let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cos * (arctan + arccot )) & Z c= ].(- 1),1.[ implies ( cos * (arctan + arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan + arccot )) `| Z) . x = 0 ) ) )
assume that
A1: Z c= dom (cos * (arctan + arccot )) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( cos * (arctan + arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan + arccot )) `| Z) . x = 0 ) )
A3: arctan + arccot is_differentiable_on Z by A2, Th37;
A4: for x being Real st x in Z holds
cos * (arctan + arccot ) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cos * (arctan + arccot ) is_differentiable_in x )
assume x in Z ; :: thesis: cos * (arctan + arccot ) is_differentiable_in x
then A5: arctan + arccot is_differentiable_in x by A3, FDIFF_1:16;
cos is_differentiable_in (arctan + arccot ) . x by SIN_COS:68;
hence cos * (arctan + arccot ) is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: cos * (arctan + arccot ) is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((cos * (arctan + arccot )) `| Z) . x = 0

proof

let x be Real; :: thesis: ( x in Z implies ((cos * (arctan + arccot )) `| Z) . x = 0 )
A7: cos is_differentiable_in (arctan + arccot ) . x by SIN_COS:68;
assume A8: x in Z ; :: thesis: ((cos * (arctan + arccot )) `| Z) . x = 0
then A9: arctan + arccot is_differentiable_in x by A3, FDIFF_1:16;
((cos * (arctan + arccot )) `| Z) . x = diff (cos * (arctan + arccot )),x by A6, A8, FDIFF_1:def 8
.= (diff cos ,((arctan + arccot ) . x)) * (diff (arctan + arccot ),x) by A9, A7, FDIFF_2:13
.= (- (sin . ((arctan + arccot ) . x))) * (diff (arctan + arccot ),x) by SIN_COS:68
.= (- (sin . ((arctan + arccot ) . x))) * (((arctan + arccot ) `| Z) . x) by A3, A8, FDIFF_1:def 8
.= (- (sin . ((arctan + arccot ) . x))) * 0 by A2, A8, Th37
.= 0 ;
hence ((cos * (arctan + arccot )) `| Z) . x = 0 ; :: thesis: verum

end;

hence ( cos * (arctan + arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * (arctan + arccot )) `| Z) . x = 0 ) ) by A1, A4, FDIFF_1:16; :: thesis: verum