let Z be open Subset of REAL ; :: thesis: ( Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x <> 0 ) implies ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan ^ ) `| Z) . x = - (1 / (((arctan . x) ^2 ) * (1 + (x ^2 )))) ) ) )
assume that
A1: Z c= ].(- 1),1.[ and
A2: for x being Real st x in Z holds
arctan . x <> 0 ; :: thesis: ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan ^ ) `| Z) . x = - (1 / (((arctan . x) ^2 ) * (1 + (x ^2 )))) ) )
A3: arctan is_differentiable_on Z by A1, SIN_COS9:81;
then A4: arctan ^ is_differentiable_on Z by A2, FDIFF_2:22;
for x being Real st x in Z holds
((arctan ^ ) `| Z) . x = - (1 / (((arctan . x) ^2 ) * (1 + (x ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((arctan ^ ) `| Z) . x = - (1 / (((arctan . x) ^2 ) * (1 + (x ^2 )))) )
assume A5: x in Z ; :: thesis: ((arctan ^ ) `| Z) . x = - (1 / (((arctan . x) ^2 ) * (1 + (x ^2 ))))
then A6: arctan . x <> 0 by A2;
A7: arctan is_differentiable_in x by A3, A5, FDIFF_1:16;
((arctan ^ ) `| Z) . x = diff (arctan ^ ),x by A4, A5, FDIFF_1:def 8
.= - ((diff arctan ,x) / ((arctan . x) ^2 )) by A6, A7, FDIFF_2:15
.= - (((arctan `| Z) . x) / ((arctan . x) ^2 )) by A3, A5, FDIFF_1:def 8
.= - ((1 / (1 + (x ^2 ))) / ((arctan . x) ^2 )) by A1, A5, SIN_COS9:81
.= - (1 / (((arctan . x) ^2 ) * (1 + (x ^2 )))) by XCMPLX_1:79 ;
hence ((arctan ^ ) `| Z) . x = - (1 / (((arctan . x) ^2 ) * (1 + (x ^2 )))) ; :: thesis: verum

end;

hence ( arctan ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan ^ ) `| Z) . x = - (1 / (((arctan . x) ^2 ) * (1 + (x ^2 )))) ) ) by A2, A3, FDIFF_2:22; :: thesis: verum