let Z be open Subset of REAL ; :: thesis: ( Z c= ].(- 1),1.[ implies ( (arctan + arccot ) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan + arccot ) / exp_R ) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; :: thesis: ( (arctan + arccot ) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan + arccot ) / exp_R ) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) )
then A2: arctan + arccot is_differentiable_on Z by Th37;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then Z c= dom arctan by A1, XBOOLE_1:1;
then Z c= (dom arctan ) /\ (dom arccot ) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan + arccot ) by VALUED_1:def 1;
A6: ( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
exp_R . x <> 0 ) ) by FDIFF_1:34, SIN_COS:59, TAYLOR_1:16;
then A7: (arctan + arccot ) / exp_R is_differentiable_on Z by A2, FDIFF_2:21;
for x being Real st x in Z holds
(((arctan + arccot ) / exp_R ) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x))

proof

let x be Real; :: thesis: ( x in Z implies (((arctan + arccot ) / exp_R ) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) )
A8: exp_R is_differentiable_in x by SIN_COS:70;
A9: exp_R . x <> 0 by SIN_COS:59;
assume A10: x in Z ; :: thesis: (((arctan + arccot ) / exp_R ) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x))
then A11: arctan + arccot is_differentiable_in x by A2, FDIFF_1:16;
(((arctan + arccot ) / exp_R ) `| Z) . x = diff ((arctan + arccot ) / exp_R ),x by A7, A10, FDIFF_1:def 8
.= (((diff (arctan + arccot ),x) * (exp_R . x)) - ((diff exp_R ,x) * ((arctan + arccot ) . x))) / ((exp_R . x) ^2 ) by A11, A8, A9, FDIFF_2:14
.= (((((arctan + arccot ) `| Z) . x) * (exp_R . x)) - ((diff exp_R ,x) * ((arctan + arccot ) . x))) / ((exp_R . x) ^2 ) by A2, A10, FDIFF_1:def 8
.= ((0 * (exp_R . x)) - ((diff exp_R ,x) * ((arctan + arccot ) . x))) / ((exp_R . x) ^2 ) by A1, A10, Th37
.= - (((diff exp_R ,x) * ((arctan + arccot ) . x)) / ((exp_R . x) ^2 ))
.= - (((exp_R . x) * ((arctan + arccot ) . x)) / ((exp_R . x) ^2 )) by SIN_COS:70
.= - ((((arctan . x) + (arccot . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x))) by A5, A10, VALUED_1:def 1
.= - (((arctan . x) + (arccot . x)) * ((exp_R . x) / ((exp_R . x) * (exp_R . x))))
.= - (((arctan . x) + (arccot . x)) * (((exp_R . x) / (exp_R . x)) / (exp_R . x))) by XCMPLX_1:79
.= - (((arctan . x) + (arccot . x)) * (1 / (exp_R . x))) by A9, XCMPLX_1:60
.= - (((arctan . x) + (arccot . x)) / (exp_R . x)) ;
hence (((arctan + arccot ) / exp_R ) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ; :: thesis: verum

end;

hence ( (arctan + arccot ) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan + arccot ) / exp_R ) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) ) by A2, A6, FDIFF_2:21; :: thesis: verum