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

proof

let x be Real; :: thesis: ( x in Z implies arctan * arccot is_differentiable_in x )
assume A4: x in Z ; :: thesis: arctan * arccot is_differentiable_in x
arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A5: arccot is_differentiable_in x by A4, FDIFF_1:16;
( arccot . x > - 1 & arccot . x < 1 ) by A3, A4;
hence arctan * arccot is_differentiable_in x by A5, SIN_COS9:85; :: thesis: verum

end;

then A6: arctan * arccot is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((arctan * arccot ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (1 + ((arccot . x) ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((arctan * arccot ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (1 + ((arccot . x) ^2 )))) )
assume A7: x in Z ; :: thesis: ((arctan * arccot ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (1 + ((arccot . x) ^2 ))))
A8: arccot is_differentiable_on Z by A2, SIN_COS9:82;
then A9: arccot is_differentiable_in x by A7, FDIFF_1:16;
A10: ( arccot . x > - 1 & arccot . x < 1 ) by A3, A7;
((arctan * arccot ) `| Z) . x = diff (arctan * arccot ),x by A6, A7, FDIFF_1:def 8
.= (diff arccot ,x) / (1 + ((arccot . x) ^2 )) by A9, A10, SIN_COS9:85
.= ((arccot `| Z) . x) / (1 + ((arccot . x) ^2 )) by A7, A8, FDIFF_1:def 8
.= (- (1 / (1 + (x ^2 )))) / (1 + ((arccot . x) ^2 )) by A2, A7, SIN_COS9:82
.= - ((1 / (1 + (x ^2 ))) / (1 + ((arccot . x) ^2 )))
.= - (1 / ((1 + (x ^2 )) * (1 + ((arccot . x) ^2 )))) by XCMPLX_1:79 ;
hence ((arctan * arccot ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (1 + ((arccot . x) ^2 )))) ; :: thesis: verum

end;

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