) + (1 / ((sin . x) * (1 + (x ^2 )))) by XCMPLX_1:103 ;
hence ((cosec (#) arctan ) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2 ))) + (1 / ((sin . x) * (1 + (x ^2 )))) ; :: thesis: verum

end;

hence ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan ) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2 ))) + (1 / ((sin . x) * (1 + (x ^2 )))) ) ) by A1, A4, A5, FDIFF_1:29; :: thesis: verum