let Z be open Subset of REAL ; :: thesis: ( Z c= dom ((#Z 2) * arccot ) & Z c= ].(- 1),1.[ implies ( - ((1 / 2) (#) ((#Z 2) * arccot )) is_differentiable_on Z & ( for x being Real st x in Z holds
((- ((1 / 2) (#) ((#Z 2) * arccot ))) `| Z) . x = (arccot . x) / (1 + (x ^2 )) ) ) )
assume B1: ( Z c= dom ((#Z 2) * arccot ) & Z c= ].(- 1),1.[ ) ; :: thesis: ( - ((1 / 2) (#) ((#Z 2) * arccot )) is_differentiable_on Z & ( for x being Real st x in Z holds
((- ((1 / 2) (#) ((#Z 2) * arccot ))) `| Z) . x = (arccot . x) / (1 + (x ^2 )) ) )
then A2: Z c= dom ((1 / 2) (#) ((#Z 2) * arccot )) by VALUED_1:def 5;
then A1: Z c= dom (- ((1 / 2) (#) ((#Z 2) * arccot ))) by VALUED_1:8;
A3: (1 / 2) (#) ((#Z 2) * arccot ) is_differentiable_on Z by A2, B1, SIN_COS9:94;
then A4: (- 1) (#) ((1 / 2) (#) ((#Z 2) * arccot )) is_differentiable_on Z by A1, FDIFF_1:28, A;
A6: ( (#Z 2) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * arccot ) `| Z) . x = - ((2 * ((arccot . x) #Z (2 - 1))) / (1 + (x ^2 ))) ) ) by B1, SIN_COS9:92;
for x being Real st x in Z holds
((- ((1 / 2) (#) ((#Z 2) * arccot ))) `| Z) . x = (arccot . x) / (1 + (x ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((- ((1 / 2) (#) ((#Z 2) * arccot ))) `| Z) . x = (arccot . x) / (1 + (x ^2 )) )
assume A7: x in Z ; :: thesis: ((- ((1 / 2) (#) ((#Z 2) * arccot ))) `| Z) . x = (arccot . x) / (1 + (x ^2 ))
then A8: (1 / 2) (#) ((#Z 2) * arccot ) is_differentiable_in x by A3, FDIFF_1:16;
A9: (#Z 2) * arccot is_differentiable_in x by A6, A7, FDIFF_1:16;
((- ((1 / 2) (#) ((#Z 2) * arccot ))) `| Z) . x = diff (- ((1 / 2) (#) ((#Z 2) * arccot ))),x by A4, A7, FDIFF_1:def 8
.= (- 1) * (diff ((1 / 2) (#) ((#Z 2) * arccot )),x) by A8, FDIFF_1:23, A
.= (- 1) * ((1 / 2) * (diff ((#Z 2) * arccot ),x)) by A9, FDIFF_1:23, A
.= (- 1) * ((1 / 2) * ((((#Z 2) * arccot ) `| Z) . x)) by A6, A7, FDIFF_1:def 8
.= (- 1) * ((1 / 2) * (- ((2 * ((arccot . x) #Z (2 - 1))) / (1 + (x ^2 ))))) by B1, A7, SIN_COS9:92
.= (- 1) * (- ((1 / 2) * ((2 * ((arccot . x) #Z 1)) / (1 + (x ^2 )))))
.= (- 1) * (- ((1 / 2) * ((2 * (arccot . x)) / (1 + (x ^2 ))))) by PREPOWER:45
.= (arccot . x) / (1 + (x ^2 )) ;
hence ((- ((1 / 2) (#) ((#Z 2) * arccot ))) `| Z) . x = (arccot . x) / (1 + (x ^2 )) ; :: thesis: verum

end;

hence ( - ((1 / 2) (#) ((#Z 2) * arccot )) is_differentiable_on Z & ( for x being Real st x in Z holds
((- ((1 / 2) (#) ((#Z 2) * arccot ))) `| Z) . x = (arccot . x) / (1 + (x ^2 )) ) ) by A1, A3, FDIFF_1:28, A; :: thesis: verum