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

proof

let x be Real; :: thesis: ( x in Z implies cot * arctan is_differentiable_in x )
assume A3: x in Z ; :: thesis: cot * arctan is_differentiable_in x
arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A4: arctan is_differentiable_in x by A3, FDIFF_1:16;
arctan . x in dom cot by A1, A3, FUNCT_1:21;
then sin . (arctan . x) <> 0 by FDIFF_8:2;
then cot is_differentiable_in arctan . x by FDIFF_7:47;
hence cot * arctan is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((cot * arctan ) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 )))) )
assume A6: x in Z ; :: thesis: ((cot * arctan ) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 ))))
A7: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A8: arctan is_differentiable_in x by A6, FDIFF_1:16;
arctan . x in dom cot by A1, A6, FUNCT_1:21;
then A9: sin . (arctan . x) <> 0 by FDIFF_8:2;
then A10: cot is_differentiable_in arctan . x by FDIFF_7:47;
((cot * arctan ) `| Z) . x = diff (cot * arctan ),x by A5, A6, FDIFF_1:def 8
.= (diff cot ,(arctan . x)) * (diff arctan ,x) by A8, A10, FDIFF_2:13
.= (- (1 / ((sin . (arctan . x)) ^2 ))) * (diff arctan ,x) by A9, FDIFF_7:47
.= - ((1 / ((sin . (arctan . x)) ^2 )) * (diff arctan ,x))
.= - ((1 / ((sin . (arctan . x)) ^2 )) * ((arctan `| Z) . x)) by A6, A7, FDIFF_1:def 8
.= - ((1 / ((sin . (arctan . x)) ^2 )) * (1 / (1 + (x ^2 )))) by A2, A6, SIN_COS9:81
.= - (1 / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 )))) by XCMPLX_1:103 ;
hence ((cot * arctan ) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2 ) * (1 + (x ^2 )))) ; :: thesis: verum

end;

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