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 = (((2 / (1 + (x ^2 ))) - (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 = (((2 / (1 + (x ^2 ))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) )
A2: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arctan by SIN_COS9:23, XBOOLE_1:1;
then A3: Z c= dom arctan by A1, XBOOLE_1:1;
].(- 1),1.[ c= dom arccot by A2, SIN_COS9:24, XBOOLE_1:1;
then Z c= dom arccot by A1, XBOOLE_1:1;
then Z c= (dom arctan ) /\ (dom arccot ) by A3, XBOOLE_1:19;
then A4: Z c= dom (arctan - arccot ) by VALUED_1:12;
A5: ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - arccot ) `| Z) . x = 2 / (1 + (x ^2 )) ) ) by A1, Th38;
A6: exp_R is_differentiable_on Z by TAYLOR_1:16, FDIFF_1:34;
A7: for x being Real st x in Z holds
exp_R . x <> 0 by SIN_COS:59;
then A8: (arctan - arccot ) / exp_R is_differentiable_on Z by A5, A6, FDIFF_2:21;
for x being Real st x in Z holds
(((arctan - arccot ) / exp_R ) `| Z) . x = (((2 / (1 + (x ^2 ))) - (arctan . x)) + (arccot . x)) / (exp_R . x)

proof

let x be Real; :: thesis: ( x in Z implies (((arctan - arccot ) / exp_R ) `| Z) . x = (((2 / (1 + (x ^2 ))) - (arctan . x)) + (arccot . x)) / (exp_R . x) )
assume A9: x in Z ; :: thesis: (((arctan - arccot ) / exp_R ) `| Z) . x = (((2 / (1 + (x ^2 ))) - (arctan . x)) + (arccot . x)) / (exp_R . x)
A10: arctan - arccot is_differentiable_in x by A5, A9, FDIFF_1:16;
A11: exp_R is_differentiable_in x by SIN_COS:70;
A12: exp_R . x <> 0 by SIN_COS:59;
(((arctan - arccot ) / exp_R ) `| Z) . x = diff ((arctan - arccot ) / exp_R ),x by A8, A9, FDIFF_1:def 8
.= (((diff (arctan - arccot ),x) * (exp_R . x)) - ((diff exp_R ,x) * ((arctan - arccot ) . x))) / ((exp_R . x) ^2 ) by A10, A11, A12, FDIFF_2:14
.= (((((arctan - arccot ) `| Z) . x) * (exp_R . x)) - ((diff exp_R ,x) * ((arctan - arccot ) . x))) / ((exp_R . x) ^2 ) by A5, A9, FDIFF_1:def 8
.= (((2 / (1 + (x ^2 ))) * (exp_R . x)) - ((diff exp_R ,x) * ((arctan - arccot ) . x))) / ((exp_R . x) ^2 ) by A1, A9, Th38
.= (((2 / (1 + (x ^2 ))) * (exp_R . x)) - ((exp_R . x) * ((arctan - arccot ) . x))) / ((exp_R . x) ^2 ) by SIN_COS:70
.= (((2 / (1 + (x ^2 ))) * (exp_R . x)) - ((exp_R . x) * ((arctan . x) - (arccot . x)))) / ((exp_R . x) ^2 ) by A4, A9, VALUED_1:13
.= ((2 / (1 + (x ^2 ))) - ((arctan . x) - (arccot . x))) * ((exp_R . x) / ((exp_R . x) * (exp_R . x)))
.= ((2 / (1 + (x ^2 ))) - ((arctan . x) - (arccot . x))) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) by XCMPLX_1:79
.= ((2 / (1 + (x ^2 ))) - ((arctan . x) - (arccot . x))) * (1 / (exp_R . x)) by A12, XCMPLX_1:60
.= (((2 / (1 + (x ^2 ))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ;
hence (((arctan - arccot ) / exp_R ) `| Z) . x = (((2 / (1 + (x ^2 ))) - (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 = (((2 / (1 + (x ^2 ))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) ) by A5, A6, A7, FDIFF_2:21; :: thesis: verum