let Z be open Subset of REAL ; :: thesis: ( not 0 in Z & Z c= dom (arctan * ((id Z) ^ )) & ( for x being Real st x in Z holds
( ((id Z) ^ ) . x > - 1 & ((id Z) ^ ) . x < 1 ) ) implies ( arctan * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ((id Z) ^ )) `| Z) . x = - (1 / (1 + (x ^2 ))) ) ) )
set f = id Z;
assume that
A1: ( not 0 in Z & Z c= dom (arctan * ((id Z) ^ )) ) and
A2: for x being Real st x in Z holds
( ((id Z) ^ ) . x > - 1 & ((id Z) ^ ) . x < 1 ) ; :: thesis: ( arctan * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ((id Z) ^ )) `| Z) . x = - (1 / (1 + (x ^2 ))) ) )
dom (arctan * ((id Z) ^ )) c= dom ((id Z) ^ ) by RELAT_1:44;
then A3: Z c= dom ((id Z) ^ ) by A1, XBOOLE_1:1;
A6: ( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by A1, FDIFF_5:4;
A7: for x being Real st x in Z holds
arctan * ((id Z) ^ ) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies arctan * ((id Z) ^ ) is_differentiable_in x )
assume A8: x in Z ; :: thesis: arctan * ((id Z) ^ ) is_differentiable_in x
then A9: (id Z) ^ is_differentiable_in x by A6, FDIFF_1:16;
( ((id Z) ^ ) . x > - 1 & ((id Z) ^ ) . x < 1 ) by A2, A8;
hence arctan * ((id Z) ^ ) is_differentiable_in x by A9, Th83; :: thesis: verum

end;

then A11: arctan * ((id Z) ^ ) is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((arctan * ((id Z) ^ )) `| Z) . x = - (1 / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((arctan * ((id Z) ^ )) `| Z) . x = - (1 / (1 + (x ^2 ))) )
assume A12: x in Z ; :: thesis: ((arctan * ((id Z) ^ )) `| Z) . x = - (1 / (1 + (x ^2 )))
then A13: (id Z) ^ is_differentiable_in x by A6, FDIFF_1:16;
A14: ( ((id Z) ^ ) . x > - 1 & ((id Z) ^ ) . x < 1 ) by A2, A12;
(id Z) . x = x by A12, FUNCT_1:35;
then x <> 0 by A3, A12, RFUNCT_1:13;
then A15: x ^2 <> 0 by SQUARE_1:74;
((arctan * ((id Z) ^ )) `| Z) . x = diff (arctan * ((id Z) ^ )),x by A11, A12, FDIFF_1:def 8
.= (diff ((id Z) ^ ),x) / (1 + ((((id Z) ^ ) . x) ^2 )) by A13, A14, Th83
.= ((((id Z) ^ ) `| Z) . x) / (1 + ((((id Z) ^ ) . x) ^2 )) by A6, A12, FDIFF_1:def 8
.= (- (1 / (x ^2 ))) / (1 + ((((id Z) ^ ) . x) ^2 )) by A1, A12, FDIFF_5:4
.= (- (1 / (x ^2 ))) / (1 + ((((id Z) . x) " ) ^2 )) by A3, A12, RFUNCT_1:def 8
.= (- (1 / (x ^2 ))) / (1 + ((1 / x) ^2 )) by A12, FUNCT_1:35
.= - ((1 / (x ^2 )) / (1 + ((1 / x) ^2 )))
.= - (1 / ((x ^2 ) * (1 + ((1 / x) ^2 )))) by XCMPLX_1:79
.= - (1 / (((x ^2 ) * 1) + ((x ^2 ) * ((1 / x) ^2 ))))
.= - (1 / ((x ^2 ) + ((x ^2 ) * (1 / (x * x))))) by XCMPLX_1:103
.= - (1 / ((x ^2 ) + (((x ^2 ) * 1) / (x ^2 ))))
.= - (1 / (1 + (x ^2 ))) by A15, XCMPLX_1:60 ;
hence ((arctan * ((id Z) ^ )) `| Z) . x = - (1 / (1 + (x ^2 ))) ; :: thesis: verum

end;

hence ( arctan * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ((id Z) ^ )) `| Z) . x = - (1 / (1 + (x ^2 ))) ) ) by A1, A7, FDIFF_1:16; :: thesis: verum