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

proof

let x be Real; :: thesis: ( x in Z implies (#Z n) * arctan is_differentiable_in x )
assume A4: x in Z ; :: thesis: (#Z n) * arctan is_differentiable_in x
arctan is_differentiable_on Z by A2, Th81;
then arctan is_differentiable_in x by A4, FDIFF_1:16;
hence (#Z n) * arctan is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;

then A5: (#Z n) * arctan is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
(((#Z n) * arctan ) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies (((#Z n) * arctan ) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 )) )
assume A6: x in Z ; :: thesis: (((#Z n) * arctan ) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 ))
then A7: - 1 < x by A2, XXREAL_1:4;
arctan is_differentiable_on Z by A2, Th81;
then A8: arctan is_differentiable_in x by A6, FDIFF_1:16;
A9: x < 1 by A2, A6, XXREAL_1:4;
(((#Z n) * arctan ) `| Z) . x = diff ((#Z n) * arctan ),x by A5, A6, FDIFF_1:def 8
.= (n * ((arctan . x) #Z (n - 1))) * (diff arctan ,x) by A8, TAYLOR_1:3
.= (n * ((arctan . x) #Z (n - 1))) * (1 / (1 + (x ^2 ))) by A7, A9, Th75
.= (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 )) ;
hence (((#Z n) * arctan ) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 )) ; :: thesis: verum

end;

hence ( (#Z n) * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z n) * arctan ) `| Z) . x = (n * ((arctan . x) #Z (n - 1))) / (1 + (x ^2 )) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum