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 . (arctan . x)) / (((sin . (arctan . x)) ^2 ) * (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 . (arctan . x)) / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 )))) ) )
A3: for x being Real st x in Z holds
sin . (arctan . x) <> 0

proof

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

end;

A4: for x being Real st x in Z holds
cosec * arctan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cosec * arctan is_differentiable_in x )
assume A5: x in Z ; :: thesis: cosec * arctan is_differentiable_in x
then sin . (arctan . x) <> 0 by A3;
then A6: cosec is_differentiable_in arctan . x by FDIFF_9:2;
arctan is_differentiable_on Z by A2, SIN_COS9:81;
then arctan is_differentiable_in x by A5, FDIFF_1:16;
hence cosec * arctan is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: cosec * arctan is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((cosec * arctan ) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((cosec * arctan ) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 )))) )
assume A8: x in Z ; :: thesis: ((cosec * arctan ) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 ))))
then A9: sin . (arctan . x) <> 0 by A3;
sin . (arctan . x) <> 0 by A3, A8;
then A10: cosec is_differentiable_in arctan . x by FDIFF_9:2;
A11: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A12: arctan is_differentiable_in x by A8, FDIFF_1:16;
((cosec * arctan ) `| Z) . x = diff (cosec * arctan ),x by A7, A8, FDIFF_1:def 8
.= (diff cosec ,(arctan . x)) * (diff arctan ,x) by A12, A10, FDIFF_2:13
.= (- ((cos . (arctan . x)) / ((sin . (arctan . x)) ^2 ))) * (diff arctan ,x) by A9, FDIFF_9:2
.= - (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2 )) * (diff arctan ,x))
.= - (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2 )) * ((arctan `| Z) . x)) by A8, A11, FDIFF_1:def 8
.= - (((cos . (arctan . x)) / ((sin . (arctan . x)) ^2 )) * (1 / (1 + (x ^2 )))) by A2, A8, SIN_COS9:81
.= - (((cos . (arctan . x)) * 1) / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 )))) by XCMPLX_1:77
.= - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 )))) ;
hence ((cosec * arctan ) `| Z) . x = - ((cos . (arctan . x)) / (((sin . (arctan . x)) ^2 ) * (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 . (arctan . x)) / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 )))) ) ) by A1, A4, FDIFF_1:16; :: thesis: verum