let Z be open Subset of REAL ; :: thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * arctan )) & Z c= ].(- 1),1.[ implies ( (1 / 2) (#) ((#Z 2) * arctan ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = (arctan . x) / (1 + (x ^2 )) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) ((#Z 2) * arctan )) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( (1 / 2) (#) ((#Z 2) * arctan ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = (arctan . x) / (1 + (x ^2 )) ) )
A3: Z c= dom ((#Z 2) * arctan ) by A1, VALUED_1:def 5;
then A4: ( (#Z 2) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * arctan ) `| Z) . x = (2 * ((arctan . x) #Z (2 - 1))) / (1 + (x ^2 )) ) ) by A2, Th89;
for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = (arctan . x) / (1 + (x ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = (arctan . x) / (1 + (x ^2 )) )
assume A5: x in Z ; :: thesis: (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = (arctan . x) / (1 + (x ^2 ))
then (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = (1 / 2) * (diff ((#Z 2) * arctan ),x) by A1, A4, FDIFF_1:28
.= (1 / 2) * ((((#Z 2) * arctan ) `| Z) . x) by A4, A5, FDIFF_1:def 8
.= (1 / 2) * ((2 * ((arctan . x) #Z (2 - 1))) / (1 + (x ^2 ))) by A2, A3, A5, Th89
.= (1 / 2) * ((2 * (arctan . x)) / (1 + (x ^2 ))) by PREPOWER:45
.= (arctan . x) / (1 + (x ^2 )) ;
hence (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = (arctan . x) / (1 + (x ^2 )) ; :: thesis: verum

end;

hence ( (1 / 2) (#) ((#Z 2) * arctan ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = (arctan . x) / (1 + (x ^2 )) ) ) by A1, A4, FDIFF_1:28; :: thesis: verum