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

proof

let x be Real; :: thesis: ( x in Z implies exp_R * (arctan + arccot) is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * (arctan + arccot) is_differentiable_in x
then A5: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9;
exp_R is_differentiable_in (arctan + arccot) . x by SIN_COS:65;
hence exp_R * (arctan + arccot) is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: exp_R * (arctan + arccot) is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((exp_R * (arctan + arccot)) `| Z) . x = 0

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * (arctan + arccot)) `| Z) . x = 0 )
A7: exp_R is_differentiable_in (arctan + arccot) . x by SIN_COS:65;
assume A8: x in Z ; :: thesis: ((exp_R * (arctan + arccot)) `| Z) . x = 0
then A9: arctan + arccot is_differentiable_in x by A3, FDIFF_1:9;
((exp_R * (arctan + arccot)) `| Z) . x = diff ((exp_R * (arctan + arccot)),x) by A6, A8, FDIFF_1:def 7
.= (diff (exp_R,((arctan + arccot) . x))) * (diff ((arctan + arccot),x)) by A9, A7, FDIFF_2:13
.= (exp_R . ((arctan + arccot) . x)) * (diff ((arctan + arccot),x)) by SIN_COS:65
.= (exp_R . ((arctan + arccot) . x)) * (((arctan + arccot) `| Z) . x) by A3, A8, FDIFF_1:def 7
.= (exp_R . ((arctan + arccot) . x)) * 0 by A2, A8, Th37
.= 0 ;
hence ((exp_R * (arctan + arccot)) `| Z) . x = 0 ; :: thesis: verum

end;

hence ( exp_R * (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (arctan + arccot)) `| Z) . x = 0 ) ) by A1, A4, FDIFF_1:9; :: thesis: verum