let Z be open Subset of REAL ; :: thesis: ( Z c= dom ((- arccot ) - (id Z)) & Z c= ].(- 1),1.[ implies ( (- arccot ) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot ) - (id Z)) `| Z) . x = - ((x ^2 ) / (1 + (x ^2 ))) ) ) )
assume that
A1: Z c= dom ((- arccot ) - (id Z)) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( (- arccot ) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot ) - (id Z)) `| Z) . x = - ((x ^2 ) / (1 + (x ^2 ))) ) )
A3: arccot is_differentiable_on Z by A2, Th82;
A4: Z c= (dom (- arccot )) /\ (dom (id Z)) by A1, VALUED_1:12;
then A5: Z c= dom (id Z) by XBOOLE_1:18;
A6: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:35;
then A7: id Z is_differentiable_on Z by A5, FDIFF_1:31;
A8: Z c= dom ((- 1) (#) arccot ) by A4, XBOOLE_1:18;
then A9: - arccot is_differentiable_on Z by A3, FDIFF_1:28;
for x being Real st x in Z holds
(((- arccot ) - (id Z)) `| Z) . x = - ((x ^2 ) / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies (((- arccot ) - (id Z)) `| Z) . x = - ((x ^2 ) / (1 + (x ^2 ))) )
A10: 1 + (x ^2 ) > 0 by XREAL_1:36, XREAL_1:65;
assume A11: x in Z ; :: thesis: (((- arccot ) - (id Z)) `| Z) . x = - ((x ^2 ) / (1 + (x ^2 )))
then (((- arccot ) - (id Z)) `| Z) . x = (diff (- arccot ),x) - (diff (id Z),x) by A1, A7, A9, FDIFF_1:27
.= (((- arccot ) `| Z) . x) - (diff (id Z),x) by A9, A11, FDIFF_1:def 8
.= ((- 1) * (diff arccot ,x)) - (diff (id Z),x) by A8, A3, A11, FDIFF_1:28
.= ((- 1) * ((arccot `| Z) . x)) - (diff (id Z),x) by A3, A11, FDIFF_1:def 8
.= ((- 1) * (- (1 / (1 + (x ^2 ))))) - (diff (id Z),x) by A2, A11, Th82
.= (1 / (1 + (x ^2 ))) - (((id Z) `| Z) . x) by A7, A11, FDIFF_1:def 8
.= (1 / (1 + (x ^2 ))) - 1 by A5, A6, A11, FDIFF_1:31
.= (1 / (1 + (x ^2 ))) - ((1 + (x ^2 )) / (1 + (x ^2 ))) by A10, XCMPLX_1:60
.= - ((x ^2 ) / (1 + (x ^2 ))) ;
hence (((- arccot ) - (id Z)) `| Z) . x = - ((x ^2 ) / (1 + (x ^2 ))) ; :: thesis: verum

end;

hence ( (- arccot ) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot ) - (id Z)) `| Z) . x = - ((x ^2 ) / (1 + (x ^2 ))) ) ) by A1, A7, A9, FDIFF_1:27; :: thesis: verum