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

proof

let x be Real; :: thesis: ( x in Z implies arccot * arctan is_differentiable_in x )
assume A4: x in Z ; :: thesis: arccot * arctan is_differentiable_in x
arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A5: arctan is_differentiable_in x by A4, FDIFF_1:16;
( arctan . x > - 1 & arctan . x < 1 ) by A3, A4;
hence arccot * arctan is_differentiable_in x by A5, SIN_COS9:86; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((arccot * arctan ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 )))) )
assume A7: x in Z ; :: thesis: ((arccot * arctan ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 ))))
A8: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A9: arctan is_differentiable_in x by A7, FDIFF_1:16;
A10: ( arctan . x > - 1 & arctan . x < 1 ) by A3, A7;
((arccot * arctan ) `| Z) . x = diff (arccot * arctan ),x by A6, A7, FDIFF_1:def 8
.= - ((diff arctan ,x) / (1 + ((arctan . x) ^2 ))) by A9, A10, SIN_COS9:86
.= - (((arctan `| Z) . x) / (1 + ((arctan . x) ^2 ))) by A7, A8, FDIFF_1:def 8
.= - ((1 / (1 + (x ^2 ))) / (1 + ((arctan . x) ^2 ))) by A2, A7, SIN_COS9:81
.= - (1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 )))) by XCMPLX_1:79 ;
hence ((arccot * arctan ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (1 + ((arctan . x) ^2 )))) ; :: thesis: verum

end;

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