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 & arctan is_differentiable_in x ) by A2, A3, FDIFF_1:9;
((arctan ^) `| Z) . x = diff ((arctan ^),x) by A4, A5, FDIFF_1:def 7
.= - ((diff (arctan,x)) / ((arctan . x) ^2)) by A6, FDIFF_2:15
.= - (((arctan `| Z) . x) / ((arctan . x) ^2)) by A3, A5, FDIFF_1:def 7
.= - ((1 / (1 + (x ^2))) / ((arctan . x) ^2)) by A1, A5, SIN_COS9:81
.= - (1 / (((arctan . x) ^2) * (1 + (x ^2)))) by XCMPLX_1:78 ;
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